{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r04-conditional-probability-0001","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16ca38562df75d25aef3650d2183fd3a4c6dd4a0dc52129001e4e4f97b2e4b4a","family_id":"qf-density10-conditional-masses-pair","seed":202610104001,"semantic_template":"qf-density10-conditional-masses-pair/[\"0/1\",\"1/3\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,2}, F=P(S), masses=[0, 1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[], B=[1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 1/3)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (0/1, 1/3)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0002","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f082aaee5c950627da702c3c9335bfbccc44961704274d8d55a0dd7122186a6","family_id":"qf-density10-conditional-masses-pair","seed":202610104002,"semantic_template":"qf-density10-conditional-masses-pair/[\"1/1\",\"1/1\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,2}, F=P(S), masses=[1, 0, 0], D=1, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (1/1, 1/1)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0003","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50ac94d9991201109eae736fa0a910799b939a6f471fbc795e31e40cad2e5923","family_id":"qf-density10-conditional-masses-pair","seed":202610104003,"semantic_template":"qf-density10-conditional-masses-pair/[\"2/5\",\"3/5\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[2, 1, 2, 0], D=5, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/5, 3/5)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (2/5, 3/5)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0004","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43570b33118bb7c118e1ed5d0f99834e4b47d6ae0d20c4c16a0a311450d99a94","family_id":"qf-density10-conditional-masses-pair","seed":202610104004,"semantic_template":"qf-density10-conditional-masses-pair/[\"1/2\",\"2/3\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[3, 1, 2, 0], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 2/3)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (1/2, 2/3)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0005","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bd403c6bc5b496abaeb88f4c26c796cdc27b08c8587568d5916b6c6ce7bdfdd","family_id":"qf-density10-conditional-masses-pair","seed":202610104005,"semantic_template":"qf-density10-conditional-masses-pair/[\"4/7\",\"5/7\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[4, 1, 2, 0], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/7, 5/7)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (4/7, 5/7)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0006","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce8fe9389647f15f38fd8eea8832f4325327f83c77b8b2d2b71b09b0a518f2ba","family_id":"qf-density10-conditional-masses-pair","seed":202610104006,"semantic_template":"qf-density10-conditional-masses-pair/[\"5/8\",\"3/4\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[5, 1, 2, 0], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/8, 3/4)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (5/8, 3/4)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0007","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd8049690e0d863d93347efcd4600a2a3b96654f615e77a626bab49609242437","family_id":"qf-density10-conditional-masses-pair","seed":202610104007,"semantic_template":"qf-density10-conditional-masses-pair/[\"2/3\",\"7/9\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[6, 1, 2, 0], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/3, 7/9)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (2/3, 7/9)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0008","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28f6511c8bea04ee006c84228e0f8a61dfcf775d3c17e3a9c6b7f98adca1193c","family_id":"qf-density10-conditional-masses-pair","seed":202610104008,"semantic_template":"qf-density10-conditional-masses-pair/[\"7/10\",\"4/5\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[7, 1, 2, 0], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/10, 4/5)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (7/10, 4/5)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0009","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b29e5d08b5cd3fa5050c2b469ce5b44db815680edd09006bfc1536f7138865a","family_id":"qf-density10-conditional-masses-pair","seed":202610104009,"semantic_template":"qf-density10-conditional-masses-pair/[\"8/11\",\"9/11\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[8, 1, 2, 0], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/11, 9/11)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (8/11, 9/11)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0010","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"293877b22d1452b59488335649370ea5fa3a0578ce3138a8555476c15e285e5c","family_id":"qf-density10-conditional-masses-pair","seed":202610104010,"semantic_template":"qf-density10-conditional-masses-pair/[\"3/4\",\"5/6\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[9, 1, 2, 0], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/4, 5/6)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (3/4, 5/6)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0011","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db5468ebdf7a5795359b711f2e3d3de8d17cd64f65e87d7c7d67fe19751f3034","family_id":"qf-density10-conditional-masses-pair","seed":202610104011,"semantic_template":"qf-density10-conditional-masses-pair/[\"10/13\",\"11/13\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[10, 1, 2, 0], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10/13, 11/13)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (10/13, 11/13)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0012","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec2d21f791856bc2782d59044982dc6b9b6c9838e8fa5b21a945b1f97cb49478","family_id":"qf-density10-conditional-masses-pair","seed":202610104012,"semantic_template":"qf-density10-conditional-masses-pair/[\"11/14\",\"6/7\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[11, 1, 2, 0], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/14, 6/7)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (11/14, 6/7)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0013","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"692354e58be61f665cae50377e74a68bf1a4eb7bca4672ae3ee542f126c52cd6","family_id":"qf-density10-conditional-masses-pair","seed":202610104013,"semantic_template":"qf-density10-conditional-masses-pair/[\"4/5\",\"13/15\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[12, 1, 2, 0], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/5, 13/15)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (4/5, 