{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261027-r06-event-independence-0001","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad223eff56baefccabf2ebbab137118240f6ca098a2c894f9ac7125fd19f006","family_id":"qf-mod27-eventind-audit","seed":202610276001,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[36,49],[6,49],[6,49],[1,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 1/7, 37/49, 1/49, 37/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 1/7, 37/49, 1/49, 37/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0002","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1498eaf424aaec8fee9334ba8cd1652f066f48dc7780528b94b56a3b27acc34b","family_id":"qf-mod27-eventind-audit","seed":202610276002,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[30,49],[5,49],[12,49],[2,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 2/7, 1/7, 2/49, 2/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 1/7, 2/49, 2/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0003","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43a14e413551ef60d4fbf2c07ddd7800c12d1d3e6f2fd307970c2faa837c70f3","family_id":"qf-mod27-eventind-audit","seed":202610276003,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[24,49],[4,49],[18,49],[3,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 3/7, 1/7, 3/49, 3/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 1/7, 3/49, 3/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0004","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0edf9fc2f4071d5bc1179d8a1bc71608fc3e688c7793fcfeea343a8e69550cf9","family_id":"qf-mod27-eventind-audit","seed":202610276004,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[18,49],[3,49],[24,49],[4,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 4/7, 1/7, 4/49, 4/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 1/7, 4/49, 4/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0005","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"516e28bd297c58843ff8dc8b4452c209abcf2e53dbe3233d88783c6108c60ef1","family_id":"qf-mod27-eventind-audit","seed":202610276005,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[12,49],[2,49],[30,49],[5,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 5/7, 17/49, 5/49, 85/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 5/7, 17/49, 5/49, 85/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0006","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bbb64a44cb590b164ca4a4abb5ee0dbc86ff5ca9ec4a514477a1d7bd1fd5e09","family_id":"qf-mod27-eventind-audit","seed":202610276006,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[30,49],[12,49],[5,49],[2,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 1/7, 2/7, 2/49, 2/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 1/7, 2/7, 2/49, 2/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0007","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce856fbaf66f11a7b49dd5d44fa22dd0725a9e0aab517e74ec8a7ee8cca6cbec","family_id":"qf-mod27-eventind-audit","seed":202610276007,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[25,49],[10,49],[10,49],[4,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 2/7, 2/7, 4/49, 4/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 2/7, 4/49, 4/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0008","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87748f7d38dae24761af03542352fd19ce75d8a6621908bcb7624f9d55817821","family_id":"qf-mod27-eventind-audit","seed":202610276008,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[8,49],[15,49],[6,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 3/7, 2/7, 6/49, 6/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 2/7, 6/49, 6/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0009","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98ce4c787c4049512915da99dbd7af3d55e3bcab0184f14c01b03e45e2824f1c","family_id":"qf-mod27-eventind-audit","seed":202610276009,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[15,49],[6,49],[20,49],[8,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 4/7, 23/49, 8/49, 92/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 4/7, 23/49, 8/49, 92/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0010","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"349c4519b89c2d286ca2c476440f3c9d0d078bf21d98eacac1add380eb5dc7ae","family_id":"qf-mod27-eventind-audit","seed":202610276010,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[10,49],[4,49],[25,49],[10,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 5/7, 2/7, 10/49, 10/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 5/7, 2/7, 10/49, 10/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0011","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30214cbfc45cdf9dd35e0f291140ed58a1db1413b96c13057b29d6104f6b155b","family_id":"qf-mod27-eventind-audit","seed":202610276011,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[24,49],[18,49],[4,49],[3,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 1/7, 3/7, 3/49, 3/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 1/7, 3/7, 3/49, 3/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0012","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"454c300c366ef70afa825b1bfbd213adc9d3b436aa1f2d63537299d92cf280c0","family_id":"qf-mod27-eventind-audit","seed":202610276012,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[15,49],[8,49],[6,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0013","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76056eec6a47685737d85746ee158ea749b53b7c7edde4375944f6b37c81adc8","family_id":"qf-mod27-eventind-audit","seed":202610276013,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[16,49],[12,49],[12,49],[9,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 3/7, 25/49, 9/49, 75/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 3/7, 25/49, 9/49, 75/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0014","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e738c4b3418a4c2e5ec25fe6626735020dbb6ea377d82882364cc0e9d319dbc5","family_id":"qf-mod27-eventind-audit","seed":202610276014,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[9,49],[16,49],[12,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 4/7, 3/7, 12/49, 12/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 3/7, 12/49, 12/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0015","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"353436d34ff68a656ef853d2a7025350ea29b64756904b9304f15d032cbab520","family_id":"qf-mod27-eventind-audit","seed":202610276015,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[8,49],[6,49],[20,49],[15,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 