{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0001","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-12, -11, -10, -9}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/4, 1/4, 2/4, 1/4]，總和恰為一。A=∅，B={-12, -11, -10, -9} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047001,"parameters":{"S":[-12,-11,-10,-9],"F":"powerset(S)","weights":[0,1,2,1],"denominator":4,"B":[-12,-11,-10,-9],"operation":"weighted_conditional","A":[],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[],\"B\":[-12,-11,-10,-9],\"F\":\"powerset(S)\",\"S\":[-12,-11,-10,-9],\"denominator\":\"4\",\"operation\":\"weighted_conditional\",\"weights\":[[-12,\"0\"],[-11,\"1\"],[-10,\"2\"],[-9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0002","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-11, -10, -9, -8}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/8, 2/8, 3/8, 2/8]，總和恰為一。A={-11, -10, -9, -8}，B={-11, -10, -9, -8} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047002,"parameters":{"S":[-11,-10,-9,-8],"F":"powerset(S)","weights":[1,2,3,2],"denominator":8,"B":[-11,-10,-9,-8],"operation":"weighted_conditional","A":[-11,-10,-9,-8],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-11,-10,-9,-8],\"B\":[-11,-10,-9,-8],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9,-8],\"denominator\":\"8\",\"operation\":\"weighted_conditional\",\"weights\":[[-11,\"1\"],[-10,\"2\"],[-9,\"3\"],[-8,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0003","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-10, -9, -8, -7}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/12, 3/12, 4/12, 3/12]，總和恰為一。A={-10, -9}，B={-9, -8} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"3/7","explanation_zh":"P(A|B)=3/7。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047003,"parameters":{"S":[-10,-9,-8,-7],"F":"powerset(S)","weights":[2,3,4,3],"denominator":12,"B":[-9,-8],"operation":"weighted_conditional","A":[-10,-9],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-10,-9],\"B\":[-9,-8],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8,-7],\"denominator\":\"12\",\"operation\":\"weighted_conditional\",\"weights\":[[-10,\"2\"],[-9,\"3\"],[-8,\"4\"],[-7,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0004","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-9, -8, -7, -6}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/13, 4/13, 5/13, 4/13]，總和恰為一。A={-9}，B={-7, -6} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047004,"parameters":{"S":[-9,-8,-7,-6],"F":"powerset(S)","weights":[0,4,5,4],"denominator":13,"B":[-7,-6],"operation":"weighted_conditional","A":[-9],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-9],\"B\":[-7,-6],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7,-6],\"denominator\":\"13\",\"operation\":\"weighted_conditional\",\"weights\":[[-9,\"0\"],[-8,\"4\"],[-7,\"5\"],[-6,\"4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0005","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-8, -7, -6, -5}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/13, 1/13, 6/13, 5/13]，總和恰為一。A={-6, -5}，B={-8, -6, -5} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"11/12","explanation_zh":"P(A|B)=11/12。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047005,"parameters":{"S":[-8,-7,-6,-5],"F":"powerset(S)","weights":[1,1,6,5],"denominator":13,"B":[-8,-6,-5],"operation":"weighted_conditional","A":[-6,-5],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-6,-5],\"B\":[-8,-6,-5],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6,-5],\"denominator\":\"13\",\"operation\":\"weighted_conditional\",\"weights\":[[-8,\"1\"],[-7,\"1\"],[-6,\"6\"],[-5,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0006","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-7, -6, -5, -4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/12, 2/12, 2/12, 6/12]，總和恰為一。A=∅，B={-7, -6, -5, -4} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047006,"parameters":{"S":[-7,-6,-5,-4],"F":"powerset(S)","weights":[2,2,2,6],"denominator":12,"B":[-7,-6,-5,-4],"operation":"weighted_conditional","A":[],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[],\"B\":[-7,-6,-5,-4],\"F\":\"powerset(S)\",\"S\":[-7,-6,-5,-4],\"denominator\":\"6\",\"operation\":\"weighted_conditional\",\"weights\":[[-7,\"1\"],[-6,\"1\"],[-5,\"1\"],[-4,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0007","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-6, -5, -4, -3}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/13, 3/13, 3/13, 7/13]，總和恰為一。A={-6, -5, -4, -3}，B={-6, -5, -4, -3} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047007,"parameters":{"S":[-6,-5,-4,-3],"F":"powerset(S)","weights":[0,3,3,7],"denominator":13,"B":[-6,-5,-4,-3],"operation":"weighted_conditional","A":[-6,-5,-4,-3],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-6,-5,-4,-3],\"B\":[-6,-5,-4,-3],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4,-3],\"denominator\":\"13\",\"operation\":\"weighted_conditional\",\"weights\":[[-6,\"0\"],[-5,\"3\"],[-4,\"3\"],[-3,\"7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0008","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-5, -4, -3, -2}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/17, 4/17, 4/17, 8/17]，總和恰為一。A={-5, -4}，B={-4, -3} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(A|B)=1/2。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047008,"parameters":{"S":[-5,-4,-3,-2],"F":"powerset(S)","weights":[1,4,4,8],"denominator":17,"B":[-4,-3],"operation":"weighted_conditional","A":[-5,-4],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-5,-4],\"B\":[-4,-3],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3,-2],\"denominator\":\"17\",\"operation\":\"weighted_conditional\",\"weights\":[[-5,\"1\"],[-4,\"4\"],[-3,\"4\"],[-2,\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0009","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-4, -3, -2, -1}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/17, 1/17, 5/17, 9/17]，總和恰為一。A={-4}，B={-2, -1} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047009,"parameters":{"S":[-4,-3,-2,-1],"F":"powerset(S)","weights":[2,1,5,9],"denominator":17,"B":[-2,-1],"operation":"weighted_conditional","A":[-4],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-4],\"B\":[-2,-1],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2,-1],\"denominator\":\"17\",\"operation\":\"weighted_conditional\",\"weights\":[[-4,\"2\"],[-3,\"1\"],[-2,\"5\"],[-1,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0010","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-3, -2, -1, 0}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/18, 2/18, 6/18, 10/18]，總和恰為一。A={-1, 0}，B={-3, -1, 0} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047010,"parameters":{"S":[-3,-2,-1,0],"F":"powerset(S)","weights":[0,2,6,10],"denominator":18,"B":[-3,-1,0],"operation":"weighted_conditional","A":[-1,0],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-1,0],\"B\":[-3,-1,0],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1,0],\"denominator\":\"9\",\"operation\":\"weighted_conditional\",\"weights\":[[-3,\"0\"],[-2,\"1\"],[-1,\"3\"],[0,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0011","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-2, -1, 0, 1}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/17, 3/17, 2/17, 11/17]，總和恰為一。A=∅，B={-2, -1, 0, 1} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047011,"parameters":{"S":[-2,-1,0,1],"F":"powerset(S)","weights":[1,3,2,11],"denominator":17,"B":[-2,-1,0,1],"operation":"weighted_conditional","A":[],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[],\"B\":[-2,-1,0,1],\"F\":\"powerset(S)\",\"S\":[-2,-1,0,1],\"denominator\":\"17\",\"operation\":\"weighted_conditional\",\"weights\":[[-2,\"1\"],[-1,\"3\"],[0,\"2\"],[1,\"11\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0012","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={-1, 0, 1, 2}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/21, 4/21, 3/21, 12/21]，總和恰為一。