{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0001","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71dbeaa52f43d10a27a3c7ed31722b19b9f6bc3451f90da10d1bb808bc92788f","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026001,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,8],[1,8],[1,8],[5,8]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/4, 1/2, 1/4, 1/8, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/4, 1/2, 1/4, 1/8, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0002","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bae2fc0921c62b43fdb99902f18f15bddff753d0c446e6f1c2c89ec3848c9a67","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026002,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,9],[1,9],[1,9],[6,9]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/9, 1/2, 2/9, 1/9, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/9, 1/2, 2/9, 1/9, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0003","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"505f1a9a861eb867415f91633f98efd12f4db78371eaafa48b78859342b3410d","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026003,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,10],[1,10],[1,10],[7,10]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/5, 1/2, 1/5, 1/10, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/5, 1/2, 1/5, 1/10, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0004","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"068161c91fe95dd87c921552d57b252db3d3c5d369fc6e2009b3fa7432624710","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026004,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,11],[1,11],[1,11],[8,11]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/11, 1/2, 2/11, 1/11, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/11, 1/2, 2/11, 1/11, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0005","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec7ddacf10a8a3cefae8978ffff420ab977550d568ad2f8e71185520db0bad99","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026005,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,12],[1,12],[1,12],[9,12]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/6, 1/2, 1/6, 1/12, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/6, 1/2, 1/6, 1/12, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0006","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2541e1c7a1abb9eb61fe8d761dcc07273d14d122f78b76c49eae658ba64e4f23","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026006,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,13],[1,13],[1,13],[10,13]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/13, 1/2, 2/13, 1/13, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/13, 1/2, 2/13, 1/13, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0007","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c319e26463249877a7f4b3094fdac267509067a5bef4fea8ed8d0702cf401d1d","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026007,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,14],[1,14],[1,14],[11,14]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/7, 1/2, 1/7, 1/14, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/7, 1/2, 1/7, 1/14, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0008","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1cdfe24bf0e1af5a8a454fc40eb7f74d1ee145349a666b91966b4aee057abcc","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026008,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,15],[1,15],[1,15],[12,15]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/15, 1/2, 2/15, 1/15, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/15, 1/2, 2/15, 1/15, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0009","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21cc1e4072b2bfa3a1ed79b51f0593983386557a16aa13fc2c48d1dc258a1c8f","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026009,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,16],[1,16],[1,16],[13,16]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/8, 1/2, 1/8, 1/16, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/8, 1/2, 1/8, 1/16, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0010","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e69a4e795ab1a53f1c6f860fa21cbd7f3ce7f04bae56157605a298f5b31fae4c","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026010,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,17],[1,17],[1,17],[14,17]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/17, 1/2, 2/17, 1/17, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/17, 1/2, 2/17, 1/17, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0011","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1691291fb9484f2c042502456d95100c628d68562d3a7c7608d549b8a9d273cd","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026011,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,18],[1,18],[1,18],[15,18]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/9, 1/2, 1/9, 1/18, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/9, 1/2, 1/9, 1/18, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0012","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2e3b14e64fb2c888a67f7feba27a01ac3059edd163d77aa79700a329218bc2a","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026012,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,19],[1,19],[1,19],[16,19]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/19, 1/2, 2/19, 1/19, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/19, 1/2, 2/19, 1/19, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0013","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6df4b57e186589bbaee86372240db46a5bcf0e1c9829fa35a03aa6d6460fb0a","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026013,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,20],[1,20],[1,20],[17,20]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/10, 1/2, 1/10, 1/20, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/10, 1/2, 1/10, 1/20, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0014","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86b0a06e2067997c3ffffe80bc191d02e3284e58411f6603b9a988aaecea13df","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026014,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,21],[1,21],[1,21],[18,21]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/21, 1/2, 2/21, 1/21, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/21, 1/2, 2/21, 