5/7, 3/7, 15/49, 15/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 5/7, 3/7, 15/49, 15/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0016","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5af5bf4c58d16cfa902a41f0a4d8d1f0ff7d55c441c20fd56841f935a327befa","family_id":"qf-mod27-eventind-audit","seed":202610276016,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[18,49],[24,49],[3,49],[4,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 1/7, 4/7, 4/49, 4/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 1/7, 4/7, 4/49, 4/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0017","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14e9554ad29c15d4fc1695aefd5f799c492d8f553372a6413157fa1f7434ec2a","family_id":"qf-mod27-eventind-audit","seed":202610276017,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[15,49],[20,49],[6,49],[8,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 2/7, 23/49, 8/49, 46/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 2/7, 23/49, 8/49, 46/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0018","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74f3bfef3f591dd9455173daf9f40c1d3ca482893706efe1027ccd6216f2b6e4","family_id":"qf-mod27-eventind-audit","seed":202610276018,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[16,49],[9,49],[12,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 3/7, 4/7, 12/49, 12/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 4/7, 12/49, 12/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0019","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3c6f82ac44dd5274361ed8505c12f4473ad3cbf87dd08047ae919a525c2b6a0","family_id":"qf-mod27-eventind-audit","seed":202610276019,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[9,49],[12,49],[12,49],[16,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 4/7, 4/7, 16/49, 16/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 4/7, 16/49, 16/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0020","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec3e00d5d44523d006422fe7e4e6cf15fb0c72f6054332b188d95d1f27f1bd0d","family_id":"qf-mod27-eventind-audit","seed":202610276020,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[6,49],[8,49],[15,49],[20,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 5/7, 4/7, 20/49, 20/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 5/7, 4/7, 20/49, 20/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0021","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b596d4e28e9cb7faaed99f6809ed25416aa28e8d9819759f12e0bc0527483e2","family_id":"qf-mod27-eventind-audit","seed":202610276021,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[12,49],[30,49],[2,49],[5,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 1/7, 17/49, 5/49, 17/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 1/7, 17/49, 5/49, 17/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0022","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ddef521c4fada2fdf49b06c8f96313d3b12a0ecfbeaee87ac2263c777585869","family_id":"qf-mod27-eventind-audit","seed":202610276022,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[10,49],[25,49],[4,49],[10,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 2/7, 5/7, 10/49, 10/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 5/7, 10/49, 10/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0023","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1297d8485313c2571f2c4801ed07d1368e337431a3bd775ee37cc4fe55ddd8f0","family_id":"qf-mod27-eventind-audit","seed":202610276023,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[8,49],[20,49],[6,49],[15,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 3/7, 5/7, 15/49, 15/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 3/7, 5/7, 15/49, 15/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0024","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c53b30f9f35369e54b227ee344b6ce605510f0a615a842e9d223d7d93637f72","family_id":"qf-mod27-eventind-audit","seed":202610276024,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[6,49],[15,49],[8,49],[20,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [1, 3], 4/7, 5/7, 20/49, 20/49, Yes)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 4/7, 5/7, 20/49, 20/49, Yes)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0025","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a2801be8b518dc0657198840e550335a346742718eb1f16a7d7bb9dfaea267a","family_id":"qf-mod27-eventind-audit","seed":202610276025,"semantic_template":"qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[0,3],\"masses\":[[4,49],[10,49],[10,49],[25,49]]}"},"fields":{"title":"Test Independence by the Exact Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).","answer":{"kind":"expression","value":"([2, 3], [0, 3], 5/7, 29/49, 25/49, 145/343, No)","explanation":"Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [0, 3], 5/7, 29/49, 25/49, 145/343, No)."},"skills":["Test Independence by the Exact Product Definition"],"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-20261027-r06-event-independence-0026","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91cc96628640efa8467ed32d9490ae2ff41fd164952387bbe6c51aee9a367226","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276026,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[36,49],[6,49],[6,49],[1,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(1/7, 1/7, 1/49, 1/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 1/7, 1/49, 1/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0027","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5366f21bc32e29069acaf5af4b1fe666643ba5bf6f943898639271464787190","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276027,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[30,49],[5,49],[12,49],[2,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(2/7, 1/7, 2/49, 2/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 1/7, 2/49, 2/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0028","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff92e501ce6592b36743db9f6869707a3954abb70e8a0587befc39e6d0121d85","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276028,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[24,49],[4,49],[18,49],[3,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(3/7, 1/7, 3/49, 3/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 1/7, 3/49, 