{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0001","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[0, 1, 2]、D=3，權重為質量/D；B=[1]，P(B)>0，A=[]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(0/1, 1/3)","explanation_zh":"交集與B概率=(0/1, 1/3)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104001,"parameters":{"masses":[0,1,2],"D":3,"A":[],"B":[1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"0/1\",\"1/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0002","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[1, 0, 0]、D=1，權重為質量/D；B=[0, 1]，P(B)>0，A=[0]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation_zh":"交集與B概率=(1/1, 1/1)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104002,"parameters":{"masses":[1,0,0],"D":1,"A":[0],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"1/1\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0003","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[2, 1, 2, 0]、D=5，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(2/5, 3/5)","explanation_zh":"交集與B概率=(2/5, 3/5)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104003,"parameters":{"masses":[2,1,2,0],"D":5,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"2/5\",\"3/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0004","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[3, 1, 2, 0]、D=6，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(1/2, 2/3)","explanation_zh":"交集與B概率=(1/2, 2/3)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104004,"parameters":{"masses":[3,1,2,0],"D":6,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"1/2\",\"2/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0005","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[4, 1, 2, 0]、D=7，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(4/7, 5/7)","explanation_zh":"交集與B概率=(4/7, 5/7)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104005,"parameters":{"masses":[4,1,2,0],"D":7,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"4/7\",\"5/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0006","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[5, 1, 2, 0]、D=8，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(5/8, 3/4)","explanation_zh":"交集與B概率=(5/8, 3/4)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104006,"parameters":{"masses":[5,1,2,0],"D":8,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"5/8\",\"3/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0007","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[6, 1, 2, 0]、D=9，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(2/3, 7/9)","explanation_zh":"交集與B概率=(2/3, 7/9)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104007,"parameters":{"masses":[6,1,2,0],"D":9,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"2/3\",\"7/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0008","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[7, 1, 2, 0]、D=10，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(7/10, 4/5)","explanation_zh":"交集與B概率=(7/10, 4/5)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104008,"parameters":{"masses":[7,1,2,0],"D":10,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"7/10\",\"4/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0009","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[8, 1, 2, 0]、D=11，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(8/11, 9/11)","explanation_zh":"交集與B概率=(8/11, 9/11)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104009,"parameters":{"masses":[8,1,2,0],"D":11,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"8/11\",\"9/11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0010","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[9, 1, 2, 0]、D=12，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(3/4, 5/6)","explanation_zh":"交集與B概率=(3/4, 5/6)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104010,"parameters":{"masses":[9,1,2,0],"D":12,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"3/4\",\"5/6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0011","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[10, 1, 2, 0]、D=13，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(10/13, 11/13)","explanation_zh":"交集與B概率=(10/13, 11/13)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104011,"parameters":{"masses":[10,1,2,0],"D":13,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"10/13\",\"11/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0012","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[11, 1, 2, 0]、D=14，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(11/14, 6/7)","explanation_zh":"交集與B概率=(11/14, 6/7)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104012,"parameters":{"masses":[11,1,2,0],"D":14,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"11/14\",\"6/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0013","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[12, 1, 2, 0]、D=15，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(4/5, 13/15)","explanation_zh":"交集與B概率=(4/5, 13/15)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104013,"parameters":{"masses":[12,1,2,0],"D":15,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"4/5\",\"13/15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0014","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[13, 1, 2, 0]、D=16，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(13/16, 7/8)","explanation_zh":"交集與B概率=(13/16, 7/8)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104014,"parameters":{"masses":[13,1,2,0],"D":16,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"13/16\",\"7/8\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0015","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[14, 1, 2, 0]、D=17，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(14/17, 15/17)","explanation_zh":"交集與B概率=(14/17, 15/17)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104015,"parameters":{"masses":[14,1,2,0],"D":17,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"14/17\",\"15/17\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0016","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[15, 1, 2, 0]、D=18，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(5/6, 