1/21, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0015","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"738b7e4eed30eb6bc663b4a0dca71164c4e772cc5e31dbf3aa637c345fe308c6","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026015,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,22],[1,22],[1,22],[19,22]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/11, 1/2, 1/11, 1/22, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/11, 1/2, 1/11, 1/22, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0016","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73bf2aaae74f8d90af8565d5a9ca161fe1b140bc532e4e171845788501b05e6e","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026016,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,23],[1,23],[1,23],[20,23]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/23, 1/2, 2/23, 1/23, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/23, 1/2, 2/23, 1/23, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0017","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f47ac4307ebb49f6af85487b7ba5d041ad1cd1d804535ee2d6c7bee18a86f1c1","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026017,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,24],[1,24],[1,24],[21,24]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/12, 1/2, 1/12, 1/24, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/12, 1/2, 1/12, 1/24, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0018","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cdfbd4681568f7ff772d99967d83d70673f42a293b10bb83c14cb5f552b6e86","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026018,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,25],[1,25],[1,25],[22,25]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/25, 1/2, 2/25, 1/25, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/25, 1/2, 2/25, 1/25, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0019","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d96bb1f3a311d1695ba395f3f827520f93ac35e2e726700860d98c2066d26f21","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026019,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,26],[1,26],[1,26],[23,26]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/13, 1/2, 1/13, 1/26, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/13, 1/2, 1/13, 1/26, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0020","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb24c673de2a4ec54120cfd1eba259391efe66bc351727bc1cc6028217bc0e4c","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026020,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,27],[1,27],[1,27],[24,27]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/27, 1/2, 2/27, 1/27, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/27, 1/2, 2/27, 1/27, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0021","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12238123bedbcc552e33a2026ff9e5bdccc9381491fec00ba1315da137df017e","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026021,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,28],[1,28],[1,28],[25,28]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/14, 1/2, 1/14, 1/28, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/14, 1/2, 1/14, 1/28, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0022","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e892342717852a9ab537c91c85ab58348a2ef7794b763d5e57b46ce5de3a9743","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026022,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/29, 1/2, 2/29, 1/29, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/29, 1/2, 2/29, 1/29, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0023","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64481baf056dde21866b00f61876e339295f57a5d39c4b3197f9d0ba08ca00a3","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026023,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,30],[1,30],[1,30],[27,30]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/15, 1/2, 1/15, 1/30, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/15, 1/2, 1/15, 1/30, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0024","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f793d80b2c5917cabc9c2dc685b1c7d13200154f4cf1b6e40d9e41337d33dcb0","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026024,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,31],[1,31],[1,31],[28,31]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(2/31, 1/2, 2/31, 1/31, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (2/31, 1/2, 2/31, 1/31, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0025","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaf75498de0d315c69e6808b58cb0dc711608f7414530eab4b6c2b952cdba217","family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026025,"semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,32],[1,32],[1,32],[29,32]]}"},"fields":{"title":"Prior and Likelihood Must Both Enter a Bayes Reversal","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal).","answer":{"kind":"expression","value":"(1/16, 1/2, 1/16, 1/32, 1/2, not_causal)","explanation":"Compute P(H), P(E|H), P(E), and the same joint P(H∩E), then form P(H|E)=joint/evidence_mass. Likelihood and posterior need not be equal; dropping the prior or reversing a zero denominator is invalid. The exact result, retaining the requested order, is (1/16, 1/2, 1/16, 1/32, 1/2, not_causal)."},"skills":["Prior and Likelihood Must Both Enter a Bayes Reversal"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0026","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb237f80163b35d0d09296445ff572deb94846d72551552be02edeaa73c3f77b","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026026,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,8],[1,8],[1,8],[5,8]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/4, 1/2, 1/4, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/4, 1/2, 1/4, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0027","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0ffcd144526487d7a5d4e403e95d4ba13c24a5f3a74f51bbaf6b232db88920c","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026027,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,9],[1,9],[1,9],[6,9]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 7/9, 1/7, 2/9, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 7/9, 1/7, 2/9, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0028","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47832fd774fc3d32a8d05d376e37c2064f1e1f10d8bbf4973ec16736eea3f4ba","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026028,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,10],[1,10],[1,10],[7,10]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/5, 