13/15)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0014","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f956dab296ee07356b9f0dc4f879f17b9d30934a991a8cbda2d844471662cf1e","family_id":"qf-density10-conditional-masses-pair","seed":202610104014,"semantic_template":"qf-density10-conditional-masses-pair/[\"13/16\",\"7/8\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[13, 1, 2, 0], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13/16, 7/8)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (13/16, 7/8)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0015","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3de8503d1a4f142f9cbd06da8c8285b6790abe39fbec5344f80f19acce153c9d","family_id":"qf-density10-conditional-masses-pair","seed":202610104015,"semantic_template":"qf-density10-conditional-masses-pair/[\"14/17\",\"15/17\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[14, 1, 2, 0], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(14/17, 15/17)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (14/17, 15/17)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0016","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df3ab99ec26358d9ffc0393576025586c66e223bc6916341055bafa1301b692d","family_id":"qf-density10-conditional-masses-pair","seed":202610104016,"semantic_template":"qf-density10-conditional-masses-pair/[\"5/6\",\"8/9\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[15, 1, 2, 0], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/6, 8/9)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (5/6, 8/9)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0017","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e4618eb884451f8cc42d93eef78b32fa85468764c1d25787f23421721c5dac4","family_id":"qf-density10-conditional-masses-pair","seed":202610104017,"semantic_template":"qf-density10-conditional-masses-pair/[\"16/19\",\"17/19\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[16, 1, 2, 0], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(16/19, 17/19)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (16/19, 17/19)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0018","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1dc6e27094b3bd9e4da4bad7367d67450d7142923c432a737b2c8238c2955dd","family_id":"qf-density10-conditional-masses-pair","seed":202610104018,"semantic_template":"qf-density10-conditional-masses-pair/[\"17/20\",\"9/10\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[17, 1, 2, 0], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(17/20, 9/10)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (17/20, 9/10)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0019","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db415653b8d0439851a7ae0f50e7856b62b89fff1446fef08748a0ae20905f70","family_id":"qf-density10-conditional-masses-pair","seed":202610104019,"semantic_template":"qf-density10-conditional-masses-pair/[\"6/7\",\"19/21\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[18, 1, 2, 0], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/7, 19/21)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (6/7, 19/21)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0020","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c34ce32f7af11930eaf21db3e23fe3074a2beddd16f259b8bc8572ef72f74021","family_id":"qf-density10-conditional-masses-pair","seed":202610104020,"semantic_template":"qf-density10-conditional-masses-pair/[\"19/22\",\"10/11\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[19, 1, 2, 0], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(19/22, 10/11)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (19/22, 10/11)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0021","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec84247f4ee4f9269ef60f5a992c42565b8cb57698e52e3f48bcd0cc20b76b2d","family_id":"qf-density10-conditional-masses-pair","seed":202610104021,"semantic_template":"qf-density10-conditional-masses-pair/[\"20/23\",\"21/23\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[20, 1, 2, 0], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(20/23, 21/23)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (20/23, 21/23)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0022","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09e56340ddabfaaca48ddbcb6e977b5b8bdaa9a2d0e950ec70ec1a918b73c50a","family_id":"qf-density10-conditional-masses-pair","seed":202610104022,"semantic_template":"qf-density10-conditional-masses-pair/[\"7/8\",\"11/12\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[21, 1, 2, 0], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/8, 11/12)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (7/8, 11/12)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0023","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ee8e9586ee81f0132028e985091046f08bcb53e0d4fd3b5a272e10d9f188cc","family_id":"qf-density10-conditional-masses-pair","seed":202610104023,"semantic_template":"qf-density10-conditional-masses-pair/[\"22/25\",\"23/25\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[22, 1, 2, 0], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(22/25, 23/25)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (22/25, 23/25)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0024","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c4615bba24bf524b44719594379ebff5084440d042f51404ac22ea9e179dcec","family_id":"qf-density10-conditional-masses-pair","seed":202610104024,"semantic_template":"qf-density10-conditional-masses-pair/[\"23/26\",\"12/13\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[23, 1, 2, 0], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(23/26, 12/13)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (23/26, 12/13)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0025","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50a9ebb710cd5d6eab569ec9a6bed9e2a88b95e0dffa40e484d9f3681a637361","family_id":"qf-density10-conditional-masses-pair","seed":202610104025,"semantic_template":"qf-density10-conditional-masses-pair/[\"8/9\",\"25/27\"]"},"fields":{"title":"Intersection and Conditioning-Event Probabilities","stem":"S={0,…,3}, F=P(S), masses=[24, 1, 2, 0], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Give (P(A∩B), P(B)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/9, 25/27)","explanation":"Sum each event mass and divide separately by D; this question does not yet divide the two probabilities, and nonempty B alone would not imply positive probability. The exact result, retaining the requested order, is (8/9, 25/27)."