A={-1, 0, 1, 2}，B={-1, 0, 1, 2} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047012,"parameters":{"S":[-1,0,1,2],"F":"powerset(S)","weights":[2,4,3,12],"denominator":21,"B":[-1,0,1,2],"operation":"weighted_conditional","A":[-1,0,1,2],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[-1,0,1,2],\"B\":[-1,0,1,2],\"F\":\"powerset(S)\",\"S\":[-1,0,1,2],\"denominator\":\"21\",\"operation\":\"weighted_conditional\",\"weights\":[[-1,\"2\"],[0,\"4\"],[1,\"3\"],[2,\"12\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0013","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={0, 1, 2, 3}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/18, 1/18, 4/18, 13/18]，總和恰為一。A={0, 1}，B={1, 2} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/5","explanation_zh":"P(A|B)=1/5。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047013,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","weights":[0,1,4,13],"denominator":18,"B":[1,2],"operation":"weighted_conditional","A":[0,1],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[0,1],\"B\":[1,2],\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"denominator\":\"18\",\"operation\":\"weighted_conditional\",\"weights\":[[0,\"0\"],[1,\"1\"],[2,\"4\"],[3,\"13\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0014","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={1, 2, 3, 4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 2/22, 5/22, 14/22]，總和恰為一。A={1}，B={3, 4} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047014,"parameters":{"S":[1,2,3,4],"F":"powerset(S)","weights":[1,2,5,14],"denominator":22,"B":[3,4],"operation":"weighted_conditional","A":[1],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[1],\"B\":[3,4],\"F\":\"powerset(S)\",\"S\":[1,2,3,4],\"denominator\":\"22\",\"operation\":\"weighted_conditional\",\"weights\":[[1,\"1\"],[2,\"2\"],[3,\"5\"],[4,\"14\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0015","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={2, 3, 4, 5}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 3/26, 6/26, 15/26]，總和恰為一。A={4, 5}，B={2, 4, 5} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"21/23","explanation_zh":"P(A|B)=21/23。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047015,"parameters":{"S":[2,3,4,5],"F":"powerset(S)","weights":[2,3,6,15],"denominator":26,"B":[2,4,5],"operation":"weighted_conditional","A":[4,5],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[4,5],\"B\":[2,4,5],\"F\":\"powerset(S)\",\"S\":[2,3,4,5],\"denominator\":\"26\",\"operation\":\"weighted_conditional\",\"weights\":[[2,\"2\"],[3,\"3\"],[4,\"6\"],[5,\"15\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0016","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={3, 4, 5, 6}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/22, 4/22, 2/22, 16/22]，總和恰為一。A=∅，B={3, 4, 5, 6} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047016,"parameters":{"S":[3,4,5,6],"F":"powerset(S)","weights":[0,4,2,16],"denominator":22,"B":[3,4,5,6],"operation":"weighted_conditional","A":[],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[],\"B\":[3,4,5,6],\"F\":\"powerset(S)\",\"S\":[3,4,5,6],\"denominator\":\"11\",\"operation\":\"weighted_conditional\",\"weights\":[[3,\"0\"],[4,\"2\"],[5,\"1\"],[6,\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0017","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={4, 5, 6, 7}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 1/22, 3/22, 17/22]，總和恰為一。A={4, 5, 6, 7}，B={4, 5, 6, 7} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047017,"parameters":{"S":[4,5,6,7],"F":"powerset(S)","weights":[1,1,3,17],"denominator":22,"B":[4,5,6,7],"operation":"weighted_conditional","A":[4,5,6,7],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[4,5,6,7],\"B\":[4,5,6,7],\"F\":\"powerset(S)\",\"S\":[4,5,6,7],\"denominator\":\"22\",\"operation\":\"weighted_conditional\",\"weights\":[[4,\"1\"],[5,\"1\"],[6,\"3\"],[7,\"17\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0018","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={5, 6, 7, 8}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 2/26, 4/26, 18/26]，總和恰為一。A={5, 6}，B={6, 7} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/3","explanation_zh":"P(A|B)=1/3。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047018,"parameters":{"S":[5,6,7,8],"F":"powerset(S)","weights":[2,2,4,18],"denominator":26,"B":[6,7],"operation":"weighted_conditional","A":[5,6],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[5,6],\"B\":[6,7],\"F\":\"powerset(S)\",\"S\":[5,6,7,8],\"denominator\":\"13\",\"operation\":\"weighted_conditional\",\"weights\":[[5,\"1\"],[6,\"1\"],[7,\"2\"],[8,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0019","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={6, 7, 8, 9}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/27, 3/27, 5/27, 19/27]，總和恰為一。A={6}，B={8, 9} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047019,"parameters":{"S":[6,7,8,9],"F":"powerset(S)","weights":[0,3,5,19],"denominator":27,"B":[8,9],"operation":"weighted_conditional","A":[6],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[6],\"B\":[8,9],\"F\":\"powerset(S)\",\"S\":[6,7,8,9],\"denominator\":\"27\",\"operation\":\"weighted_conditional\",\"weights\":[[6,\"0\"],[7,\"3\"],[8,\"5\"],[9,\"19\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0020","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={7, 8, 9, 10}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/31, 4/31, 6/31, 20/31]，總和恰為一。A={9, 10}，B={7, 9, 10} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"26/27","explanation_zh":"P(A|B)=26/27。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047020,"parameters":{"S":[7,8,9,10],"F":"powerset(S)","weights":[1,4,6,20],"denominator":31,"B":[7,9,10],"operation":"weighted_conditional","A":[9,10],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[9,10],\"B\":[7,9,10],\"F\":\"powerset(S)\",\"S\":[7,8,9,10],\"denominator\":\"31\",\"operation\":\"weighted_conditional\",\"weights\":[[7,\"1\"],[8,\"4\"],[9,\"6\"],[10,\"20\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0021","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={8, 9, 10, 11}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 1/26, 2/26, 21/26]，總和恰為一。A=∅，B={8, 9, 10, 11} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047021,"parameters":{"S":[8,9,10,11],"F":"powerset(S)","weights":[2,1,2,21],"denominator":26,"B":[8,9,10,11],"operation":"weighted_conditional","A":[],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[],\"B\":[8,9,10,11],\"F\":\"powerset(S)\",\"S\":[8,9,10,11],\"denominator\":\"26\",\"operation\":\"weighted_conditional\",\"weights\":[[8,\"2\"],[9,\"1\"],[10,\"2\"],[11,\"21\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0022","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={9, 10, 11, 12}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/27, 2/27, 3/27, 22/27]，總和恰為一。A={9, 10, 11, 12}，B={9, 10, 11, 12} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047022,"parameters":{"S":[9,10,11,12],"F":"powerset(S)","weights":[0,2,3,22],"denominator":27,"B":[9,10,11,12],"operation":"weighted_conditional","A":[9,10,11,12],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[9,10,11,12],\"B\":[9,10,11,12],\"F\":\"powerset(S)\",\"S\":[9,10,11,12],\"denominator\":\"27\",\"operation\":\"weighted_conditional\",\"weights\":[[9,\"0\"],[10,\"2\"],[11,\"3\"],[12,\"22\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0023","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={10, 11, 12, 13}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/31, 3/31, 4/31, 23/31]，總和恰為一。A={10, 11}，B={11, 12} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"3/7","explanation_zh":"P(A|B)=3/7。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047023,"parameters":{"S":[10,11,12,13],"F":"powerset(S)","weights":[1,3,4,23],"denominator":31,"B":[11,12],"operation":"weighted_conditional","A":[10,11],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[10,11],\"B\":[11,12],\"F\":\"powerset(S)\",\"S\":[10,11,12,13],\"denominator\":\"31\",\"operation\":\"weighted_conditional\",\"weights\":[[10,\"1\"],[11,\"3\"],[12,\"4\"],[13,\"23\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0024","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={11, 12, 13, 14}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/35, 4/35, 5/35, 24/35]，總和恰為一。