1/2, 1/5, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/5, 1/2, 1/5, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0029","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5fcfa3d2d0acbcc2c45dfad8ec67c4ddefc429193145c2dd55782d0e336d6d","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026029,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,11],[1,11],[1,11],[8,11]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 9/11, 1/9, 2/11, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 9/11, 1/9, 2/11, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0030","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f2cccbe1721fac35ab91d006246d060072c26a2e5f07c7541e8d06b7fadc896","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026030,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,12],[1,12],[1,12],[9,12]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/6, 1/2, 1/6, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/6, 1/2, 1/6, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0031","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21ac4e069652a6c7a084d19fc62d427222fa29789404b11499f318791afbff04","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026031,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,13],[1,13],[1,13],[10,13]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 11/13, 1/11, 2/13, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 11/13, 1/11, 2/13, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0032","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20bd289f271850300d3c4d98323fbe94961a1661df11af52acc7e1b8dee49948","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026032,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,14],[1,14],[1,14],[11,14]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/7, 1/2, 1/7, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/7, 1/2, 1/7, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0033","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b4526b634a8bfc89d4b5cb8e05476d0c712d28c4ae50e9a7871597b70fd57d3","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026033,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,15],[1,15],[1,15],[12,15]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 13/15, 1/13, 2/15, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 13/15, 1/13, 2/15, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0034","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbac159b3845437d2a03fcb37292cb54333aad86d46a63022657be2e1ca40b65","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026034,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,16],[1,16],[1,16],[13,16]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/8, 1/2, 1/8, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/8, 1/2, 1/8, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0035","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a72b9a971f567a89474ed75a8bd9865acc985597db572932608783f414327927","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026035,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,17],[1,17],[1,17],[14,17]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 15/17, 1/15, 2/17, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 15/17, 1/15, 2/17, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0036","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2151a4c6aeba02286384a5cc341b93be7d3d69801a53a73a86795a341d191673","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026036,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,18],[1,18],[1,18],[15,18]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/9, 1/2, 1/9, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/9, 1/2, 1/9, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0037","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"546e5a29f12bc513c4cd5e9ef2af5a862a7a99bf2c4b3514fc3e5f44e5a9e7f0","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026037,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,19],[1,19],[1,19],[16,19]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 17/19, 1/17, 2/19, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 17/19, 1/17, 2/19, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0038","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a1faf153b6972f0b7f6b5cadbc653c450040cacd3693bda71f88a3e08c4667d","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026038,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,20],[1,20],[1,20],[17,20]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/10, 1/2, 1/10, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/10, 1/2, 1/10, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0039","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56be9137d0b1e8f6f43f2f2b120b2a77ab3fa48599f1966734d58ba226a68895","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026039,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,21],[1,21],[1,21],[18,21]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0040","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73ba62de3f5a137cf745dc6a1fb3c0df4e7af7059f690cde0f8ca5fe31117aad","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026040,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0041","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77d6e373b29fcd5530a727bed3965d5e69ede765c67af4da2ed61a93fabeb838","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026041,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,23],[1,23],[1,23],[20,23]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 21/23, 1/21, 2/23, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 21/23, 1/21, 2/23, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0042","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1d7c5fd23d1d92a3770e0d9693357ce5382d826c87c209ef32757c7dab2c517","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026042,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,24],[1,24],[1,24],[21,24]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/12, 1/2, 1/12, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/12, 1/2, 1/12, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0043","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44cbc5e9c05176d7b3c1397e0b23506085937a869ff5943634c7a842d249828f","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026043,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,25],[1,25],[1,25],[22,25]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 23/25, 1/23, 2/25, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 23/25, 1/23, 