8/9)","explanation_zh":"交集與B概率=(5/6, 8/9)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104016,"parameters":{"masses":[15,1,2,0],"D":18,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"5/6\",\"8/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0017","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[16, 1, 2, 0]、D=19，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(16/19, 17/19)","explanation_zh":"交集與B概率=(16/19, 17/19)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104017,"parameters":{"masses":[16,1,2,0],"D":19,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"16/19\",\"17/19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0018","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[17, 1, 2, 0]、D=20，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(17/20, 9/10)","explanation_zh":"交集與B概率=(17/20, 9/10)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104018,"parameters":{"masses":[17,1,2,0],"D":20,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"17/20\",\"9/10\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0019","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[18, 1, 2, 0]、D=21，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(6/7, 19/21)","explanation_zh":"交集與B概率=(6/7, 19/21)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104019,"parameters":{"masses":[18,1,2,0],"D":21,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"6/7\",\"19/21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0020","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[19, 1, 2, 0]、D=22，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(19/22, 10/11)","explanation_zh":"交集與B概率=(19/22, 10/11)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104020,"parameters":{"masses":[19,1,2,0],"D":22,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"19/22\",\"10/11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0021","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[20, 1, 2, 0]、D=23，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(20/23, 21/23)","explanation_zh":"交集與B概率=(20/23, 21/23)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104021,"parameters":{"masses":[20,1,2,0],"D":23,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"20/23\",\"21/23\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0022","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[21, 1, 2, 0]、D=24，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(7/8, 11/12)","explanation_zh":"交集與B概率=(7/8, 11/12)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104022,"parameters":{"masses":[21,1,2,0],"D":24,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"7/8\",\"11/12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0023","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[22, 1, 2, 0]、D=25，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(22/25, 23/25)","explanation_zh":"交集與B概率=(22/25, 23/25)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104023,"parameters":{"masses":[22,1,2,0],"D":25,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"22/25\",\"23/25\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0024","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[23, 1, 2, 0]、D=26，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(23/26, 12/13)","explanation_zh":"交集與B概率=(23/26, 12/13)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104024,"parameters":{"masses":[23,1,2,0],"D":26,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"23/26\",\"12/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0025","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"交集與條件事件的概率對","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[24, 1, 2, 0]、D=27，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。依序求 (P(A∩B), P(B))，各以最簡正分母分數表示。","answer":{"kind":"expression","value":"(8/9, 25/27)","explanation_zh":"交集與B概率=(8/9, 25/27)。枚舉交集與B質量後各除D；本題尚未相除，不把B非空代替正概率。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["交集與條件事件的概率對"]},"variant":{"family_id":"qf-density10-conditional-masses-pair","seed":202610104025,"parameters":{"masses":[24,1,2,0],"D":27,"A":[0,2],"B":[0,1],"operation":"density10_conditional_masses_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-masses-pair/[\"8/9\",\"25/27\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0026","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 2]、D=3，權重為質量/D；B=[1]，P(B)>0，A=[0]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation_zh":"雙向條件值=(0/1, 0/1)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104026,"parameters":{"masses":[1,2],"D":3,"A":[0],"B":[1],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"disjoint_positive\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0027","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[1, 0, 2]、D=3，權重為質量/D；B=[0, 1]，P(B)>0，A=[0]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation_zh":"雙向條件值=(1/1, 1/1)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104027,"parameters":{"masses":[1,0,2],"D":3,"A":[0],"B":[0,1],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"same_positive_event\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0028","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[2, 1, 2]、D=5，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(1/2, 2/3)","explanation_zh":"雙向條件值=(1/2, 2/3)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104028,"parameters":{"masses":[2,1,2],"D":5,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"1/2\",\"2/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0029","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[3, 1, 2]、D=6，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(3/5, 3/4)","explanation_zh":"雙向條件值=(3/5, 3/4)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104029,"parameters":{"masses":[3,1,2],"D":6,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"3/5\",\"3/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0030","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[4, 1, 2]、D=7，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(2/3, 4/5)","explanation_zh":"雙向條件值=(2/3, 4/5)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104030,"parameters":{"masses":[4,1,2],"D":7,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"2/3\",\"4/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0031","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[5, 