},"skills":["Intersection and Conditioning-Event Probabilities"],"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-20261008-advance-20261010-r04-conditional-probability-0026","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43a4e841465af6eeb0c5f21dae281c132a30074effeaa075729a6dd0d71cefaf","family_id":"qf-density10-conditional-directions-pair","seed":202610104026,"semantic_template":"qf-density10-conditional-directions-pair/[\"disjoint_positive\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,1}, F=P(S), masses=[1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0], B=[1], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (0/1, 0/1)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0027","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f450cecbcaa5f18864cebbf61f648b226de81cf8d612813cf8c9145d4be1dc6","family_id":"qf-density10-conditional-directions-pair","seed":202610104027,"semantic_template":"qf-density10-conditional-directions-pair/[\"same_positive_event\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0], B=[0, 1], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (1/1, 1/1)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0028","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43f4e047325b2a3a759ddeae744278e64bbd547ef5dcdedb4f593f29afcfa0ad","family_id":"qf-density10-conditional-directions-pair","seed":202610104028,"semantic_template":"qf-density10-conditional-directions-pair/[\"1/2\",\"2/3\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[2, 1, 2], D=5, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 2/3)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (1/2, 2/3)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0029","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"512a4d5f695457a21a270514c06ed87057cc68c91af19feaad7f66eaeb03b633","family_id":"qf-density10-conditional-directions-pair","seed":202610104029,"semantic_template":"qf-density10-conditional-directions-pair/[\"3/5\",\"3/4\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[3, 1, 2], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/5, 3/4)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (3/5, 3/4)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0030","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9937d407dcdbd5571e15d277482389d3acf10dae052ebda84bb2087b522f0bd","family_id":"qf-density10-conditional-directions-pair","seed":202610104030,"semantic_template":"qf-density10-conditional-directions-pair/[\"2/3\",\"4/5\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[4, 1, 2], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2/3, 4/5)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (2/3, 4/5)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0031","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bec888b0c3c39f3091fa49288aad723f30ddec16c4e96b1d6836a20b59535037","family_id":"qf-density10-conditional-directions-pair","seed":202610104031,"semantic_template":"qf-density10-conditional-directions-pair/[\"5/7\",\"5/6\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[5, 1, 2], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/7, 5/6)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (5/7, 5/6)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0032","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86c68b3f7632205601d41e37675691229ac41ab55bf6d506aed1a132ce2d461a","family_id":"qf-density10-conditional-directions-pair","seed":202610104032,"semantic_template":"qf-density10-conditional-directions-pair/[\"3/4\",\"6/7\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[6, 1, 2], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/4, 6/7)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (3/4, 6/7)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0033","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be17cc5945c34826ccca6da4f4c4818616d19bce2d5e349a3aa7e04ff54e6b4f","family_id":"qf-density10-conditional-directions-pair","seed":202610104033,"semantic_template":"qf-density10-conditional-directions-pair/[\"7/9\",\"7/8\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[7, 1, 2], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/9, 7/8)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (7/9, 7/8)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0034","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0174273abcc860afe391550be5918dc517ffc3ecfefcfa3f87955953a744a4f","family_id":"qf-density10-conditional-directions-pair","seed":202610104034,"semantic_template":"qf-density10-conditional-directions-pair/[\"4/5\",\"8/9\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[8, 1, 2], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4/5, 8/9)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (4/5, 8/9)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0035","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be63bd05251cf6a4b040eb71cac9ae596adf745e8a1db01be914d59e9d5a0d72","family_id":"qf-density10-conditional-directions-pair","seed":202610104035,"semantic_template":"qf-density10-conditional-directions-pair/[\"9/11\",\"9/10\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[9, 1, 2], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(9/11, 9/10)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (9/11, 9/10)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0036","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de99dafdcc46190dbb11b1d786de5eaf1eefe26630767e78f9bd2334c8b66d4c","family_id":"qf-density10-conditional-directions-pair","seed":202610104036,"semantic_template":"qf-density10-conditional-directions-pair/[\"5/6\",\"10/11\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[10, 1, 2], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5/6, 10/11)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (5/6, 10/11)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0037","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d6b8f6dc40e63d448c1d625740ecf5a86cade98934abfbf50d289c018f334c9","family_id":"qf-density10-conditional-directions-pair","seed":202610104037,"semantic_template":"qf-density10-conditional-directions-pair/[\"11/13\",\"11/12\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[11, 1, 2], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/13, 11/12)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (11/13, 11/12)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0038","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f5d49a779a8f593fe5606fc9e99650991dbca43b2f0f176dbcf78c32c5ed1c","family_id":"qf-density10-conditional-directions-pair","seed":202610104038,"semantic_template":"qf-density10-conditional-directions-pair/[\"6/7\",\"12/13\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[12, 1, 2], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6/7, 12/13)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (6/7, 