2/25, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0044","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b922fbd19b506726c53ada923b84c419eedd8ec6048fb7f95ee134458ab0957","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026044,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,26],[1,26],[1,26],[23,26]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/13, 1/2, 1/13, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/13, 1/2, 1/13, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0045","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a6736c65891b9b4888acba9f5bdf4fdaa36182d198bd433d51daa08e7292910","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026045,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,27],[1,27],[1,27],[24,27]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 25/27, 1/25, 2/27, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 25/27, 1/25, 2/27, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0046","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b8514419e96c24fcfe701a49960b0398f69ab669e646fbe7f96ff889b5b546","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026046,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,28],[1,28],[1,28],[25,28]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/14, 1/2, 1/14, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/14, 1/2, 1/14, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0047","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d5b3da909e2ab9c9455cc279e1c548f4c69780bf63c778e0b1e10838321d094","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026047,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,29],[1,29],[1,29],[26,29]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 27/29, 1/27, 2/29, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 27/29, 1/27, 2/29, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0048","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdc10b1ca154fa71e51aa463bea2b92dcbb7dd4c4522d96301932d6afbaa37f2","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026048,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,30],[1,30],[1,30],[27,30]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/15, 1/2, 1/15, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/15, 1/2, 1/15, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0049","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92cc0e58a66a7f2728d3901a817d3c91f935bbdbdf5ed803de8b665b259a2afe","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026049,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,31],[1,31],[1,31],[28,31]],\"target_index\":1}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(1, 29/31, 1/29, 2/31, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (1, 29/31, 1/29, 2/31, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0050","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec4e5e7a28f6aed1994bf8fcd2d3babb1e38f472043caeaa02a3864112f6364d","family_id":"qf-mod33-bayes-partition-posterior","seed":202611026050,"semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,32],[1,32],[1,32],[29,32]],\"target_index\":0}"},"fields":{"title":"Normalize Partition Hypotheses by the Same Total Evidence Mass","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).","answer":{"kind":"expression","value":"(0, 1/16, 1/2, 1/16, 1/2, partition_pairwise_disjoint_exhaustive)","explanation":"Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/16, 1/2, 1/16, 1/2, partition_pairwise_disjoint_exhaustive)."},"skills":["Normalize Partition Hypotheses by the Same Total Evidence Mass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0051","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9426f57c1aeab54d0eea7398cfbbd9a68e0ebd3a4c0c8c2d22e324965a020130","family_id":"qf-mod33-bayes-total-evidence","seed":202611026051,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,8],[1,8],[1,8],[5,8]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/4, 3/4], [1/2, 1/6], [1/8, 1/8], 1/4, 1/4, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/4, 3/4], [1/2, 1/6], [1/8, 1/8], 1/4, 1/4, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0052","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f7777b11f0ecf079a3c9095b5aacc70156d11509cce57d585309d68398eac0b","family_id":"qf-mod33-bayes-total-evidence","seed":202611026052,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,9],[1,9],[1,9],[6,9]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/9, 7/9], [1/2, 1/7], [1/9, 1/9], 2/9, 2/9, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/9, 7/9], [1/2, 1/7], [1/9, 1/9], 2/9, 2/9, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0053","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2718388b6444ddacca763bc9bd844ee761c5df47822b1be12ebffd8bc917eee5","family_id":"qf-mod33-bayes-total-evidence","seed":202611026053,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,10],[1,10],[1,10],[7,10]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/5, 4/5], [1/2, 1/8], [1/10, 1/10], 1/5, 1/5, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/5, 4/5], [1/2, 1/8], [1/10, 1/10], 1/5, 1/5, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0054","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"568a26c7418c9fc4182254d2f1fb9ac9ef6a80a418b69a2ca7846656fe25cf59","family_id":"qf-mod33-bayes-total-evidence","seed":202611026054,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,11],[1,11],[1,11],[8,11]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/11, 9/11], [1/2, 1/9], [1/11, 1/11], 2/11, 2/11, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/11, 9/11], [1/2, 1/9], [1/11, 1/11], 2/11, 2/11, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0055","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1265b8c4ba7d7d4daf57d83806683859b1d671c7a296a2a3e2a1bc7968e82072","family_id":"qf-mod33-bayes-total-evidence","seed":202611026055,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,12],[1,12],[1,12],[9,12]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/6, 5/6], [1/2, 1/10], [1/12, 1/12], 1/6, 1/6, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/6, 5/6], [1/2, 1/10], [1/12, 1/12], 1/6, 1/6, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0056","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a146304ceb5a9e93acffcddc9578ce513542e50f417e0b81ad9c00d5c4916c5","family_id":"qf-mod33-bayes-total-evidence","seed":202611026056,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,13],[1,13],[1,13],[10,13]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/13, 11/13], [1/2, 1/11], [1/13, 1/13], 2/13, 2/13, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/13, 