1, 2]、D=8，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(5/7, 5/6)","explanation_zh":"雙向條件值=(5/7, 5/6)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104031,"parameters":{"masses":[5,1,2],"D":8,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"5/7\",\"5/6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0032","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[6, 1, 2]、D=9，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(3/4, 6/7)","explanation_zh":"雙向條件值=(3/4, 6/7)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104032,"parameters":{"masses":[6,1,2],"D":9,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"3/4\",\"6/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0033","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[7, 1, 2]、D=10，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(7/9, 7/8)","explanation_zh":"雙向條件值=(7/9, 7/8)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104033,"parameters":{"masses":[7,1,2],"D":10,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"7/9\",\"7/8\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0034","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[8, 1, 2]、D=11，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(4/5, 8/9)","explanation_zh":"雙向條件值=(4/5, 8/9)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104034,"parameters":{"masses":[8,1,2],"D":11,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"4/5\",\"8/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0035","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[9, 1, 2]、D=12，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(9/11, 9/10)","explanation_zh":"雙向條件值=(9/11, 9/10)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104035,"parameters":{"masses":[9,1,2],"D":12,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"9/11\",\"9/10\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0036","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[10, 1, 2]、D=13，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(5/6, 10/11)","explanation_zh":"雙向條件值=(5/6, 10/11)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104036,"parameters":{"masses":[10,1,2],"D":13,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"5/6\",\"10/11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0037","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[11, 1, 2]、D=14，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(11/13, 11/12)","explanation_zh":"雙向條件值=(11/13, 11/12)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104037,"parameters":{"masses":[11,1,2],"D":14,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"11/13\",\"11/12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0038","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 1, 2]、D=15，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(6/7, 12/13)","explanation_zh":"雙向條件值=(6/7, 12/13)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104038,"parameters":{"masses":[12,1,2],"D":15,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"6/7\",\"12/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0039","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[13, 1, 2]、D=16，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(13/15, 13/14)","explanation_zh":"雙向條件值=(13/15, 13/14)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104039,"parameters":{"masses":[13,1,2],"D":16,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"13/15\",\"13/14\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0040","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[14, 1, 2]、D=17，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(7/8, 14/15)","explanation_zh":"雙向條件值=(7/8, 14/15)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104040,"parameters":{"masses":[14,1,2],"D":17,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"7/8\",\"14/15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0041","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[15, 1, 2]、D=18，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(15/17, 15/16)","explanation_zh":"雙向條件值=(15/17, 15/16)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104041,"parameters":{"masses":[15,1,2],"D":18,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"15/17\",\"15/16\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0042","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[16, 1, 2]、D=19，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(8/9, 16/17)","explanation_zh":"雙向條件值=(8/9, 16/17)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104042,"parameters":{"masses":[16,1,2],"D":19,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"8/9\",\"16/17\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0043","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[17, 1, 2]、D=20，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(17/19, 17/18)","explanation_zh":"雙向條件值=(17/19, 17/18)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104043,"parameters":{"masses":[17,1,2],"D":20,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"17/19\",\"17/18\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0044","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[18, 1, 2]、D=21，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(9/10, 18/19)","explanation_zh":"雙向條件值=(9/10, 18/19)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104044,"parameters":{"masses":[18,1,2],"D":21,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"9/10\",\"18/19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0045","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[19, 1, 2]、D=22，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(19/21, 19/20)","explanation_zh":"雙向條件值=(19/21, 19/20)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104045,"parameters":{"masses":[19,1,2],"D":22,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"19/21\",\"19/20\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0046","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[20, 1, 2]、D=23，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(10/11, 20/21)","explanation_zh":"雙向條件值=(10/11, 