A={11}，B={13, 14} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(A|B)=0/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047024,"parameters":{"S":[11,12,13,14],"F":"powerset(S)","weights":[2,4,5,24],"denominator":35,"B":[13,14],"operation":"weighted_conditional","A":[11],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[11],\"B\":[13,14],\"F\":\"powerset(S)\",\"S\":[11,12,13,14],\"denominator\":\"35\",\"operation\":\"weighted_conditional\",\"weights\":[[11,\"2\"],[12,\"4\"],[13,\"5\"],[14,\"24\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0025","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"有限加權模型的條件機率","stem_zh":"明示非空 S={12, 13, 14, 15}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/32, 1/32, 6/32, 25/32]，總和恰為一。A={14, 15}，B={12, 14, 15} 且 P(B)>0。求精確 P(A|B)，不可自行假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(A|B)=1/1。先取 A∩B 和 B 的機率質量再相除，共同正分母相消；不是直接相乘 P(A) 與 P(B)。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["有限加權模型的條件機率"]},"variant":{"family_id":"qf-conditional-weighted-probability","seed":202610047025,"parameters":{"S":[12,13,14,15],"F":"powerset(S)","weights":[0,1,6,25],"denominator":32,"B":[12,14,15],"operation":"weighted_conditional","A":[14,15],"semantic_template":"qf-conditional-weighted-probability/{\"A\":[14,15],\"B\":[12,14,15],\"F\":\"powerset(S)\",\"S\":[12,13,14,15],\"denominator\":\"32\",\"operation\":\"weighted_conditional\",\"weights\":[[12,\"0\"],[13,\"1\"],[14,\"6\"],[15,\"25\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0026","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-12, -11, -10, -9}、事件族 F=𝒫(S)，所有基本結果明示等可能。A=∅，B={-12, -11, -10, -9} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047026,"parameters":{"S":[-12,-11,-10,-9],"F":"powerset(S)","A":[],"B":[-12,-11,-10,-9],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[],\"B\":[-12,-11,-10,-9],\"F\":\"powerset(S)\",\"S\":[-12,-11,-10,-9],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0027","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-11, -10, -9, -8, -7}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-11, -10, -9, -8, -7}，B={-11, -10} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047027,"parameters":{"S":[-11,-10,-9,-8,-7],"F":"powerset(S)","A":[-11,-10,-9,-8,-7],"B":[-11,-10],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-11,-10,-9,-8,-7],\"B\":[-11,-10],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9,-8,-7],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0028","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-10, -9, -8, -7, -6, -5}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-10}，B={-6, -5} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047028,"parameters":{"S":[-10,-9,-8,-7,-6,-5],"F":"powerset(S)","A":[-10],"B":[-6,-5],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-10],\"B\":[-6,-5],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8,-7,-6,-5],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0029","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-9, -8, -7, -6}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-9, -8, -7}，B={-9, -8, -7, -6} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"3/4","explanation_zh":"交集數=3；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=3/4；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047029,"parameters":{"S":[-9,-8,-7,-6],"F":"powerset(S)","A":[-9,-8,-7],"B":[-9,-8,-7,-6],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-9,-8,-7],\"B\":[-9,-8,-7,-6],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7,-6],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0030","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-8, -7, -6, -5, -4}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-8, -7}，B={-8, -7} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047030,"parameters":{"S":[-8,-7,-6,-5,-4],"F":"powerset(S)","A":[-8,-7],"B":[-8,-7],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-8,-7],\"B\":[-8,-7],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6,-5,-4],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0031","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-7, -6, -5, -4, -3, -2}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-7, -5, -3}，B={-3, -2} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/2","explanation_zh":"交集數=1；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/2；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047031,"parameters":{"S":[-7,-6,-5,-4,-3,-2],"F":"powerset(S)","A":[-7,-5,-3],"B":[-3,-2],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-7,-5,-3],\"B\":[-3,-2],\"F\":\"powerset(S)\",\"S\":[-7,-6,-5,-4,-3,-2],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0032","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-6, -5, -4, -3}、事件族 F=𝒫(S)，所有基本結果明示等可能。A=∅，B={-6, -5, -4, -3} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047032,"parameters":{"S":[-6,-5,-4,-3],"F":"powerset(S)","A":[],"B":[-6,-5,-4,-3],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[],\"B\":[-6,-5,-4,-3],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4,-3],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0033","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-5, -4, -3, -2, -1}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-5, -4, -3, -2, -1}，B={-5, -4} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047033,"parameters":{"S":[-5,-4,-3,-2,-1],"F":"powerset(S)","A":[-5,-4,-3,-2,-1],"B":[-5,-4],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-5,-4,-3,-2,-1],\"B\":[-5,-4],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3,-2,-1],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0034","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-4, -3, -2, -1, 0, 1}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-4}，B={0, 1} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047034,"parameters":{"S":[-4,-3,-2,-1,0,1],"F":"powerset(S)","A":[-4],"B":[0,1],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-4],\"B\":[0,1],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2,-1,0,1],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0035","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-3, -2, -1, 0}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-3, -2, -1}，B={-3, -2, -1, 0} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"3/4","explanation_zh":"交集數=3；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=3/4；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047035,"parameters":{"S":[-3,-2,-1,0],"F":"powerset(S)","A":[-3,-2,-1],"B":[-3,-2,-1,0],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-3,-2,-1],\"B\":[-3,-2,-1,0],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1,0],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0036","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-2, -1, 0, 1, 2}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-2, -1}，B={-2, -1} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047036,"parameters":{"S":[-2,-1,0,1,2],"F":"powerset(S)","A":[-2,-1],"B":[-2,-1],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-2,-1],\"B\":[-2,-1],\"F\":\"powerset(S)\",\"S\":[-2,-1,0,1,2],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0037","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={-1, 0, 1, 2, 3, 4}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={-1, 1, 3}，B={3, 4} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/2","explanation_zh":"交集數=1；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/2；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047037,"parameters":{"S":[-1,0,1,2,3,4],"F":"powerset(S)","A":[-1,1,3],"B":[3,4],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[-1,1,3],\"B\":[3,4],\"F\":\"powerset(S)\",\"S\":[-1,0,1,2,3,4],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0038","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={0, 1, 2, 3}、事件族 F=𝒫(S)，所有基本結果明示等可能。