3/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0029","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cd49a9ab05d0c61a130def0f876c8be7f045d8288a11eb4a334c5681f35dfaa","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276029,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[18,49],[3,49],[24,49],[4,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(4/7, 1/7, 4/49, 4/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 1/7, 4/49, 4/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0030","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3189dd5418679bffb8a3c93f0df766be49f2c0ad9c7ecc32c10b9cf58897a756","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276030,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[2,49],[30,49],[5,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(5/7, 1/7, 5/49, 5/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 1/7, 5/49, 5/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0031","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a64b59b3e32ff0f795df77329fe08ba217ca901fd6c8032fbf0b84ce5691cf9","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276031,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[30,49],[12,49],[5,49],[2,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(1/7, 2/7, 2/49, 1/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 2/7, 2/49, 1/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0032","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57410aa95603a2d8f6a37ec221cfd6e77b6ca5787bf807a278f710038add31c4","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276032,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[25,49],[10,49],[10,49],[4,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(2/7, 2/7, 4/49, 2/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 2/7, 4/49, 2/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0033","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7014a30d5db2f789d978f63ae24e93f5ccc43ac7d4f895493af6aa637965fff9","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276033,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[8,49],[15,49],[6,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(3/7, 2/7, 6/49, 3/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 2/7, 6/49, 3/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0034","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afb7ade4e2a159c44ab82b2a0bff47552362a389dc91e0b38f4e10651c979203","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276034,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[15,49],[6,49],[20,49],[8,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(4/7, 2/7, 8/49, 4/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 2/7, 8/49, 4/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0035","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32400089ae85968f53306b8ba09d53480256543cbc2414b7db5e85bd3ba45034","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276035,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[10,49],[4,49],[25,49],[10,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(5/7, 2/7, 10/49, 5/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 2/7, 10/49, 5/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0036","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f3bf90ac3f79edb4b3bb58aef4a18310e62baa1aaa19396dcf6f846efa71af1","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276036,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[24,49],[18,49],[4,49],[3,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(1/7, 3/7, 3/49, 1/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 3/7, 3/49, 1/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0037","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba097a481fc89f15f13222ac9f00bfd61e1aa6a83826ae2c7471d8353c782bbf","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276037,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[15,49],[8,49],[6,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(2/7, 3/7, 6/49, 2/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 3/7, 6/49, 2/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0038","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"371984ca2fc94fcb2d1920a6f2bc652b0b6b92a0c84aff18718879bbec87f5a9","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276038,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[16,49],[12,49],[12,49],[9,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(3/7, 3/7, 9/49, 3/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 3/7, 9/49, 3/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0039","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95c8b2933331b1207cb5d8bce1eb5bab7e7d09cf3ef3e501c3c1fe1f278d5544","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276039,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[9,49],[16,49],[12,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(4/7, 3/7, 12/49, 4/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 3/7, 12/49, 4/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0040","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f9ed1656340cb60a734f7a908320db14be35858405bc2be1f2dff8627dea3bb","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276040,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[8,49],[6,49],[20,49],[15,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(5/7, 3/7, 15/49, 5/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 3/7, 15/49, 5/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0041","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"011267011f83ef328a8e71d57437ed08c08142df394c4747ef2b5d6d96a2af19","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276041,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[18,49],[24,49],[3,49],[4,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(1/7, 4/7, 4/49, 1/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 4/7, 4/49, 1/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0042","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39c79cc05cfd70e5ea384c6cd2bd96be9130bee60388ab7c2d1f1fbddfd6f146","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276042,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[15,49],[20,49],[6,49],[8,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(2/7, 4/7, 8/49, 2/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 4/7, 8/49, 2/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0043","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc3857eceb209107fa006f566b13132c7a5f5487ec29475f5de4704ccb8b6efe","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276043,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[16,49],[9,49],[12,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(3/7, 