20/21)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104046,"parameters":{"masses":[20,1,2],"D":23,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"10/11\",\"20/21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0047","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 1, 2]、D=24，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(21/23, 21/22)","explanation_zh":"雙向條件值=(21/23, 21/22)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104047,"parameters":{"masses":[21,1,2],"D":24,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"21/23\",\"21/22\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0048","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[22, 1, 2]、D=25，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(11/12, 22/23)","explanation_zh":"雙向條件值=(11/12, 22/23)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104048,"parameters":{"masses":[22,1,2],"D":25,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"11/12\",\"22/23\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0049","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[23, 1, 2]、D=26，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(23/25, 23/24)","explanation_zh":"雙向條件值=(23/25, 23/24)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104049,"parameters":{"masses":[23,1,2],"D":26,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"23/25\",\"23/24\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0050","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"同一模型的雙向條件值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[24, 1, 2]、D=27，權重為質量/D；B=[0, 2]，P(B)>0，A=[0, 1]。另有P(A)>0。依序求 (P(A|B), P(B|A))，不要交換分母。","answer":{"kind":"expression","value":"(12/13, 24/25)","explanation_zh":"雙向條件值=(12/13, 24/25)。共同交集質量分別除B與A質量；雙向通常不同，沒有假設獨立。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一模型的雙向條件值"]},"variant":{"family_id":"qf-density10-conditional-directions-pair","seed":202610104050,"parameters":{"masses":[24,1,2],"D":27,"A":[0,1],"B":[0,2],"operation":"density10_conditional_directions_pair","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-directions-pair/[\"12/13\",\"24/25\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0051","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[0, 1, 2]、D=3，權重為質量/D；B=[1]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[0/1, 1/1, 0/1]","explanation_zh":"逐點條件權重=[0/1, 1/1, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104051,"parameters":{"masses":[0,1,2],"D":3,"B":[1],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"0\",\"1\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0052","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,0}，F=𝒫(S)，質量=[1]、D=1，權重為質量/D；B=[0]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[1/1]","explanation_zh":"逐點條件權重=[1/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104052,"parameters":{"masses":[1],"D":1,"B":[0],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0053","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[2, 1, 2, 3]、D=8，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[2/5, 1/5, 2/5, 0/1]","explanation_zh":"逐點條件權重=[2/5, 1/5, 2/5, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104053,"parameters":{"masses":[2,1,2,3],"D":8,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"2\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0054","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[3, 1, 2, 3]、D=9，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[1/2, 1/6, 1/3, 0/1]","explanation_zh":"逐點條件權重=[1/2, 1/6, 1/3, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104054,"parameters":{"masses":[3,1,2,3],"D":9,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"3\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0055","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[4, 1, 2, 3]、D=10，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[4/7, 1/7, 2/7, 0/1]","explanation_zh":"逐點條件權重=[4/7, 1/7, 2/7, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104055,"parameters":{"masses":[4,1,2,3],"D":10,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"4\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0056","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[5, 1, 2, 3]、D=11，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[5/8, 1/8, 1/4, 0/1]","explanation_zh":"逐點條件權重=[5/8, 1/8, 1/4, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104056,"parameters":{"masses":[5,1,2,3],"D":11,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"5\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0057","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[6, 1, 2, 3]、D=12，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[2/3, 1/9, 2/9, 0/1]","explanation_zh":"逐點條件權重=[2/3, 1/9, 2/9, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104057,"parameters":{"masses":[6,1,2,3],"D":12,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"6\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0058","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[7, 1, 2, 3]、D=13，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[7/10, 1/10, 1/5, 0/1]","explanation_zh":"逐點條件權重=[7/10, 1/10, 1/5, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104058,"parameters":{"masses":[7,1,2,3],"D":13,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"7\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0059","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[8, 1, 2, 3]、D=14，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[8/11, 1/11, 2/11, 0/1]","explanation_zh":"逐點條件權重=[8/11, 1/11, 2/11, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104059,"parameters":{"masses":[8,1,2,3],"D":14,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"8\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0060","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[9, 1, 2, 3]、D=15，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[3/4, 1/12, 1/6, 0/1]","explanation_zh":"逐點條件權重=[3/4, 1/12, 1/6, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104060,"parameters":{"masses":[9,1,2,3],"D":15,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"9\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0061","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[10, 