A=∅，B={0, 1, 2, 3} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047038,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","A":[],"B":[0,1,2,3],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[],\"B\":[0,1,2,3],\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0039","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={1, 2, 3, 4, 5}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={1, 2, 3, 4, 5}，B={1, 2} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047039,"parameters":{"S":[1,2,3,4,5],"F":"powerset(S)","A":[1,2,3,4,5],"B":[1,2],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[1,2,3,4,5],\"B\":[1,2],\"F\":\"powerset(S)\",\"S\":[1,2,3,4,5],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0040","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={2, 3, 4, 5, 6, 7}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={2}，B={6, 7} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047040,"parameters":{"S":[2,3,4,5,6,7],"F":"powerset(S)","A":[2],"B":[6,7],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[2],\"B\":[6,7],\"F\":\"powerset(S)\",\"S\":[2,3,4,5,6,7],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0041","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={3, 4, 5, 6}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={3, 4, 5}，B={3, 4, 5, 6} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"3/4","explanation_zh":"交集數=3；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=3/4；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047041,"parameters":{"S":[3,4,5,6],"F":"powerset(S)","A":[3,4,5],"B":[3,4,5,6],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[3,4,5],\"B\":[3,4,5,6],\"F\":\"powerset(S)\",\"S\":[3,4,5,6],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0042","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={4, 5, 6, 7, 8}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={4, 5}，B={4, 5} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047042,"parameters":{"S":[4,5,6,7,8],"F":"powerset(S)","A":[4,5],"B":[4,5],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[4,5],\"B\":[4,5],\"F\":\"powerset(S)\",\"S\":[4,5,6,7,8],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0043","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={5, 6, 7, 8, 9, 10}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={5, 7, 9}，B={9, 10} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/2","explanation_zh":"交集數=1；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/2；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047043,"parameters":{"S":[5,6,7,8,9,10],"F":"powerset(S)","A":[5,7,9],"B":[9,10],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[5,7,9],\"B\":[9,10],\"F\":\"powerset(S)\",\"S\":[5,6,7,8,9,10],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0044","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={6, 7, 8, 9}、事件族 F=𝒫(S)，所有基本結果明示等可能。A=∅，B={6, 7, 8, 9} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047044,"parameters":{"S":[6,7,8,9],"F":"powerset(S)","A":[],"B":[6,7,8,9],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[],\"B\":[6,7,8,9],\"F\":\"powerset(S)\",\"S\":[6,7,8,9],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0045","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={7, 8, 9, 10, 11}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={7, 8, 9, 10, 11}，B={7, 8} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047045,"parameters":{"S":[7,8,9,10,11],"F":"powerset(S)","A":[7,8,9,10,11],"B":[7,8],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[7,8,9,10,11],\"B\":[7,8],\"F\":\"powerset(S)\",\"S\":[7,8,9,10,11],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0046","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={8, 9, 10, 11, 12, 13}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={8}，B={12, 13} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047046,"parameters":{"S":[8,9,10,11,12,13],"F":"powerset(S)","A":[8],"B":[12,13],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[8],\"B\":[12,13],\"F\":\"powerset(S)\",\"S\":[8,9,10,11,12,13],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0047","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={9, 10, 11, 12}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={9, 10, 11}，B={9, 10, 11, 12} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"3/4","explanation_zh":"交集數=3；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=3/4；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047047,"parameters":{"S":[9,10,11,12],"F":"powerset(S)","A":[9,10,11],"B":[9,10,11,12],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[9,10,11],\"B\":[9,10,11,12],\"F\":\"powerset(S)\",\"S\":[9,10,11,12],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0048","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={10, 11, 12, 13, 14}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={10, 11}，B={10, 11} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/1","explanation_zh":"交集數=2；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047048,"parameters":{"S":[10,11,12,13,14],"F":"powerset(S)","A":[10,11],"B":[10,11],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[10,11],\"B\":[10,11],\"F\":\"powerset(S)\",\"S\":[10,11,12,13,14],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0049","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={11, 12, 13, 14, 15, 16}、事件族 F=𝒫(S)，所有基本結果明示等可能。A={11, 13, 15}，B={15, 16} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"1/2","explanation_zh":"交集數=1；條件數=2。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=1/2；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047049,"parameters":{"S":[11,12,13,14,15,16],"F":"powerset(S)","A":[11,13,15],"B":[15,16],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[11,13,15],\"B\":[15,16],\"F\":\"powerset(S)\",\"S\":[11,12,13,14,15,16],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0050","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"明示等可能的條件機率","stem_zh":"明示非空 S={12, 13, 14, 15}、事件族 F=𝒫(S)，所有基本結果明示等可能。A=∅，B={12, 13, 14, 15} 且 B 非空。求 P(A|B)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"交集數=0；條件數=4。因明示等可能且 B 非空，P(A|B)=|A∩B|/|B|=0/1；𝒫 是全冪集，P 是機率律。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的條件機率"]},"variant":{"family_id":"qf-conditional-uniform-probability","seed":202610047050,"parameters":{"S":[12,13,14,15],"F":"powerset(S)","A":[],"B":[12,13,14,15],"equiprobable":true,"operation":"uniform_conditional","semantic_template":"qf-conditional-uniform-probability/{\"A\":[],\"B\":[12,13,14,15],\"F\":\"powerset(S)\",\"S\":[12,13,14,15],\"equiprobable\":true,\"operation\":\"uniform_conditional\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0051","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-12, -11, -10, -9}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/4, 1/4, 2/4, 1/4]，總和恰為一。B={-12} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047051,"parameters":{"S":[-12,-11,-10,-9],"F":"powerset(S)","weights":[0,1,2,1],"denominator":4,"B":[-12],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-12],\"F\":\"powerset(S)\",\"S\":[-12,-11,-10,-9],\"denominator\":\"4\",\"operation\":\"condition_validity\",\"weights\":[[-12,\"0\"],[-11,\"1\"],[-10,\"2\"],[-9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0052","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-11, -10, -9, -8}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/8, 2/8, 3/8, 2/8]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047052,"parameters":{"S":[-11,-10,-9,-8],"F":"powerset(S)","weights":[1,2,3,2],"denominator":8,"B":[],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9,-8],\"denominator\":\"8\",\"operation\":\"condition_validity\",\"weights\":[[-11,\"1\"],[-10,\"2\"],[-9,\"3\"],[-8,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0053","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-10, -9, -8, -7}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/12, 3/12, 4/12, 3/12]，總和恰為一。B={-9} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=3。