12/13)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0039","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0dd5eedcd0db3f657c2ff3c1d11e6611e340552689ee2c93f9c5da0a864074e","family_id":"qf-density10-conditional-directions-pair","seed":202610104039,"semantic_template":"qf-density10-conditional-directions-pair/[\"13/15\",\"13/14\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[13, 1, 2], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13/15, 13/14)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (13/15, 13/14)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0040","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26394df2d8cc0e79090ffc31c70c82f10e1084e9147327131f256e0b1f755615","family_id":"qf-density10-conditional-directions-pair","seed":202610104040,"semantic_template":"qf-density10-conditional-directions-pair/[\"7/8\",\"14/15\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[14, 1, 2], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7/8, 14/15)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (7/8, 14/15)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0041","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d08c375f6473cb0308c7dc7826419ca3f4da32f96a2e9438e7476eeb95fd92e6","family_id":"qf-density10-conditional-directions-pair","seed":202610104041,"semantic_template":"qf-density10-conditional-directions-pair/[\"15/17\",\"15/16\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[15, 1, 2], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(15/17, 15/16)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (15/17, 15/16)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0042","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"053fbd73bdc95983e459858ffd94fedd24790a42a4371f7d1c008c9384d9d2e7","family_id":"qf-density10-conditional-directions-pair","seed":202610104042,"semantic_template":"qf-density10-conditional-directions-pair/[\"8/9\",\"16/17\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[16, 1, 2], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8/9, 16/17)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (8/9, 16/17)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0043","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36bdbd40cb59a9a54d6f22c76305629a1d133adf9fe3e603b6544b113b430dae","family_id":"qf-density10-conditional-directions-pair","seed":202610104043,"semantic_template":"qf-density10-conditional-directions-pair/[\"17/19\",\"17/18\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[17, 1, 2], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(17/19, 17/18)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (17/19, 17/18)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0044","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c47124a524d4fdbcd007639909f5fafae4c0d66b77198b52712f0c576ed7a1d","family_id":"qf-density10-conditional-directions-pair","seed":202610104044,"semantic_template":"qf-density10-conditional-directions-pair/[\"9/10\",\"18/19\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[18, 1, 2], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(9/10, 18/19)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (9/10, 18/19)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0045","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"124e0e369ef535b50f7b8d8b1fc49e6c819e20fd4ef5a542519baa5e5288200e","family_id":"qf-density10-conditional-directions-pair","seed":202610104045,"semantic_template":"qf-density10-conditional-directions-pair/[\"19/21\",\"19/20\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[19, 1, 2], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(19/21, 19/20)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (19/21, 19/20)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0046","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df2efe76b72968faf508926d20f408f0e2c2c3b4bb3f91df24978317beea2abc","family_id":"qf-density10-conditional-directions-pair","seed":202610104046,"semantic_template":"qf-density10-conditional-directions-pair/[\"10/11\",\"20/21\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[20, 1, 2], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10/11, 20/21)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (10/11, 20/21)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0047","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e576440babfc9be9e698f5e115e35e312a4ac0f4fa31d8ee9b12e5ef69299887","family_id":"qf-density10-conditional-directions-pair","seed":202610104047,"semantic_template":"qf-density10-conditional-directions-pair/[\"21/23\",\"21/22\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[21, 1, 2], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(21/23, 21/22)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (21/23, 21/22)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0048","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c26f368ebbd90e62cc8d4bcd5179e584f96a38c8d59c809246baa72b2ac11bae","family_id":"qf-density10-conditional-directions-pair","seed":202610104048,"semantic_template":"qf-density10-conditional-directions-pair/[\"11/12\",\"22/23\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[22, 1, 2], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11/12, 22/23)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (11/12, 22/23)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0049","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c41a855238d59504dabfb25725cfe74a1723d65e5571080969a0c5b756474ce4","family_id":"qf-density10-conditional-directions-pair","seed":202610104049,"semantic_template":"qf-density10-conditional-directions-pair/[\"23/25\",\"23/24\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[23, 1, 2], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(23/25, 23/24)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (23/25, 23/24)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0050","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45a8d24111ab4ec28850d2b718a3fab37bebe958e161f6c17b7f67744ba94a58","family_id":"qf-density10-conditional-directions-pair","seed":202610104050,"semantic_template":"qf-density10-conditional-directions-pair/[\"12/13\",\"24/25\"]"},"fields":{"title":"Both Conditional-Probability Directions","stem":"S={0,…,2}, F=P(S), masses=[24, 1, 2], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 1], B=[0, 2], P(A)>0 and P(B)>0. Give (P(A|B), P(B|A)). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(12/13, 24/25)","explanation":"Divide the shared intersection mass by the B mass and the A mass respectively; the two directions generally differ, and independence is not assumed. The exact result, retaining the requested order, is (12/13, 24/25)."