11/13], [1/2, 1/11], [1/13, 1/13], 2/13, 2/13, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0057","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c5ab151b91f0eac98b737ae55bbb44de1a2d4b6c85fbda6dc4ca8be9d11644","family_id":"qf-mod33-bayes-total-evidence","seed":202611026057,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,14],[1,14],[1,14],[11,14]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/7, 6/7], [1/2, 1/12], [1/14, 1/14], 1/7, 1/7, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/7, 6/7], [1/2, 1/12], [1/14, 1/14], 1/7, 1/7, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0058","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"340d9c4edcfbcf978eeda831d1996dbcc140205e751ce771e0024a73ae545583","family_id":"qf-mod33-bayes-total-evidence","seed":202611026058,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,15],[1,15],[1,15],[12,15]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0059","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0309c8eefd8172d1cbec491f64108ed22b969d15a09eaf4b961801a71e66210","family_id":"qf-mod33-bayes-total-evidence","seed":202611026059,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,16],[1,16],[1,16],[13,16]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/8, 7/8], [1/2, 1/14], [1/16, 1/16], 1/8, 1/8, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/8, 7/8], [1/2, 1/14], [1/16, 1/16], 1/8, 1/8, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0060","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"367d8a771a09702c813f5ae3b17e6658d480ac0f96f3e9d52d4854d606af39f1","family_id":"qf-mod33-bayes-total-evidence","seed":202611026060,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,17],[1,17],[1,17],[14,17]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/17, 15/17], [1/2, 1/15], [1/17, 1/17], 2/17, 2/17, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/17, 15/17], [1/2, 1/15], [1/17, 1/17], 2/17, 2/17, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0061","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c942c30e14162c9e1905d77f9400969e9ce06b7091214ee1014fb1447f73a8c4","family_id":"qf-mod33-bayes-total-evidence","seed":202611026061,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,18],[1,18],[1,18],[15,18]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/9, 8/9], [1/2, 1/16], [1/18, 1/18], 1/9, 1/9, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/9, 8/9], [1/2, 1/16], [1/18, 1/18], 1/9, 1/9, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0062","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"168cb9bd1aea65c6fb44b31c92b4e10e91c676c7da6eadf2ecf5d1f7707683fc","family_id":"qf-mod33-bayes-total-evidence","seed":202611026062,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,19],[1,19],[1,19],[16,19]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/19, 17/19], [1/2, 1/17], [1/19, 1/19], 2/19, 2/19, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/19, 17/19], [1/2, 1/17], [1/19, 1/19], 2/19, 2/19, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0063","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fe5f2d6297abc3d091609ffa249f2420bf4564ea71ad51f48cd34a45f73b24a","family_id":"qf-mod33-bayes-total-evidence","seed":202611026063,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,20],[1,20],[1,20],[17,20]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/10, 9/10], [1/2, 1/18], [1/20, 1/20], 1/10, 1/10, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/10, 9/10], [1/2, 1/18], [1/20, 1/20], 1/10, 1/10, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0064","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef8059aecc062ca9a744eaace05ca56c36391c1d337f927fa66960bf8c5dba43","family_id":"qf-mod33-bayes-total-evidence","seed":202611026064,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,21],[1,21],[1,21],[18,21]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/21, 19/21], [1/2, 1/19], [1/21, 1/21], 2/21, 2/21, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/21, 19/21], [1/2, 1/19], [1/21, 1/21], 2/21, 2/21, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0065","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8805b8df8a575317dfed4c5b462ea6f87642316a24a95806a2695682f1f52e3","family_id":"qf-mod33-bayes-total-evidence","seed":202611026065,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,22],[1,22],[1,22],[19,22]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/11, 10/11], [1/2, 1/20], [1/22, 1/22], 1/11, 1/11, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/11, 10/11], [1/2, 1/20], [1/22, 1/22], 1/11, 1/11, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0066","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"805590142e779e9ea4efd2369f0542caa9c9121bfea85ba5daadfc13bf0235c4","family_id":"qf-mod33-bayes-total-evidence","seed":202611026066,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,23],[1,23],[1,23],[20,23]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/23, 21/23], [1/2, 1/21], [1/23, 1/23], 2/23, 2/23, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/23, 21/23], [1/2, 1/21], [1/23, 1/23], 2/23, 2/23, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0067","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6956ef35b52207ddecf945bf7b46e9545a685264b55cdf273e944466cac0b771","family_id":"qf-mod33-bayes-total-evidence","seed":202611026067,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,24],[1,24],[1,24],[21,24]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/12, 11/12], [1/2, 1/22], [1/24, 1/24], 1/12, 1/12, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/12, 11/12], [1/2, 1/22], [1/24, 1/24], 1/12, 1/12, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0068","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee2d83583e93c7b700cc0e4b976acc841c68cbfc5a7363b6f9687b415723bdf","family_id":"qf-mod33-bayes-total-evidence","seed":202611026068,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,25],[1,25],[1,25],[22,25]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/25, 23/25], [1/2, 1/23], [1/25, 1/25], 2/25, 2/25, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/25, 23/25], [1/2, 1/23], [1/25, 1/25], 2/25, 2/25, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0069","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c3e6b80b280c8a382ed71ec3c8fba0597d3215125daefe31f24f838efa6ab57","family_id":"qf-mod33-bayes-total-evidence","seed":202611026069,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,26],[1,26],[1,26],[23,26]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/13, 12/13], [1/2, 1/24], [1/26, 1/26], 1/13, 