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047053,"parameters":{"S":[-10,-9,-8,-7],"F":"powerset(S)","weights":[2,3,4,3],"denominator":12,"B":[-9],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-9],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8,-7],\"denominator\":\"12\",\"operation\":\"condition_validity\",\"weights\":[[-10,\"2\"],[-9,\"3\"],[-8,\"4\"],[-7,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0054","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-9, -8, -7, -6}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/13, 4/13, 5/13, 4/13]，總和恰為一。B={-9, -8, -7, -6} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=13。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047054,"parameters":{"S":[-9,-8,-7,-6],"F":"powerset(S)","weights":[0,4,5,4],"denominator":13,"B":[-9,-8,-7,-6],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-9,-8,-7,-6],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7,-6],\"denominator\":\"13\",\"operation\":\"condition_validity\",\"weights\":[[-9,\"0\"],[-8,\"4\"],[-7,\"5\"],[-6,\"4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0055","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-8, -7, -6, -5}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/13, 1/13, 6/13, 5/13]，總和恰為一。B={-8} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=1。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047055,"parameters":{"S":[-8,-7,-6,-5],"F":"powerset(S)","weights":[1,1,6,5],"denominator":13,"B":[-8],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-8],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6,-5],\"denominator\":\"13\",\"operation\":\"condition_validity\",\"weights\":[[-8,\"1\"],[-7,\"1\"],[-6,\"6\"],[-5,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0056","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-7, -6, -5, -4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/12, 2/12, 2/12, 6/12]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047056,"parameters":{"S":[-7,-6,-5,-4],"F":"powerset(S)","weights":[2,2,2,6],"denominator":12,"B":[],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[-7,-6,-5,-4],\"denominator\":\"6\",\"operation\":\"condition_validity\",\"weights\":[[-7,\"1\"],[-6,\"1\"],[-5,\"1\"],[-4,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0057","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-6, -5, -4, -3}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/13, 3/13, 3/13, 7/13]，總和恰為一。B={-5} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=3。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047057,"parameters":{"S":[-6,-5,-4,-3],"F":"powerset(S)","weights":[0,3,3,7],"denominator":13,"B":[-5],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-5],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4,-3],\"denominator\":\"13\",\"operation\":\"condition_validity\",\"weights\":[[-6,\"0\"],[-5,\"3\"],[-4,\"3\"],[-3,\"7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0058","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-5, -4, -3, -2}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/17, 4/17, 4/17, 8/17]，總和恰為一。B={-5, -4, -3, -2} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=17。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047058,"parameters":{"S":[-5,-4,-3,-2],"F":"powerset(S)","weights":[1,4,4,8],"denominator":17,"B":[-5,-4,-3,-2],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-5,-4,-3,-2],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3,-2],\"denominator\":\"17\",\"operation\":\"condition_validity\",\"weights\":[[-5,\"1\"],[-4,\"4\"],[-3,\"4\"],[-2,\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0059","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-4, -3, -2, -1}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/17, 1/17, 5/17, 9/17]，總和恰為一。B={-4} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=2。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047059,"parameters":{"S":[-4,-3,-2,-1],"F":"powerset(S)","weights":[2,1,5,9],"denominator":17,"B":[-4],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-4],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2,-1],\"denominator\":\"17\",\"operation\":\"condition_validity\",\"weights\":[[-4,\"2\"],[-3,\"1\"],[-2,\"5\"],[-1,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0060","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-3, -2, -1, 0}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/18, 2/18, 6/18, 10/18]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047060,"parameters":{"S":[-3,-2,-1,0],"F":"powerset(S)","weights":[0,2,6,10],"denominator":18,"B":[],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1,0],\"denominator\":\"9\",\"operation\":\"condition_validity\",\"weights\":[[-3,\"0\"],[-2,\"1\"],[-1,\"3\"],[0,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0061","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-2, -1, 0, 1}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/17, 3/17, 2/17, 11/17]，總和恰為一。B={-1} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=3。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047061,"parameters":{"S":[-2,-1,0,1],"F":"powerset(S)","weights":[1,3,2,11],"denominator":17,"B":[-1],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-1],\"F\":\"powerset(S)\",\"S\":[-2,-1,0,1],\"denominator\":\"17\",\"operation\":\"condition_validity\",\"weights\":[[-2,\"1\"],[-1,\"3\"],[0,\"2\"],[1,\"11\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0062","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={-1, 0, 1, 2}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/21, 4/21, 3/21, 12/21]，總和恰為一。B={-1, 0, 1, 2} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=21。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047062,"parameters":{"S":[-1,0,1,2],"F":"powerset(S)","weights":[2,4,3,12],"denominator":21,"B":[-1,0,1,2],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[-1,0,1,2],\"F\":\"powerset(S)\",\"S\":[-1,0,1,2],\"denominator\":\"21\",\"operation\":\"condition_validity\",\"weights\":[[-1,\"2\"],[0,\"4\"],[1,\"3\"],[2,\"12\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0063","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={0, 1, 2, 3}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/18, 1/18, 4/18, 13/18]，總和恰為一。B={0} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047063,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","weights":[0,1,4,13],"denominator":18,"B":[0],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[0],\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"denominator\":\"18\",\"operation\":\"condition_validity\",\"weights\":[[0,\"0\"],[1,\"1\"],[2,\"4\"],[3,\"13\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0064","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={1, 2, 3, 4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 2/22, 5/22, 14/22]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047064,"parameters":{"S":[1,2,3,4],"F":"powerset(S)","weights":[1,2,5,14],"denominator":22,"B":[],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[1,2,3,4],\"denominator\":\"22\",\"operation\":\"condition_validity\",\"weights\":[[1,\"1\"],[2,\"2\"],[3,\"5\"],[4,\"14\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0065","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={2, 3, 4, 5}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 3/26, 6/26, 15/26]，總和恰為一。B={3} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=3。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047065,"parameters":{"S":[2,3,4,5],"F":"powerset(S)","weights":[2,3,6,15],"denominator":26,"B":[3],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[3],\"F\":\"powerset(S)\",\"S\":[2,3,4,5],\"denominator\":\"26\",\"operation\":\"condition_validity\",\"weights\":[[2,\"2\"],[3,\"3\"],[4,\"6\"],[5,\"15\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0066","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={3, 4, 5, 6}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/22, 4/22, 2/22, 16/22]，總和恰為一。B={3, 4, 5, 6} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=22。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047066,"parameters":{"S":[3,4,5,6],"F":"powerset(S)","weights":[0,4,2,16],"denominator":22,"B":[3,4,5,6],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[3,4,5,6],\"F\":\"powerset(S)\",\"S\":[3,4,5,6],\"denominator\":\"11\",\"operation\":\"condition_validity\",\"weights\":[[3,\"0\"],[4,\"2\"],[5,\"1\"],[6,\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0067","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={4, 5, 6, 7}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 1/22, 3/22, 17/22]，總和恰為一。