},"skills":["Both Conditional-Probability Directions"],"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-20261008-advance-20261010-r04-conditional-probability-0051","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e8167934a4257b203cab19c375ac979492b76f75bbbba9fff791f88845a34da","family_id":"qf-density10-conditional-point-law","seed":202610104051,"semantic_template":"qf-density10-conditional-point-law/[\"0\",\"1\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,2}, F=P(S), masses=[0, 1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[1], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[0/1, 1/1, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [0/1, 1/1, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0052","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6575cee2a4df2f8dfb7ea8559a396d55fb3c1834b289095fff64a6d81584cc53","family_id":"qf-density10-conditional-point-law","seed":202610104052,"semantic_template":"qf-density10-conditional-point-law/[\"1\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,0}, F=P(S), masses=[1], D=1, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[1/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [1/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0053","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c0302eac506fdee28120c542edfb6ca3c75292271d6b9c19b868990ba2893ec","family_id":"qf-density10-conditional-point-law","seed":202610104053,"semantic_template":"qf-density10-conditional-point-law/[\"2\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[2, 1, 2, 3], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[2/5, 1/5, 2/5, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [2/5, 1/5, 2/5, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0054","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28d5d4e8df5f300c5f92d6fb05cb99009e2c2176b8d5c59bd1405264b36d30dc","family_id":"qf-density10-conditional-point-law","seed":202610104054,"semantic_template":"qf-density10-conditional-point-law/[\"3\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[3, 1, 2, 3], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[1/2, 1/6, 1/3, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [1/2, 1/6, 1/3, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0055","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeeb42c033b19709f218fbfb1d5cb0171a531d7d3a593fd1194b2b096c3c3a6e","family_id":"qf-density10-conditional-point-law","seed":202610104055,"semantic_template":"qf-density10-conditional-point-law/[\"4\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[4, 1, 2, 3], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[4/7, 1/7, 2/7, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [4/7, 1/7, 2/7, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0056","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85c37ebb8a94dfdad6479365d102e3da35ee3fe66a35b48fdc314fa4f0af4b4c","family_id":"qf-density10-conditional-point-law","seed":202610104056,"semantic_template":"qf-density10-conditional-point-law/[\"5\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[5, 1, 2, 3], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[5/8, 1/8, 1/4, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [5/8, 1/8, 1/4, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0057","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b09a59c3ce5eb8565cb5653d103acc1427c10b7c051fb39d497800a57f0f3ab","family_id":"qf-density10-conditional-point-law","seed":202610104057,"semantic_template":"qf-density10-conditional-point-law/[\"6\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[6, 1, 2, 3], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[2/3, 1/9, 2/9, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [2/3, 1/9, 2/9, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0058","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1f00ba9a4fda16022b3192bbde900800f5282668cf3786fd331e5c02529b09a","family_id":"qf-density10-conditional-point-law","seed":202610104058,"semantic_template":"qf-density10-conditional-point-law/[\"7\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[7, 1, 2, 3], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[7/10, 1/10, 1/5, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [7/10, 1/10, 1/5, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0059","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e717e9e331c0842737ef9853bd7f4b5eda97bec2a12ca0c858dc6fc55a3ee723","family_id":"qf-density10-conditional-point-law","seed":202610104059,"semantic_template":"qf-density10-conditional-point-law/[\"8\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[8, 1, 2, 3], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[8/11, 1/11, 2/11, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [8/11, 1/11, 2/11, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0060","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c16a1dfa8a12f86079dcebcc4e9f8a6a9a7ce59ad5c5d159e5f11a0ef61ba87e","family_id":"qf-density10-conditional-point-law","seed":202610104060,"semantic_template":"qf-density10-conditional-point-law/[\"9\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[9, 1, 2, 3], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[3/4, 1/12, 1/6, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [3/4, 1/12, 1/6, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0061","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99a6753a7ab4afbaf37429486d86b1160029c5038f2f8b32d4baab4fde86d7e3","family_id":"qf-density10-conditional-point-law","seed":202610104061,"semantic_template":"qf-density10-conditional-point-law/[\"10\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[10, 1, 2, 3], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[10/13, 1/13, 2/13, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [10/13, 1/13, 2/13, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0062","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a2bdc85d7c38b83b1972913fdaa9d7240283debe50d29ce399ab4ae77430097","family_id":"qf-density10-conditional-point-law","seed":202610104062,"semantic_template":"qf-density10-conditional-point-law/[\"11\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[11, 1, 2, 3], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[11/14, 1/14, 1/7, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [11/14, 1/14, 1/7, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0063","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b9b9d40f8d199d9806c94344286a5b5d1402a22655eb21296a5655de65b713c","family_id":"qf-density10-conditional-point-law","seed":202610104063,"semantic_template":"qf-density10-conditional-point-law/[\"12\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[12, 1, 2, 3], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[4/5, 1/15, 2/15, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [4/5, 1/15, 