1/13, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/13, 12/13], [1/2, 1/24], [1/26, 1/26], 1/13, 1/13, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0070","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49f24195ef2a6edd68b4b6036579df15800e5b36ea23970b95b97ca83487f268","family_id":"qf-mod33-bayes-total-evidence","seed":202611026070,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,27],[1,27],[1,27],[24,27]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/27, 25/27], [1/2, 1/25], [1/27, 1/27], 2/27, 2/27, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/27, 25/27], [1/2, 1/25], [1/27, 1/27], 2/27, 2/27, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0071","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa3bfaef0b3e4ee9424ecea886da5145d87939b475bff7c0a6ce422dd21126c9","family_id":"qf-mod33-bayes-total-evidence","seed":202611026071,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,28],[1,28],[1,28],[25,28]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/14, 13/14], [1/2, 1/26], [1/28, 1/28], 1/14, 1/14, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/14, 13/14], [1/2, 1/26], [1/28, 1/28], 1/14, 1/14, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0072","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f01dcea321f8d1fd3f3408b05086a1a73a324d9bbe3def6f891086043ac583b9","family_id":"qf-mod33-bayes-total-evidence","seed":202611026072,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/29, 27/29], [1/2, 1/27], [1/29, 1/29], 2/29, 2/29, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/29, 27/29], [1/2, 1/27], [1/29, 1/29], 2/29, 2/29, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0073","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d4d4fdb97003ea24f82865d34ce723785954e4af5ea66cebefaae4a7f0e700d","family_id":"qf-mod33-bayes-total-evidence","seed":202611026073,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,30],[1,30],[1,30],[27,30]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0074","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74fd2b3692a63f3136a58fd9be38055575ee26fa8316f0f4ea0bd69c124c4fe2","family_id":"qf-mod33-bayes-total-evidence","seed":202611026074,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,31],[1,31],[1,31],[28,31]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([2/31, 29/31], [1/2, 1/29], [1/31, 1/31], 2/31, 2/31, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([2/31, 29/31], [1/2, 1/29], [1/31, 1/31], 2/31, 2/31, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0075","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f37cd684d09eb2fea598cbdb8cc32cd14a9bb437df76cfa817a6dc8cb6bdc9fc","family_id":"qf-mod33-bayes-total-evidence","seed":202611026075,"semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,32],[1,32],[1,32],[29,32]]}"},"fields":{"title":"The Law of Total Probability Supplies the Bayes Denominator","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[0, 2]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Ordered exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.","answer":{"kind":"expression","value":"([1/16, 15/16], [1/2, 1/30], [1/32, 1/32], 1/16, 1/16, Yes)","explanation":"Compute each prior, conditional likelihood and joint, sum all joints and compare with the direct mass of E. An overlapping or incomplete hypothesis list is rejected, not normalized as if it were a partition. The exact result, retaining the requested order, is ([1/16, 15/16], [1/2, 1/30], [1/32, 1/32], 1/16, 1/16, Yes)."},"skills":["The Law of Total Probability Supplies the Bayes Denominator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0076","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df0976f70d9e5495f0c82f75b87f4b698fda89e43822a2ec29bc3987803478b9","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026076,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,8],[1,8],[1,8],[5,8]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/8, 1/8, 1/8, 5/8], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0077","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2dc64fa19f93f3c5451a42996a55aaa37b410cb0434351cf679145d91bd40cf","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026077,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,9],[1,9],[1,9],[6,9]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/9, 1/9, 1/9, 6/9], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0078","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d07a82c96e59e52c062dad2f7c98ac94d4a0cb97d18a72d2b1a9495416b828b9","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026078,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,10],[1,10],[1,10],[7,10]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/10, 1/10, 1/10, 7/10], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0079","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b905177902f7e5d49e6fbc415a1719202b68ddf945b01e482a5c4a2e792bac5c","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026079,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,11],[1,11],[1,11],[8,11]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0080","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"130ac0c2abab610cb43f00117cffbc92bfd90b1c272308adbdab76e73baf1a8c","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026080,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,12],[1,12],[1,12],[9,12]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/12, 1/12, 1/12, 9/12], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0081","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9535f02eba690e9d2dcbca539e43dded3eaa5f3481850ca132847b766672c2bf","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026081,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,13],[1,13],[1,13],[10,13]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/13, 1/13, 1/13, 10/13], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0082","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e29d8844cf15290962e3d983957700b5a64fe2bed04ab476dc6c203f3aedfe9e","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026082,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,14],[1,14],[1,14],[11,14]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/14, 1/14, 1/14, 11/14], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0083","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f26a6d5fbd28db16b9ba46c0adb0717044b0f1e2499789372ba1d736ee95d75","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026083,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,15],[1,15],[1,15],[12,15]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0084","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7360e52fa7e3180a79d97c5ec20c4aef401de42fffa618e1bcb79f1239313794","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026084,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,16],[1,16],[1,16],[13,16]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/16, 