B={4} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=1。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047067,"parameters":{"S":[4,5,6,7],"F":"powerset(S)","weights":[1,1,3,17],"denominator":22,"B":[4],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[4],\"F\":\"powerset(S)\",\"S\":[4,5,6,7],\"denominator\":\"22\",\"operation\":\"condition_validity\",\"weights\":[[4,\"1\"],[5,\"1\"],[6,\"3\"],[7,\"17\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0068","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={5, 6, 7, 8}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 2/26, 4/26, 18/26]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047068,"parameters":{"S":[5,6,7,8],"F":"powerset(S)","weights":[2,2,4,18],"denominator":26,"B":[],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[5,6,7,8],\"denominator\":\"13\",\"operation\":\"condition_validity\",\"weights\":[[5,\"1\"],[6,\"1\"],[7,\"2\"],[8,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0069","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={6, 7, 8, 9}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/27, 3/27, 5/27, 19/27]，總和恰為一。B={7} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=3。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047069,"parameters":{"S":[6,7,8,9],"F":"powerset(S)","weights":[0,3,5,19],"denominator":27,"B":[7],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[7],\"F\":\"powerset(S)\",\"S\":[6,7,8,9],\"denominator\":\"27\",\"operation\":\"condition_validity\",\"weights\":[[6,\"0\"],[7,\"3\"],[8,\"5\"],[9,\"19\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0070","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={7, 8, 9, 10}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/31, 4/31, 6/31, 20/31]，總和恰為一。B={7, 8, 9, 10} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=31。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047070,"parameters":{"S":[7,8,9,10],"F":"powerset(S)","weights":[1,4,6,20],"denominator":31,"B":[7,8,9,10],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[7,8,9,10],\"F\":\"powerset(S)\",\"S\":[7,8,9,10],\"denominator\":\"31\",\"operation\":\"condition_validity\",\"weights\":[[7,\"1\"],[8,\"4\"],[9,\"6\"],[10,\"20\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0071","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={8, 9, 10, 11}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 1/26, 2/26, 21/26]，總和恰為一。B={8} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=2。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047071,"parameters":{"S":[8,9,10,11],"F":"powerset(S)","weights":[2,1,2,21],"denominator":26,"B":[8],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[8],\"F\":\"powerset(S)\",\"S\":[8,9,10,11],\"denominator\":\"26\",\"operation\":\"condition_validity\",\"weights\":[[8,\"2\"],[9,\"1\"],[10,\"2\"],[11,\"21\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0072","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={9, 10, 11, 12}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/27, 2/27, 3/27, 22/27]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047072,"parameters":{"S":[9,10,11,12],"F":"powerset(S)","weights":[0,2,3,22],"denominator":27,"B":[],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[9,10,11,12],\"denominator\":\"27\",\"operation\":\"condition_validity\",\"weights\":[[9,\"0\"],[10,\"2\"],[11,\"3\"],[12,\"22\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0073","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={10, 11, 12, 13}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/31, 3/31, 4/31, 23/31]，總和恰為一。B={11} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=3。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047073,"parameters":{"S":[10,11,12,13],"F":"powerset(S)","weights":[1,3,4,23],"denominator":31,"B":[11],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[11],\"F\":\"powerset(S)\",\"S\":[10,11,12,13],\"denominator\":\"31\",\"operation\":\"condition_validity\",\"weights\":[[10,\"1\"],[11,\"3\"],[12,\"4\"],[13,\"23\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0074","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={11, 12, 13, 14}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/35, 4/35, 5/35, 24/35]，總和恰為一。B={11, 12, 13, 14} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"是","explanation_zh":"條件質量分子=35。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047074,"parameters":{"S":[11,12,13,14],"F":"powerset(S)","weights":[2,4,5,24],"denominator":35,"B":[11,12,13,14],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[11,12,13,14],\"F\":\"powerset(S)\",\"S\":[11,12,13,14],\"denominator\":\"35\",\"operation\":\"condition_validity\",\"weights\":[[11,\"2\"],[12,\"4\"],[13,\"5\"],[14,\"24\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0075","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"條件事件的正機率資格","stem_zh":"明示非空 S={12, 13, 14, 15}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/32, 1/32, 6/32, 25/32]，總和恰為一。B={12} 是否具備定義 P(A|B) 所需的正機率條件資格？","answer":{"kind":"short_text","value":"否","explanation_zh":"條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["條件事件的正機率資格"]},"variant":{"family_id":"qf-conditional-event-validity","seed":202610047075,"parameters":{"S":[12,13,14,15],"F":"powerset(S)","weights":[0,1,6,25],"denominator":32,"B":[12],"operation":"condition_validity","semantic_template":"qf-conditional-event-validity/{\"B\":[12],\"F\":\"powerset(S)\",\"S\":[12,13,14,15],\"denominator\":\"32\",\"operation\":\"condition_validity\",\"weights\":[[12,\"0\"],[13,\"1\"],[14,\"6\"],[15,\"25\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0076","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-12, -11, -10, -9}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/4, 1/4, 2/4, 1/4]，總和恰為一。A={-11}，B={-10} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=0/1；P(B|A)=0/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047076,"parameters":{"S":[-12,-11,-10,-9],"F":"powerset(S)","weights":[0,1,2,1],"denominator":4,"B":[-10],"operation":"compare_direction","A":[-11],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-11],\"B\":[-10],\"F\":\"powerset(S)\",\"S\":[-12,-11,-10,-9],\"denominator\":\"4\",\"operation\":\"compare_direction\",\"weights\":[[-12,\"0\"],[-11,\"1\"],[-10,\"2\"],[-9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0077","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-11, -10, -9, -8}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/8, 2/8, 3/8, 2/8]，總和恰為一。A={-10, -9}，B={-9, -8} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=3/5；P(B|A)=3/5。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047077,"parameters":{"S":[-11,-10,-9,-8],"F":"powerset(S)","weights":[1,2,3,2],"denominator":8,"B":[-9,-8],"operation":"compare_direction","A":[-10,-9],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-10,-9],\"B\":[-9,-8],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9,-8],\"denominator\":\"8\",\"operation\":\"compare_direction\",\"weights\":[[-11,\"1\"],[-10,\"2\"],[-9,\"3\"],[-8,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0078","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-10, -9, -8, -7}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/12, 3/12, 4/12, 3/12]，總和恰為一。A={-9, -8}，B={-9, -8} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=1/1；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047078,"parameters":{"S":[-10,-9,-8,-7],"F":"powerset(S)","weights":[2,3,4,3],"denominator":12,"B":[-9,-8],"operation":"compare_direction","A":[-9,-8],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-9,-8],\"B\":[-9,-8],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8,-7],\"denominator\":\"12\",\"operation\":\"compare_direction\",\"weights\":[[-10,\"2\"],[-9,\"3\"],[-8,\"4\"],[-7,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0079","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-9, -8, -7, -6}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/13, 4/13, 5/13, 4/13]，總和恰為一。