2/15, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0064","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f14c0bd7624307d59bd43a12581e094516a7188a4b18b6f443425595be58eb0d","family_id":"qf-density10-conditional-point-law","seed":202610104064,"semantic_template":"qf-density10-conditional-point-law/[\"13\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[13, 1, 2, 3], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[13/16, 1/16, 1/8, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [13/16, 1/16, 1/8, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0065","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81b470c6ec178ef44f0107e74073a79b2e7320b558cb2cd31739514922a04a0f","family_id":"qf-density10-conditional-point-law","seed":202610104065,"semantic_template":"qf-density10-conditional-point-law/[\"14\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[14, 1, 2, 3], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[14/17, 1/17, 2/17, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [14/17, 1/17, 2/17, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0066","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fad03a71c1188041da326291bb87b6f0779cdd94c693cb93c04ec5351b2f5dd","family_id":"qf-density10-conditional-point-law","seed":202610104066,"semantic_template":"qf-density10-conditional-point-law/[\"15\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[15, 1, 2, 3], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[5/6, 1/18, 1/9, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [5/6, 1/18, 1/9, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0067","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aadf774c67cc283d73adbe442760acc468221ade4884b59df869c7c62d097fe","family_id":"qf-density10-conditional-point-law","seed":202610104067,"semantic_template":"qf-density10-conditional-point-law/[\"16\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[16, 1, 2, 3], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[16/19, 1/19, 2/19, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [16/19, 1/19, 2/19, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0068","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ba77bd45ed63f5e73d615b37482008473a9fb39df716041e22a2f3e0537edf","family_id":"qf-density10-conditional-point-law","seed":202610104068,"semantic_template":"qf-density10-conditional-point-law/[\"17\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[17, 1, 2, 3], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[17/20, 1/20, 1/10, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [17/20, 1/20, 1/10, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0069","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0702ce0c6bdfb3f0babd3bd5cc0f183604a14629189b64c06d86ada5ba150108","family_id":"qf-density10-conditional-point-law","seed":202610104069,"semantic_template":"qf-density10-conditional-point-law/[\"18\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[18, 1, 2, 3], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[6/7, 1/21, 2/21, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [6/7, 1/21, 2/21, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0070","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0340783f49a2e51d9f34bebfd64f084975dd0fd3e4122f41e7f20a591b3618a8","family_id":"qf-density10-conditional-point-law","seed":202610104070,"semantic_template":"qf-density10-conditional-point-law/[\"19\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[19, 1, 2, 3], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[19/22, 1/22, 1/11, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [19/22, 1/22, 1/11, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0071","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cee4f65c68c19c7b1e77f6fae5af6131bfe0b99fb0b7593e274aa6541c55c9f1","family_id":"qf-density10-conditional-point-law","seed":202610104071,"semantic_template":"qf-density10-conditional-point-law/[\"20\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[20, 1, 2, 3], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[20/23, 1/23, 2/23, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [20/23, 1/23, 2/23, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0072","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aab300f8f057d38b7e04e93cf02632c896a377ee42b1c31a71e6397430099bf7","family_id":"qf-density10-conditional-point-law","seed":202610104072,"semantic_template":"qf-density10-conditional-point-law/[\"21\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[21, 1, 2, 3], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[7/8, 1/24, 1/12, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [7/8, 1/24, 1/12, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0073","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48ec4c10d4e49498f249190e505dc6ddfd3476a8f0c46d500b6d629623b1aba8","family_id":"qf-density10-conditional-point-law","seed":202610104073,"semantic_template":"qf-density10-conditional-point-law/[\"22\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[22, 1, 2, 3], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[22/25, 1/25, 2/25, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [22/25, 1/25, 2/25, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0074","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e70e1d10dbf825f0947db3a2ef149e4515d78127283b1e4d8f92110462125a9","family_id":"qf-density10-conditional-point-law","seed":202610104074,"semantic_template":"qf-density10-conditional-point-law/[\"23\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[23, 1, 2, 3], D=29, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[23/26, 1/26, 1/13, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [23/26, 1/26, 1/13, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0075","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c80e92740c05befd23e4cdf004ee6c8e2feb34b23201b5814ca0a5a66fdad97","family_id":"qf-density10-conditional-point-law","seed":202610104075,"semantic_template":"qf-density10-conditional-point-law/[\"24\",\"1\",\"2\",\"0\"]"},"fields":{"title":"Complete Pointwise Conditional Law","stem":"S={0,…,3}, F=P(S), masses=[24, 1, 2, 3], D=30, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[8/9, 1/27, 2/27, 0/1]","explanation":"For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [8/9, 1/27, 2/27, 0/1]."},"skills":["Complete Pointwise Conditional Law"],"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-20261008-advance-20261010-r04-conditional-probability-0076","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddad560715341ecca0e2693c74a0ef0e009cacf210b0cb4ce96825984503b11e","family_id":"qf-density10-conditional-candidate-defect","seed":202610104076,"semantic_template":"qf-density10-conditional-candidate-defect/[\"1/2\",\"1/2\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0], B=[0, 1], P(B)>0. Candidate=1/2. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0077","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8876b01d2c552087c40d8b612c1a9dd2e6f4b870017c1a3bae63bc8e2e699d9","family_id":"qf-density10-conditional-candidate-defect","seed":202610104077,"semantic_template":"qf-density10-conditional-candidate-defect/[\"1/2\",\"-1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0], B=[0, 1], P(B)>0. Candidate=-1/1. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -3/2)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -3/2)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0078","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6566efa1962c72f1d6454c24ff21b1ebc4e40d62511110b3601358bb9a2736e2","family_id":"qf-density10-conditional-candidate-defect","seed":202610104078,"semantic_template":"qf-density10-conditional-candidate-defect/[\"1/2\",\"2/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0], B=[0, 1], P(B)>0. Candidate=2/1. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 3/2)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 3/2)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0079","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"166a0fa7cdc2fd83602ce06742796dc52fecde6708ae7ef4dfd4f540bd4a44f6","family_id":"qf-density10-conditional-candidate-defect","seed":202610104079,"semantic_template":"qf-density10-conditional-candidate-defect/[\"3/4\",\"3/4\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[3, 1, 2, 0], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=3/4. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0080","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c479df6b5757a9aefae592f759ea4fe6342dbc609bb9fe863ee3a6137cad3652","family_id":"qf-density10-conditional-candidate-defect","seed":202610104080,"semantic_template":"qf-density10-conditional-candidate-defect/[\"4/5\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[4, 1, 2, 0], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=5/5. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/5)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/5)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0081","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d84202e8f88264e880f4ce22f31d7d058d2a8dcee2f42d3416cb0c72af183fa9","family_id":"qf-density10-conditional-candidate-defect","seed":202610104081,"semantic_template":"qf-density10-conditional-candidate-defect/[\"5/6\",\"2/3\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[5, 1, 2, 0], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=4/6. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/6)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/6)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0082","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4c381d3aa0801c5995a5f6e740bb048d9ea9e15c47cf726f7712ebc5c89c27e","family_id":"qf-density10-conditional-candidate-defect","seed":202610104082,"semantic_template":"qf-density10-conditional-candidate-defect/[\"6/7\",\"6/7\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[6, 1, 2, 0], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=6/7. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0083","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0171054b5ffc5b019f0db7c0cb1f3304226c30849bd3790afc7a33c6bf3bc81","family_id":"qf-density10-conditional-candidate-defect","seed":202610104083,"semantic_template":"qf-density10-conditional-candidate-defect/[\"7/8\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[7, 1, 2, 0], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=8/8. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/8)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/8)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0084","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e9e14c65ca7bf95bb9eb6b867c5ba6c1da184dc3bbba791bc0e08ba452955f","family_id":"qf-density10-conditional-candidate-defect","seed":202610104084,"semantic_template":"qf-density10-conditional-candidate-defect/[\"8/9\",\"7/9\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[8, 1, 2, 0], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=7/9. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/9)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/9)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0085","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cbe663d07ce2f47d918f73ee9770e0bc135e656fd68ee4f49b6c87a8caa075d","family_id":"qf-density10-conditional-candidate-defect","seed":202610104085,"semantic_template":"qf-density10-conditional-candidate-defect/[\"9/10\",\"9/10\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[9, 1, 2, 0], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=9/10. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0086","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0e6fcc16db047aabf705724152b2f5d2441480c9cac1af6c417a41e5f905b5b","family_id":"qf-density10-conditional-candidate-defect","seed":202610104086,"semantic_template":"qf-density10-conditional-candidate-defect/[\"10/11\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[10, 1, 2, 0], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=11/11. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/11)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/11)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0087","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f4a9e0cd2b61624f23b09626c301dab9a9bae3d865d5a765ecd703a71c427f9","family_id":"qf-density10-conditional-candidate-defect","seed":202610104087,"semantic_template":"qf-density10-conditional-candidate-defect/[\"11/12\",\"5/6\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[11, 1, 2, 0], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=10/12. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/12)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/12)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0088","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81e3d51aabe4453c64067227e8b223148d1133f971e1d91ff3422e3a8060e9c2","family_id":"qf-density10-conditional-candidate-defect","seed":202610104088,"semantic_template":"qf-density10-conditional-candidate-defect/[\"12/13\",\"12/13\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[12, 1, 2, 0], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=12/13. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0089","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5857f2b8858b73c2224472ae2f74741097377f0bd06e7078a6f99abda6801c88","family_id":"qf-density10-conditional-candidate-defect","seed":202610104089,"semantic_template":"qf-density10-conditional-candidate-defect/[\"13/14\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[13, 1, 2, 0], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=14/14. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/14)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/14)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0090","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e0c63b65075666ccb1e35442ca9a32852515f8078def66146ffd0eaa866022","family_id":"qf-density10-conditional-candidate-defect","seed":202610104090,"semantic_template":"qf-density10-conditional-candidate-defect/[\"14/15\",\"13/15\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[14, 1, 2, 0], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=13/15. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/15)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/15)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0091","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9c93ad3ff60b1f502e048ec76a3f2ed759d3f880e73de61510edab864e6d014","family_id":"qf-density10-conditional-candidate-defect","seed":202610104091,"semantic_template":"qf-density10-conditional-candidate-defect/[\"15/16\",\"15/16\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[15, 1, 2, 0], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=15/16. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0092","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b1edbb5604d4160044e588ae6f624b9eb45664d2a0b4ddeaad0f7ac4e63a57b","family_id":"qf-density10-conditional-candidate-defect","seed":202610104092,"semantic_template":"qf-density10-conditional-candidate-defect/[\"16/17\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[16, 1, 2, 0], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=17/17. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/17)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/17)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0093","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f78f6571061ac5435e5ae02c524c63fad4bac94d7a8cc141ae6a254190f35ee9","family_id":"qf-density10-conditional-candidate-defect","seed":202610104093,"semantic_template":"qf-density10-conditional-candidate-defect/[\"17/18\",\"8/9\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[17, 1, 2, 0], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=16/18. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/18)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/18)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0094","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c59ee85604d33fff6e08c6756ac9b1400632a7a88117a284dcc0de00c9064dc","family_id":"qf-density10-conditional-candidate-defect","seed":202610104094,"semantic_template":"qf-density10-conditional-candidate-defect/[\"18/19\",\"18/19\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[18, 1, 2, 0], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=18/19. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0095","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"729c1fa9b2aaa5f585d0884eea095b22ac3bdaf942ea3eabae0ed7d4c60f266a","family_id":"qf-density10-conditional-candidate-defect","seed":202610104095,"semantic_template":"qf-density10-conditional-candidate-defect/[\"19/20\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[19, 1, 2, 0], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=20/20. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/20)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/20)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0096","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c40cf00d417a57e3225a30bc3920a42ce9cec3e66e9e733fd1a17332fb73b342","family_id":"qf-density10-conditional-candidate-defect","seed":202610104096,"semantic_template":"qf-density10-conditional-candidate-defect/[\"20/21\",\"19/21\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[20, 1, 2, 0], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=19/21. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/21)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/21)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0097","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d6eebbbe911163137cad3bb68344e39db3f6f8d05784f3f51ab0d2f5174fce1","family_id":"qf-density10-conditional-candidate-defect","seed":202610104097,"semantic_template":"qf-density10-conditional-candidate-defect/[\"21/22\",\"21/22\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[21, 1, 2, 0], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=21/22. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0098","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3904c5316d40fb2b924b85e34625cd8d16b8dc1412ff94dde90335587b61eda4","family_id":"qf-density10-conditional-candidate-defect","seed":202610104098,"semantic_template":"qf-density10-conditional-candidate-defect/[\"22/23\",\"1/1\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[22, 1, 2, 0], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=23/23. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, 1/23)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, 1/23)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0099","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13f4d8ef134340fd1d940b3805d436bf52fae21b59707662e55c86055ec2e37b","family_id":"qf-density10-conditional-candidate-defect","seed":202610104099,"semantic_template":"qf-density10-conditional-candidate-defect/[\"23/24\",\"11/12\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[23, 1, 2, 0], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=22/24. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(No, -1/24)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (No, -1/24)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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-20261008-advance-20261010-r04-conditional-probability-0100","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f93e680bf58265f80c8f1ea62d0cd54be2df36cf19d3d46498a483d824f644b7","family_id":"qf-density10-conditional-candidate-defect","seed":202610104100,"semantic_template":"qf-density10-conditional-candidate-defect/[\"24/25\",\"24/25\"]"},"fields":{"title":"Conditional Candidate Eligibility and Signed Defect","stem":"S={0,…,3}, F=P(S), masses=[24, 1, 2, 0], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=24/25. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.","answer":{"kind":"expression","value":"(Yes, 0/1)","explanation":"Compute the actual conditional probability from the events, then subtract; only zero defect passes. An invalid probability candidate does not invalidate the underlying model. The exact result, retaining the requested order, is (Yes, 0/1)."},"skills":["Conditional Candidate Eligibility and Signed Defect"],"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."}}