1/16, 1/16, 13/16], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0085","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2da5bff347e2f22eddc9e8a5211cbeee7c72d70ae03045d42e5fe8711f05743","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026085,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,17],[1,17],[1,17],[14,17]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/17, 1/17, 1/17, 14/17], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0086","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9e098f0173ac2da2868bdabcbdc78fe37ec3a2939097257fbaf1bcc490af89d","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026086,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,18],[1,18],[1,18],[15,18]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/18, 1/18, 1/18, 15/18], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0087","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7be9960871590904922aadb296e30d5a9034a9c4678e04c5d92d8e5390bfa7bb","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026087,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,19],[1,19],[1,19],[16,19]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/19, 1/19, 1/19, 16/19], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0088","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68cbdea8121cd50696eb7a31f1c9f6fb5780bea0b5d15e6fec5e72dcf846ec55","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026088,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,20],[1,20],[1,20],[17,20]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/20, 1/20, 1/20, 17/20], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0089","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5c75bfeb14ff4dfd8b1a96c9c69b4cb3b617decd3b5ee27dc7f55bef7ae8211","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026089,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,21],[1,21],[1,21],[18,21]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0090","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59b4704486dafb4399c561e26f735c383e2489c30811fc9f5bab565341fdaa98","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026090,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,22],[1,22],[1,22],[19,22]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0091","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93101682907f24797aa8de69a9f28726660bc81671726a89426799b6b19dbd36","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026091,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,23],[1,23],[1,23],[20,23]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/23, 1/23, 1/23, 20/23], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0092","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b893b7d8c5ebc496e1ad60e107b135c23d20222007bbfa004f4b4c534bde3fba","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026092,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,24],[1,24],[1,24],[21,24]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/24, 1/24, 1/24, 21/24], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0093","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d0ca94e5bba6dd30bbb8cda5886e2eaa861bc81fbd58f9b9b437a60688a43a6","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026093,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,25],[1,25],[1,25],[22,25]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/25, 1/25, 1/25, 22/25], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0094","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7258204ead2202bca329370e154560e3609bd2b00675d94357685397c0bd9486","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026094,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,26],[1,26],[1,26],[23,26]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0095","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2edad4ec50eb5e7e22baf8ef83c41b38e03116c5450c61c7a6f90901abdc4f1","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026095,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,27],[1,27],[1,27],[24,27]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0096","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ae485dc978f0f16a2c5984c3a943de525e3d6bc3beb27c926c49a368f7d4c8b","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026096,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,28],[1,28],[1,28],[25,28]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/28, 1/28, 1/28, 25/28], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0097","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"997d482972828adc31f421a89c0cc9a31e980ba98c32289b7e423329f6842437","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026097,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0098","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c617a9210bd082971ae9122b61b30b9672bb74a0d1035c9c574d40ad7bc0645","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026098,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,30],[1,30],[1,30],[27,30]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0099","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c8dc31f4dd3469c08b58e186465d1c7855652a4ad57d1b6c0493b2134ae4ab7","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026099,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,31],[1,31],[1,31],[28,31]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/31, 1/31, 1/31, 28/31], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r06-bayes-theorem-0100","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b0d167848c0141ac267fb0a3d7e85a1083d32cf53170c5c3b96c05bd6a23cd2","family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026100,"semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,32],[1,32],[1,32],[29,32]]}"},"fields":{"title":"Zero Evidence Mass Makes This Conditional Probability Undefined","stem":"Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/32, 1/32, 1/32, 29/32], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).","answer":{"kind":"expression","value":"(0, undefined, No, conditional_probability_requires_positive_denominator)","explanation":"Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)."},"skills":["Zero Evidence Mass Makes This Conditional Probability Undefined"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}