4/7, 12/49, 3/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 4/7, 12/49, 3/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0044","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82d3c4b68200b9a31a72929c75c76c2b6004f35bf71c869eb37a617431a6f1e8","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276044,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[9,49],[12,49],[12,49],[16,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(4/7, 4/7, 16/49, 4/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 4/7, 16/49, 4/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0045","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75834771227094321cda0b005a664ba0194a23fcb0d4c40eb554716cc642eb71","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276045,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[6,49],[8,49],[15,49],[20,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(5/7, 4/7, 20/49, 5/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 4/7, 20/49, 5/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0046","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94d43bc6048c26714937764c7564fe3c23e527b3134c30cc03019615fc31c8d0","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276046,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[30,49],[2,49],[5,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(1/7, 5/7, 5/49, 1/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 5/7, 5/49, 1/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0047","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6052f3a9d85ed1fdc89794660e979e329c670c91054368f15acf2c90505f4637","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276047,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[10,49],[25,49],[4,49],[10,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(2/7, 5/7, 10/49, 2/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 5/7, 10/49, 2/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0048","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f059fbd8eea002a72fdb2664a3f5a7d1a455ed6a6e1d352ab61fbc350b3301e4","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276048,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[8,49],[20,49],[6,49],[15,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(3/7, 5/7, 15/49, 3/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 5/7, 15/49, 3/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0049","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e499fca5926927539dee87b9f20eaf6097037c67cd59e58ae58666d982451e3","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276049,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[6,49],[15,49],[8,49],[20,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(4/7, 5/7, 20/49, 4/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 5/7, 20/49, 4/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0050","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6181b43ef7021d333197f82aec5fdcbcf5ee25a2b1cbdec1dbb11943cc3431a3","family_id":"qf-mod27-eventind-conditional-equivalence","seed":202610276050,"semantic_template":"qf-mod27-eventind-conditional-equivalence/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[4,49],[10,49],[10,49],[25,49]]}"},"fields":{"title":"Use Conditional Equivalence Only with Positive Denominator","stem":"Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)>0. Give (P_A,P_B,P_intersection,P_A_given_B,conditional_equal?,independent?).","answer":{"kind":"expression","value":"(5/7, 5/7, 25/49, 5/7, Yes, Yes)","explanation":"Divide the intersection mass by P(B) only after verifying it is positive. In this qualified case equality to P(A) is equivalent to the primary product definition. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 5/7, 25/49, 5/7, Yes, Yes)."},"skills":["Use Conditional Equivalence Only with Positive 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-20261027-r06-event-independence-0051","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca7178f958df7e779fa3a142c74e155d2ad2fd5c88c3e445b5a6ecdbced18ab1","family_id":"qf-mod27-eventind-complement","seed":202610276051,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[36,49],[6,49],[6,49],[1,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 6/7, 6/49, 6/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 6/49, 6/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0052","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cde5bc1dac55a1f5cf8eeb8048d2fc4f2235f2b428b78ac2a534ed0572d3cd7","family_id":"qf-mod27-eventind-complement","seed":202610276052,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[30,49],[5,49],[12,49],[2,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 5/7, 5/49, 5/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 5/49, 5/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0053","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba93f3c79de4184d2864c7bb7e28539e471f9d1d30daf7d955f021bd00c9b5c3","family_id":"qf-mod27-eventind-complement","seed":202610276053,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[24,49],[4,49],[18,49],[3,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 4/7, 4/49, 4/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 4/49, 4/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0054","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3262bc9f4ff6420f760cd18978b036f29de21e0697e2e32e46ee62668e5bde37","family_id":"qf-mod27-eventind-complement","seed":202610276054,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[18,49],[3,49],[24,49],[4,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 3/7, 3/49, 3/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 3/49, 3/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0055","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4f6e92c29063e1c91e8e7a87f1f2bf51f4d27e6ab7413fadf4f0f655b774489","family_id":"qf-mod27-eventind-complement","seed":202610276055,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[2,49],[30,49],[5,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 2/7, 2/49, 2/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 2/49, 2/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0056","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b01e863c80b460a17ab86fa84f6f10a64b6e8956078b74a6c892703799d8425d","family_id":"qf-mod27-eventind-complement","seed":202610276056,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[30,49],[12,49],[5,49],[2,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 6/7, 12/49, 12/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 