A={-9, -8, -7, -6}，B={-8, -7} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=1/1；P(B|A)=9/13。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047079,"parameters":{"S":[-9,-8,-7,-6],"F":"powerset(S)","weights":[0,4,5,4],"denominator":13,"B":[-8,-7],"operation":"compare_direction","A":[-9,-8,-7,-6],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-9,-8,-7,-6],\"B\":[-8,-7],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7,-6],\"denominator\":\"13\",\"operation\":\"compare_direction\",\"weights\":[[-9,\"0\"],[-8,\"4\"],[-7,\"5\"],[-6,\"4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0080","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-8, -7, -6, -5}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/13, 1/13, 6/13, 5/13]，總和恰為一。A={-7, -5}，B={-8, -7, -6, -5} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=6/13；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047080,"parameters":{"S":[-8,-7,-6,-5],"F":"powerset(S)","weights":[1,1,6,5],"denominator":13,"B":[-8,-7,-6,-5],"operation":"compare_direction","A":[-7,-5],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-7,-5],\"B\":[-8,-7,-6,-5],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6,-5],\"denominator\":\"13\",\"operation\":\"compare_direction\",\"weights\":[[-8,\"1\"],[-7,\"1\"],[-6,\"6\"],[-5,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0081","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-7, -6, -5, -4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/12, 2/12, 2/12, 6/12]，總和恰為一。A={-6}，B={-5} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=0/1；P(B|A)=0/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047081,"parameters":{"S":[-7,-6,-5,-4],"F":"powerset(S)","weights":[2,2,2,6],"denominator":12,"B":[-5],"operation":"compare_direction","A":[-6],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-6],\"B\":[-5],\"F\":\"powerset(S)\",\"S\":[-7,-6,-5,-4],\"denominator\":\"6\",\"operation\":\"compare_direction\",\"weights\":[[-7,\"1\"],[-6,\"1\"],[-5,\"1\"],[-4,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0082","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-6, -5, -4, -3}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/13, 3/13, 3/13, 7/13]，總和恰為一。A={-5, -4}，B={-4, -3} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=3/10；P(B|A)=1/2。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047082,"parameters":{"S":[-6,-5,-4,-3],"F":"powerset(S)","weights":[0,3,3,7],"denominator":13,"B":[-4,-3],"operation":"compare_direction","A":[-5,-4],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-5,-4],\"B\":[-4,-3],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4,-3],\"denominator\":\"13\",\"operation\":\"compare_direction\",\"weights\":[[-6,\"0\"],[-5,\"3\"],[-4,\"3\"],[-3,\"7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0083","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-5, -4, -3, -2}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/17, 4/17, 4/17, 8/17]，總和恰為一。A={-4, -3}，B={-4, -3} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=1/1；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047083,"parameters":{"S":[-5,-4,-3,-2],"F":"powerset(S)","weights":[1,4,4,8],"denominator":17,"B":[-4,-3],"operation":"compare_direction","A":[-4,-3],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-4,-3],\"B\":[-4,-3],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3,-2],\"denominator\":\"17\",\"operation\":\"compare_direction\",\"weights\":[[-5,\"1\"],[-4,\"4\"],[-3,\"4\"],[-2,\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0084","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-4, -3, -2, -1}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/17, 1/17, 5/17, 9/17]，總和恰為一。A={-4, -3, -2, -1}，B={-3, -2} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=1/1；P(B|A)=6/17。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047084,"parameters":{"S":[-4,-3,-2,-1],"F":"powerset(S)","weights":[2,1,5,9],"denominator":17,"B":[-3,-2],"operation":"compare_direction","A":[-4,-3,-2,-1],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-4,-3,-2,-1],\"B\":[-3,-2],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2,-1],\"denominator\":\"17\",\"operation\":\"compare_direction\",\"weights\":[[-4,\"2\"],[-3,\"1\"],[-2,\"5\"],[-1,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0085","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-3, -2, -1, 0}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/18, 2/18, 6/18, 10/18]，總和恰為一。A={-2, 0}，B={-3, -2, -1, 0} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=2/3；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047085,"parameters":{"S":[-3,-2,-1,0],"F":"powerset(S)","weights":[0,2,6,10],"denominator":18,"B":[-3,-2,-1,0],"operation":"compare_direction","A":[-2,0],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-2,0],\"B\":[-3,-2,-1,0],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1,0],\"denominator\":\"9\",\"operation\":\"compare_direction\",\"weights\":[[-3,\"0\"],[-2,\"1\"],[-1,\"3\"],[0,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0086","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-2, -1, 0, 1}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/17, 3/17, 2/17, 11/17]，總和恰為一。A={-1}，B={0} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=0/1；P(B|A)=0/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047086,"parameters":{"S":[-2,-1,0,1],"F":"powerset(S)","weights":[1,3,2,11],"denominator":17,"B":[0],"operation":"compare_direction","A":[-1],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[-1],\"B\":[0],\"F\":\"powerset(S)\",\"S\":[-2,-1,0,1],\"denominator\":\"17\",\"operation\":\"compare_direction\",\"weights\":[[-2,\"1\"],[-1,\"3\"],[0,\"2\"],[1,\"11\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0087","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={-1, 0, 1, 2}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/21, 4/21, 3/21, 12/21]，總和恰為一。A={0, 1}，B={1, 2} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=1/5；P(B|A)=3/7。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047087,"parameters":{"S":[-1,0,1,2],"F":"powerset(S)","weights":[2,4,3,12],"denominator":21,"B":[1,2],"operation":"compare_direction","A":[0,1],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[0,1],\"B\":[1,2],\"F\":\"powerset(S)\",\"S\":[-1,0,1,2],\"denominator\":\"21\",\"operation\":\"compare_direction\",\"weights\":[[-1,\"2\"],[0,\"4\"],[1,\"3\"],[2,\"12\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0088","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={0, 1, 2, 3}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/18, 1/18, 4/18, 13/18]，總和恰為一。A={1, 2}，B={1, 2} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=1/1；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047088,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","weights":[0,1,4,13],"denominator":18,"B":[1,2],"operation":"compare_direction","A":[1,2],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[1,2],\"B\":[1,2],\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"denominator\":\"18\",\"operation\":\"compare_direction\",\"weights\":[[0,\"0\"],[1,\"1\"],[2,\"4\"],[3,\"13\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0089","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={1, 2, 3, 4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 2/22, 5/22, 14/22]，總和恰為一。A={1, 2, 3, 4}，B={2, 3} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=1/1；P(B|A)=7/22。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047089,"parameters":{"S":[1,2,3,4],"F":"powerset(S)","weights":[1,2,5,14],"denominator":22,"B":[2,3],"operation":"compare_direction","A":[1,2,3,4],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[1,2,3,4],\"B\":[2,3],\"F\":\"powerset(S)\",\"S\":[1,2,3,4],\"denominator\":\"22\",\"operation\":\"compare_direction\",\"weights\":[[1,\"1\"],[2,\"2\"],[3,\"5\"],[4,\"14\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0090","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={2, 3, 4, 5}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 3/26, 6/26, 15/26]，總和恰為一。