1, 2, 3]、D=16，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[10/13, 1/13, 2/13, 0/1]","explanation_zh":"逐點條件權重=[10/13, 1/13, 2/13, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104061,"parameters":{"masses":[10,1,2,3],"D":16,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"10\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0062","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[11, 1, 2, 3]、D=17，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[11/14, 1/14, 1/7, 0/1]","explanation_zh":"逐點條件權重=[11/14, 1/14, 1/7, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104062,"parameters":{"masses":[11,1,2,3],"D":17,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"11\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0063","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[12, 1, 2, 3]、D=18，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[4/5, 1/15, 2/15, 0/1]","explanation_zh":"逐點條件權重=[4/5, 1/15, 2/15, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104063,"parameters":{"masses":[12,1,2,3],"D":18,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"12\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0064","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[13, 1, 2, 3]、D=19，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[13/16, 1/16, 1/8, 0/1]","explanation_zh":"逐點條件權重=[13/16, 1/16, 1/8, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104064,"parameters":{"masses":[13,1,2,3],"D":19,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"13\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0065","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[14, 1, 2, 3]、D=20，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[14/17, 1/17, 2/17, 0/1]","explanation_zh":"逐點條件權重=[14/17, 1/17, 2/17, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104065,"parameters":{"masses":[14,1,2,3],"D":20,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"14\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0066","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[15, 1, 2, 3]、D=21，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[5/6, 1/18, 1/9, 0/1]","explanation_zh":"逐點條件權重=[5/6, 1/18, 1/9, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104066,"parameters":{"masses":[15,1,2,3],"D":21,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"15\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0067","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[16, 1, 2, 3]、D=22，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[16/19, 1/19, 2/19, 0/1]","explanation_zh":"逐點條件權重=[16/19, 1/19, 2/19, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104067,"parameters":{"masses":[16,1,2,3],"D":22,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"16\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0068","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[17, 1, 2, 3]、D=23，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[17/20, 1/20, 1/10, 0/1]","explanation_zh":"逐點條件權重=[17/20, 1/20, 1/10, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104068,"parameters":{"masses":[17,1,2,3],"D":23,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"17\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0069","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[18, 1, 2, 3]、D=24，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[6/7, 1/21, 2/21, 0/1]","explanation_zh":"逐點條件權重=[6/7, 1/21, 2/21, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104069,"parameters":{"masses":[18,1,2,3],"D":24,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"18\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0070","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[19, 1, 2, 3]、D=25，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[19/22, 1/22, 1/11, 0/1]","explanation_zh":"逐點條件權重=[19/22, 1/22, 1/11, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104070,"parameters":{"masses":[19,1,2,3],"D":25,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"19\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0071","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[20, 1, 2, 3]、D=26，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[20/23, 1/23, 2/23, 0/1]","explanation_zh":"逐點條件權重=[20/23, 1/23, 2/23, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104071,"parameters":{"masses":[20,1,2,3],"D":26,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"20\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0072","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[21, 1, 2, 3]、D=27，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[7/8, 1/24, 1/12, 0/1]","explanation_zh":"逐點條件權重=[7/8, 1/24, 1/12, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104072,"parameters":{"masses":[21,1,2,3],"D":27,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"21\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0073","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[22, 1, 2, 3]、D=28，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[22/25, 1/25, 2/25, 0/1]","explanation_zh":"逐點條件權重=[22/25, 1/25, 2/25, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104073,"parameters":{"masses":[22,1,2,3],"D":28,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"22\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0074","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[23, 1, 2, 3]、D=29，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[23/26, 1/26, 1/13, 0/1]","explanation_zh":"逐點條件權重=[23/26, 1/26, 1/13, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104074,"parameters":{"masses":[23,1,2,3],"D":29,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"23\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0075","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"全部基本結果的條件權重","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[24, 1, 2, 3]、D=30，權重為質量/D；B=[0, 1, 2]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。","answer":{"kind":"expression","value":"[8/9, 1/27, 2/27, 0/1]","explanation_zh":"逐點條件權重=[8/9, 1/27, 2/27, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全部基本結果的條件權重"]},"variant":{"family_id":"qf-density10-conditional-point-law","seed":202610104075,"parameters":{"masses":[24,1,2,3],"D":30,"B":[0,1,2],"operation":"density10_conditional_point_law","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-point-law/[\"24\",\"1\",\"2\",\"0\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0076","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 1]、D=2，權重為質量/D；B=[0, 1]，P(B)>0，A=[0]。