12/49, 12/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0057","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62e76e2520cae27c3cca740fbf271d0247527a688aefe3e879ba33d4d3e3448c","family_id":"qf-mod27-eventind-complement","seed":202610276057,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[25,49],[10,49],[10,49],[4,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 5/7, 10/49, 10/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 10/49, 10/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0058","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b68de8672803ce95ac39f118f4ef96fa89c8fca95ded8308299e6d6e8879b0a9","family_id":"qf-mod27-eventind-complement","seed":202610276058,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[8,49],[15,49],[6,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 4/7, 8/49, 8/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 8/49, 8/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0059","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0f75b3926eedcf110de08d505bb93c6ec80a1d4ec5918a55a5d10ed6227972e","family_id":"qf-mod27-eventind-complement","seed":202610276059,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[15,49],[6,49],[20,49],[8,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 3/7, 6/49, 6/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 6/49, 6/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0060","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5975a03d7513f0583f10c991d1054737fec66847fad6b5642bd8c4c7f934723a","family_id":"qf-mod27-eventind-complement","seed":202610276060,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[10,49],[4,49],[25,49],[10,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 2/7, 4/49, 4/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 4/49, 4/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0061","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"955a82697f22c3aa48deddda4217edc448da254d72c5638c55d81d50da0041cb","family_id":"qf-mod27-eventind-complement","seed":202610276061,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[24,49],[18,49],[4,49],[3,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 6/7, 18/49, 18/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 18/49, 18/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0062","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7441274eb8246df14d9092d7d86a59df8f62aa791ea9bca99c9946195acacc9","family_id":"qf-mod27-eventind-complement","seed":202610276062,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[15,49],[8,49],[6,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 5/7, 15/49, 15/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 15/49, 15/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0063","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de107a41161dee2722e81ef5e1badc56b542f3c53c14419b6994219d058b70ed","family_id":"qf-mod27-eventind-complement","seed":202610276063,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[16,49],[12,49],[12,49],[9,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 4/7, 12/49, 12/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 12/49, 12/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0064","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070b3eb0b0417cb24b6b15a81ab30d3bdf0222abb64524e264ed2a05984d9a1f","family_id":"qf-mod27-eventind-complement","seed":202610276064,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[9,49],[16,49],[12,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 3/7, 9/49, 9/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 9/49, 9/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0065","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"009c49566542a495004e2763b6840a42ac3d90ab2ff8482ea37125c3ff212f31","family_id":"qf-mod27-eventind-complement","seed":202610276065,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[8,49],[6,49],[20,49],[15,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 2/7, 6/49, 6/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 6/49, 6/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0066","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46f15a93256831045f47cbddf4ea604400536338bf9c34b1ae01367cc7bc3567","family_id":"qf-mod27-eventind-complement","seed":202610276066,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[18,49],[24,49],[3,49],[4,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 6/7, 24/49, 24/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 24/49, 24/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0067","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1ad97c46b349cddecfa486ae932d9746c999635867cb07401e7fb3608f17a49","family_id":"qf-mod27-eventind-complement","seed":202610276067,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[15,49],[20,49],[6,49],[8,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 5/7, 20/49, 20/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 20/49, 20/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0068","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3667c2a64b09ea196b3e5837752a2e5b6c3ef980b74166e7ebc669047551d8e","family_id":"qf-mod27-eventind-complement","seed":202610276068,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[16,49],[9,49],[12,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 4/7, 16/49, 16/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 16/49, 16/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0069","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"102e1228b01ebe207063ddf16d98bdc7c09db46c24159b475e1c2580a395e0e7","family_id":"qf-mod27-eventind-complement","seed":202610276069,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[9,49],[12,49],[12,49],[16,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 3/7, 12/49, 12/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 12/49, 12/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0070","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091ccce0859d3b7dfabc25b20c0a6fa1a21afe043c010bc8385bdeb4c3b90068","family_id":"qf-mod27-eventind-complement","seed":202610276070,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[6,49],[8,49],[15,49],[20,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 2/7, 8/49, 8/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 8/49, 8/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0071","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95c7d8fbcbf53143df0cd3b411e783c518036dd9ca34c2d9884219d7c5839a13","family_id":"qf-mod27-eventind-complement","seed":202610276071,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[12,49],[30,49],[2,49],[5,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 