A={3, 5}，B={2, 3, 4, 5} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=9/13；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047090,"parameters":{"S":[2,3,4,5],"F":"powerset(S)","weights":[2,3,6,15],"denominator":26,"B":[2,3,4,5],"operation":"compare_direction","A":[3,5],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[3,5],\"B\":[2,3,4,5],\"F\":\"powerset(S)\",\"S\":[2,3,4,5],\"denominator\":\"26\",\"operation\":\"compare_direction\",\"weights\":[[2,\"2\"],[3,\"3\"],[4,\"6\"],[5,\"15\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0091","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={3, 4, 5, 6}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/22, 4/22, 2/22, 16/22]，總和恰為一。A={4}，B={5} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=0/1；P(B|A)=0/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047091,"parameters":{"S":[3,4,5,6],"F":"powerset(S)","weights":[0,4,2,16],"denominator":22,"B":[5],"operation":"compare_direction","A":[4],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[4],\"B\":[5],\"F\":\"powerset(S)\",\"S\":[3,4,5,6],\"denominator\":\"11\",\"operation\":\"compare_direction\",\"weights\":[[3,\"0\"],[4,\"2\"],[5,\"1\"],[6,\"8\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0092","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={4, 5, 6, 7}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 1/22, 3/22, 17/22]，總和恰為一。A={5, 6}，B={6, 7} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=3/20；P(B|A)=3/4。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047092,"parameters":{"S":[4,5,6,7],"F":"powerset(S)","weights":[1,1,3,17],"denominator":22,"B":[6,7],"operation":"compare_direction","A":[5,6],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[5,6],\"B\":[6,7],\"F\":\"powerset(S)\",\"S\":[4,5,6,7],\"denominator\":\"22\",\"operation\":\"compare_direction\",\"weights\":[[4,\"1\"],[5,\"1\"],[6,\"3\"],[7,\"17\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0093","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={5, 6, 7, 8}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 2/26, 4/26, 18/26]，總和恰為一。A={6, 7}，B={6, 7} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=1/1；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047093,"parameters":{"S":[5,6,7,8],"F":"powerset(S)","weights":[2,2,4,18],"denominator":26,"B":[6,7],"operation":"compare_direction","A":[6,7],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[6,7],\"B\":[6,7],\"F\":\"powerset(S)\",\"S\":[5,6,7,8],\"denominator\":\"13\",\"operation\":\"compare_direction\",\"weights\":[[5,\"1\"],[6,\"1\"],[7,\"2\"],[8,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0094","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={6, 7, 8, 9}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/27, 3/27, 5/27, 19/27]，總和恰為一。A={6, 7, 8, 9}，B={7, 8} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=1/1；P(B|A)=8/27。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047094,"parameters":{"S":[6,7,8,9],"F":"powerset(S)","weights":[0,3,5,19],"denominator":27,"B":[7,8],"operation":"compare_direction","A":[6,7,8,9],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[6,7,8,9],\"B\":[7,8],\"F\":\"powerset(S)\",\"S\":[6,7,8,9],\"denominator\":\"27\",\"operation\":\"compare_direction\",\"weights\":[[6,\"0\"],[7,\"3\"],[8,\"5\"],[9,\"19\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0095","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={7, 8, 9, 10}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/31, 4/31, 6/31, 20/31]，總和恰為一。A={8, 10}，B={7, 8, 9, 10} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=24/31；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047095,"parameters":{"S":[7,8,9,10],"F":"powerset(S)","weights":[1,4,6,20],"denominator":31,"B":[7,8,9,10],"operation":"compare_direction","A":[8,10],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[8,10],\"B\":[7,8,9,10],\"F\":\"powerset(S)\",\"S\":[7,8,9,10],\"denominator\":\"31\",\"operation\":\"compare_direction\",\"weights\":[[7,\"1\"],[8,\"4\"],[9,\"6\"],[10,\"20\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0096","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={8, 9, 10, 11}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/26, 1/26, 2/26, 21/26]，總和恰為一。A={9}，B={10} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=0/1；P(B|A)=0/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047096,"parameters":{"S":[8,9,10,11],"F":"powerset(S)","weights":[2,1,2,21],"denominator":26,"B":[10],"operation":"compare_direction","A":[9],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[9],\"B\":[10],\"F\":\"powerset(S)\",\"S\":[8,9,10,11],\"denominator\":\"26\",\"operation\":\"compare_direction\",\"weights\":[[8,\"2\"],[9,\"1\"],[10,\"2\"],[11,\"21\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0097","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={9, 10, 11, 12}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/27, 2/27, 3/27, 22/27]，總和恰為一。A={10, 11}，B={11, 12} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=3/25；P(B|A)=3/5。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047097,"parameters":{"S":[9,10,11,12],"F":"powerset(S)","weights":[0,2,3,22],"denominator":27,"B":[11,12],"operation":"compare_direction","A":[10,11],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[10,11],\"B\":[11,12],\"F\":\"powerset(S)\",\"S\":[9,10,11,12],\"denominator\":\"27\",\"operation\":\"compare_direction\",\"weights\":[[9,\"0\"],[10,\"2\"],[11,\"3\"],[12,\"22\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0098","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={10, 11, 12, 13}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/31, 3/31, 4/31, 23/31]，總和恰為一。A={11, 12}，B={11, 12} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"是","explanation_zh":"P(A|B)=1/1；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047098,"parameters":{"S":[10,11,12,13],"F":"powerset(S)","weights":[1,3,4,23],"denominator":31,"B":[11,12],"operation":"compare_direction","A":[11,12],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[11,12],\"B\":[11,12],\"F\":\"powerset(S)\",\"S\":[10,11,12,13],\"denominator\":\"31\",\"operation\":\"compare_direction\",\"weights\":[[10,\"1\"],[11,\"3\"],[12,\"4\"],[13,\"23\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0099","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={11, 12, 13, 14}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [2/35, 4/35, 5/35, 24/35]，總和恰為一。A={11, 12, 13, 14}，B={12, 13} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=1/1；P(B|A)=9/35。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047099,"parameters":{"S":[11,12,13,14],"F":"powerset(S)","weights":[2,4,5,24],"denominator":35,"B":[12,13],"operation":"compare_direction","A":[11,12,13,14],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[11,12,13,14],\"B\":[12,13],\"F\":\"powerset(S)\",\"S\":[11,12,13,14],\"denominator\":\"35\",\"operation\":\"compare_direction\",\"weights\":[[11,\"2\"],[12,\"4\"],[13,\"5\"],[14,\"24\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r07-conditional-probability-0100","batch_id":"batch-20261003-advance-20261004-r07-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"兩方向條件機率的數值比較","stem_zh":"明示非空 S={12, 13, 14, 15}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [0/32, 1/32, 6/32, 25/32]，總和恰為一。A={13, 15}，B={12, 13, 14, 15} 且 P(A)、P(B)>0。P(A|B) 與 P(B|A) 是否相等？","answer":{"kind":"short_text","value":"否","explanation_zh":"P(A|B)=13/16；P(B|A)=1/1。各自使用其正機率條件分母；兩方向數值比較不代表獨立性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["兩方向條件機率的數值比較"]},"variant":{"family_id":"qf-conditional-direction-comparison","seed":202610047100,"parameters":{"S":[12,13,14,15],"F":"powerset(S)","weights":[0,1,6,25],"denominator":32,"B":[12,13,14,15],"operation":"compare_direction","A":[13,15],"semantic_template":"qf-conditional-direction-comparison/{\"A\":[13,15],\"B\":[12,13,14,15],\"F\":\"powerset(S)\",\"S\":[12,13,14,15],\"denominator\":\"32\",\"operation\":\"compare_direction\",\"weights\":[[12,\"0\"],[13,\"1\"],[14,\"6\"],[15,\"25\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}