待核候選=1/2。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104076,"parameters":{"masses":[1,1],"D":2,"A":[0],"B":[0,1],"candidate_n":1,"candidate_d":2,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"1/2\",\"1/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0077","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 1]、D=2，權重為質量/D；B=[0, 1]，P(B)>0，A=[0]。待核候選=-1/1。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -3/2)","explanation_zh":"候選資格與有號缺陷=(否, -3/2)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104077,"parameters":{"masses":[1,1],"D":2,"A":[0],"B":[0,1],"candidate_n":-1,"candidate_d":1,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"1/2\",\"-1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0078","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 1]、D=2，權重為質量/D；B=[0, 1]，P(B)>0，A=[0]。待核候選=2/1。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 3/2)","explanation_zh":"候選資格與有號缺陷=(否, 3/2)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104078,"parameters":{"masses":[1,1],"D":2,"A":[0],"B":[0,1],"candidate_n":2,"candidate_d":1,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"1/2\",\"2/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0079","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[3, 1, 2, 0]、D=6，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=3/4。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104079,"parameters":{"masses":[3,1,2,0],"D":6,"A":[0,2],"B":[0,1],"candidate_n":3,"candidate_d":4,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"3/4\",\"3/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0080","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[4, 1, 2, 0]、D=7，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=5/5。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/5)","explanation_zh":"候選資格與有號缺陷=(否, 1/5)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104080,"parameters":{"masses":[4,1,2,0],"D":7,"A":[0,2],"B":[0,1],"candidate_n":5,"candidate_d":5,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"4/5\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0081","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[5, 1, 2, 0]、D=8，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=4/6。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/6)","explanation_zh":"候選資格與有號缺陷=(否, -1/6)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104081,"parameters":{"masses":[5,1,2,0],"D":8,"A":[0,2],"B":[0,1],"candidate_n":4,"candidate_d":6,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"5/6\",\"2/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0082","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[6, 1, 2, 0]、D=9，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=6/7。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104082,"parameters":{"masses":[6,1,2,0],"D":9,"A":[0,2],"B":[0,1],"candidate_n":6,"candidate_d":7,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"6/7\",\"6/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0083","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[7, 1, 2, 0]、D=10，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=8/8。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/8)","explanation_zh":"候選資格與有號缺陷=(否, 1/8)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104083,"parameters":{"masses":[7,1,2,0],"D":10,"A":[0,2],"B":[0,1],"candidate_n":8,"candidate_d":8,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"7/8\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0084","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[8, 1, 2, 0]、D=11，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=7/9。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/9)","explanation_zh":"候選資格與有號缺陷=(否, -1/9)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104084,"parameters":{"masses":[8,1,2,0],"D":11,"A":[0,2],"B":[0,1],"candidate_n":7,"candidate_d":9,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"8/9\",\"7/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0085","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[9, 1, 2, 0]、D=12，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=9/10。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104085,"parameters":{"masses":[9,1,2,0],"D":12,"A":[0,2],"B":[0,1],"candidate_n":9,"candidate_d":10,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"9/10\",\"9/10\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0086","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[10, 1, 2, 0]、D=13，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=11/11。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/11)","explanation_zh":"候選資格與有號缺陷=(否, 1/11)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104086,"parameters":{"masses":[10,1,2,0],"D":13,"A":[0,2],"B":[0,1],"candidate_n":11,"candidate_d":11,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"10/11\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0087","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[11, 1, 2, 0]、D=14，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=10/12。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/12)","explanation_zh":"候選資格與有號缺陷=(否, -1/12)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104087,"parameters":{"masses":[11,1,2,0],"D":14,"A":[0,2],"B":[0,1],"candidate_n":10,"candidate_d":12,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"11/12\",\"5/6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0088","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[12, 1, 2, 0]、D=15，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=12/13。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104088,"parameters":{"masses":[12,1,2,0],"D":15,"A":[0,2],"B":[0,1],"candidate_n":12,"candidate_d":13,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"12/13\",\"12/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0089","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[13, 1, 2, 0]、D=16，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=14/14。