6/7, 30/49, 30/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 6/7, 30/49, 30/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0072","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb7dbfa4431e24dbb4915876904104a6f46af969b260e116513f7545cd0d64f1","family_id":"qf-mod27-eventind-complement","seed":202610276072,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[10,49],[25,49],[4,49],[10,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 5/7, 25/49, 25/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 5/7, 25/49, 25/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0073","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"879148d58a1ccb6b5ada60f853f844cff22330faf15cf39054e83f96b3f52516","family_id":"qf-mod27-eventind-complement","seed":202610276073,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[8,49],[20,49],[6,49],[15,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 4/7, 20/49, 20/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 4/7, 20/49, 20/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0074","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e794b5f869e3e910a330f1695af08eab656cbf9088a47e6b15f80cd89d58d4","family_id":"qf-mod27-eventind-complement","seed":202610276074,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[6,49],[15,49],[8,49],[20,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 3/7, 15/49, 15/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 3/7, 15/49, 15/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0075","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08335acedb050ef180d47f14d4876a0bef630161afdea44df2c6dc6c3c547267","family_id":"qf-mod27-eventind-complement","seed":202610276075,"semantic_template":"qf-mod27-eventind-complement/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[4,49],[10,49],[10,49],[25,49]]}"},"fields":{"title":"Audit Complement Independence without Assuming the Original Flag","stem":"Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).","answer":{"kind":"expression","value":"(Yes, [0, 1], 2/7, 10/49, 10/49, Yes)","explanation":"Check the original product equation and its complemented counterpart, retaining both flags even when independence fails. The complement is a carrier-relative set, not a complement of the listed positive-mass outcomes alone. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], 2/7, 10/49, 10/49, Yes)."},"skills":["Audit Complement Independence without Assuming the Original Flag"],"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-20261027-r06-event-independence-0076","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c7573e57838545ec12d418319b3143923f733d1ebe631efb5297154e25f7887","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276076,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[36,49],[6,49],[6,49],[1,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[36/49, 6/49, 6/49, 1/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0077","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"424a407de548c136c73cded88c141bd06292051841d02675fe628216c9e0f19b","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276077,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[30,49],[5,49],[12,49],[2,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 5/49, 12/49, 2/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0078","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebd69c1f0350a968133116569577811c1aa5a8614d465692be6b60a4a2c4d300","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276078,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[24,49],[4,49],[18,49],[3,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 4/49, 18/49, 3/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0079","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"613880b2f6776b392f486c19ec049e3913378082b8fa228b203af6af220c0151","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276079,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[18,49],[3,49],[24,49],[4,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 3/49, 24/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0080","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f4752345eac5c527ee27d7bfea7d09587d4bcffaf197a6747857a1f712114f","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276080,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[12,49],[2,49],[30,49],[5,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 2/49, 30/49, 5/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0081","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d20acf9bee7c047a5696ed0970be3e979fb4df52b90a2cb39e17a9c64c9eafbc","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276081,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[30,49],[12,49],[5,49],[2,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[30/49, 12/49, 5/49, 2/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0082","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc4c0a608cb26672dc11be5d61c0e4078d268cf57008ccef55e6d88473e2cff2","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276082,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[25,49],[10,49],[10,49],[4,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[25/49, 10/49, 10/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0083","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a78d251438f6b4729ff91510e15dc04b5e3b4dd9ebd3e9d765bf547920d766ec","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276083,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[20,49],[8,49],[15,49],[6,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 8/49, 15/49, 6/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0084","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3fa65b1766190cf5c5bb2c9d6e8f97a1722237c853ecb9623e890b440db43f","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276084,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[15,49],[6,49],[20,49],[8,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 6/49, 20/49, 8/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0085","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff302c47ee3220de1efc24306bd46bb618ad0b200ad20c576fb1eca9fafcdd18","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276085,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[10,49],[4,49],[25,49],[10,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 4/49, 25/49, 10/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0086","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63138de045afb013822b88c8d9bf4ee6029b0e8eddf11c5ae1bdf249ee818d86","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276086,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[24,49],[18,49],[4,49],[3,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[24/49, 18/49, 4/49, 3/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0087","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a3ab95ee565b548a64d5ddbda116377cde422f7d921fabef0636882850ab19b","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276087,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[20,49],[15,49],[8,49],[6,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0088","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73548dfd82db6b3624f3cd067a6d41dabadce6fae3c2dc9b0e8384c348b7e777","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276088,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[16,49],[12,49],[12,49],[9,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0089","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4016469e83046c7aa565a3cf76dbed78e4eab9805f576242895dcfdc62c6353","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276089,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[12,49],[9,49],[16,49],[12,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 9/49, 16/49, 12/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0090","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f24719937056725f1ed9a3f68cac4baa77c4e69a4043c024715163b6fd88d5b","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276090,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[8,49],[6,49],[20,49],[15,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 6/49, 20/49, 15/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0091","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13301dcd06589f03c549d83e9fb8a4601126ea83b6ef54305ee5a624b22f7f08","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276091,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[18,49],[24,49],[3,49],[4,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0092","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edee84bc84ac970736e12cd4b105989397035261f93be40f782a99640dba9308","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276092,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[15,49],[20,49],[6,49],[8,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[15/49, 20/49, 6/49, 8/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0093","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b74eaf96ef44ed27bbc6eae2ac515c37c39257a76e3d0585c65cb1cb0a285fb4","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276093,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[12,49],[16,49],[9,49],[12,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 16/49, 9/49, 12/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0094","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"718670aa004794b4ba2d5b43a85ec7da924cd96f37972fb9cdefe391ce5952a4","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276094,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[9,49],[12,49],[12,49],[16,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[9/49, 12/49, 12/49, 16/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0095","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f8eefc210f87a1d0aa2171b31609c4f54803a48279c1de53ac37665271eeaf2","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276095,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[6,49],[8,49],[15,49],[20,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 8/49, 15/49, 20/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0096","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81787882bee963dd5db6ef323c701ebaaa219e8fb9f6212d28bf5d7e4876ca9e","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276096,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[12,49],[30,49],[2,49],[5,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[12/49, 30/49, 2/49, 5/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0097","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35a09813f135a13a76f04fa8ec19bd950dcdf9d2118dbfa8ed03e3f28b7cce50","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276097,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[10,49],[25,49],[4,49],[10,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[10/49, 25/49, 4/49, 10/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (2/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0098","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15921eedb354195961b80ad47045b36b4b5a8ba68fce8bb311b7ec68679643a","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276098,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[8,49],[20,49],[6,49],[15,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (3/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0099","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be2b918188661c48dba4fb5a5e22a63134845d03cd123116f63ba05c2f68698b","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276099,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[6,49],[15,49],[8,49],[20,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[6/49, 15/49, 8/49, 20/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (4/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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-20261027-r06-event-independence-0100","batch_id":"batch-20261014-advance-20261027-r06-event-independence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04bd01bc08d450da7537ea9048c2a420079f3c868145dca98fcd930fca3da435","family_id":"qf-mod27-eventind-zero-probability-boundary","seed":202610276100,"semantic_template":"qf-mod27-eventind-zero-probability-boundary/{\"A\":[2,3],\"B\":[4],\"masses\":[[4,49],[10,49],[10,49],[25,49],[0,1]]}"},"fields":{"title":"A Zero-Mass Conditioning Event Still Has a Product Definition","stem":"Use one complete finite probability model with aligned exact masses=[4/49, 10/49, 10/49, 25/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).","answer":{"kind":"expression","value":"(5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)","explanation":"The intersection has zero mass, as does P(A)P(B), so the two events are independent by the product definition. The elementary ratio P(A|B) is undefined; do not replace it by zero or perform a zero division. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (5/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)."},"skills":["A Zero-Mass Conditioning Event Still Has a Product Definition"],"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."}}