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/14)","explanation_zh":"候選資格與有號缺陷=(否, 1/14)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104089,"parameters":{"masses":[13,1,2,0],"D":16,"A":[0,2],"B":[0,1],"candidate_n":14,"candidate_d":14,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"13/14\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0090","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[14, 1, 2, 0]、D=17，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=13/15。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/15)","explanation_zh":"候選資格與有號缺陷=(否, -1/15)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104090,"parameters":{"masses":[14,1,2,0],"D":17,"A":[0,2],"B":[0,1],"candidate_n":13,"candidate_d":15,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"14/15\",\"13/15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0091","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[15, 1, 2, 0]、D=18，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=15/16。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104091,"parameters":{"masses":[15,1,2,0],"D":18,"A":[0,2],"B":[0,1],"candidate_n":15,"candidate_d":16,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"15/16\",\"15/16\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0092","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[16, 1, 2, 0]、D=19，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=17/17。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/17)","explanation_zh":"候選資格與有號缺陷=(否, 1/17)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104092,"parameters":{"masses":[16,1,2,0],"D":19,"A":[0,2],"B":[0,1],"candidate_n":17,"candidate_d":17,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"16/17\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0093","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[17, 1, 2, 0]、D=20，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=16/18。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/18)","explanation_zh":"候選資格與有號缺陷=(否, -1/18)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104093,"parameters":{"masses":[17,1,2,0],"D":20,"A":[0,2],"B":[0,1],"candidate_n":16,"candidate_d":18,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"17/18\",\"8/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0094","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[18, 1, 2, 0]、D=21，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=18/19。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104094,"parameters":{"masses":[18,1,2,0],"D":21,"A":[0,2],"B":[0,1],"candidate_n":18,"candidate_d":19,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"18/19\",\"18/19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0095","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[19, 1, 2, 0]、D=22，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=20/20。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/20)","explanation_zh":"候選資格與有號缺陷=(否, 1/20)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104095,"parameters":{"masses":[19,1,2,0],"D":22,"A":[0,2],"B":[0,1],"candidate_n":20,"candidate_d":20,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"19/20\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0096","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[20, 1, 2, 0]、D=23，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=19/21。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/21)","explanation_zh":"候選資格與有號缺陷=(否, -1/21)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104096,"parameters":{"masses":[20,1,2,0],"D":23,"A":[0,2],"B":[0,1],"candidate_n":19,"candidate_d":21,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"20/21\",\"19/21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0097","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[21, 1, 2, 0]、D=24，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=21/22。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104097,"parameters":{"masses":[21,1,2,0],"D":24,"A":[0,2],"B":[0,1],"candidate_n":21,"candidate_d":22,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"21/22\",\"21/22\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0098","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[22, 1, 2, 0]、D=25，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=23/23。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, 1/23)","explanation_zh":"候選資格與有號缺陷=(否, 1/23)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104098,"parameters":{"masses":[22,1,2,0],"D":25,"A":[0,2],"B":[0,1],"candidate_n":23,"candidate_d":23,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"22/23\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0099","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[23, 1, 2, 0]、D=26，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=22/24。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(否, -1/24)","explanation_zh":"候選資格與有號缺陷=(否, -1/24)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104099,"parameters":{"masses":[23,1,2,0],"D":26,"A":[0,2],"B":[0,1],"candidate_n":22,"candidate_d":24,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"23/24\",\"11/12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r04-conditional-probability-0100","batch_id":"batch-20261008-advance-20261010-r04-conditional-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-conditional-probability"],"title_zh":"候選條件值的有號缺陷","stem_zh":"有限歸一化模型；S={0,…,3}，F=𝒫(S)，質量=[24, 1, 2, 0]、D=27，權重為質量/D；B=[0, 1]，P(B)>0，A=[0, 2]。待核候選=24/25。依序答 (候選是否等於P(A|B), 候選−P(A|B))；候選可以是待反駁的域外數值。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"候選資格與有號缺陷=(是, 0/1)。先由事件算真值再相減，缺陷零才通過；域外候選不是實際概率，不改模型合法性。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選條件值的有號缺陷"]},"variant":{"family_id":"qf-density10-conditional-candidate-defect","seed":202610104100,"parameters":{"masses":[24,1,2,0],"D":27,"A":[0,2],"B":[0,1],"candidate_n":24,"candidate_d":25,"operation":"density10_conditional_candidate_defect","domain":"finite_normalized_probability","semantic_template":"qf-density10-conditional-candidate-defect/[\"24/25\",\"24/25\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}