{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0001","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,8],[1,8],[1,8],[5,8]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/4, 1/2, 1/4, 1/8, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/4, 1/2, 1/4, 1/8, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026001,"parameters":{"masses":[[1,8],[1,8],[1,8],[5,8]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,8],[1,8],[1,8],[5,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0002","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,9],[1,9],[1,9],[6,9]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/9, 1/2, 2/9, 1/9, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/9, 1/2, 2/9, 1/9, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026002,"parameters":{"masses":[[1,9],[1,9],[1,9],[6,9]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,9],[1,9],[1,9],[6,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0003","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,10],[1,10],[1,10],[7,10]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/5, 1/2, 1/5, 1/10, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/5, 1/2, 1/5, 1/10, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026003,"parameters":{"masses":[[1,10],[1,10],[1,10],[7,10]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,10],[1,10],[1,10],[7,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0004","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,11],[1,11],[1,11],[8,11]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/11, 1/2, 2/11, 1/11, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/11, 1/2, 2/11, 1/11, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026004,"parameters":{"masses":[[1,11],[1,11],[1,11],[8,11]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,11],[1,11],[1,11],[8,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0005","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,12],[1,12],[1,12],[9,12]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/6, 1/2, 1/6, 1/12, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/6, 1/2, 1/6, 1/12, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026005,"parameters":{"masses":[[1,12],[1,12],[1,12],[9,12]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,12],[1,12],[1,12],[9,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0006","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,13],[1,13],[1,13],[10,13]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/13, 1/2, 2/13, 1/13, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/13, 1/2, 2/13, 1/13, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026006,"parameters":{"masses":[[1,13],[1,13],[1,13],[10,13]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,13],[1,13],[1,13],[10,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0007","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,14],[1,14],[1,14],[11,14]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/7, 1/2, 1/7, 1/14, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/7, 1/2, 1/7, 1/14, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026007,"parameters":{"masses":[[1,14],[1,14],[1,14],[11,14]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,14],[1,14],[1,14],[11,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0008","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,15],[1,15],[1,15],[12,15]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/15, 1/2, 2/15, 1/15, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/15, 1/2, 2/15, 1/15, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026008,"parameters":{"masses":[[1,15],[1,15],[1,15],[12,15]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,15],[1,15],[1,15],[12,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0009","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,16],[1,16],[1,16],[13,16]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/8, 1/2, 1/8, 1/16, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/8, 1/2, 1/8, 1/16, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026009,"parameters":{"masses":[[1,16],[1,16],[1,16],[13,16]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,16],[1,16],[1,16],[13,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0010","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,17],[1,17],[1,17],[14,17]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/17, 1/2, 2/17, 1/17, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/17, 1/2, 2/17, 1/17, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026010,"parameters":{"masses":[[1,17],[1,17],[1,17],[14,17]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,17],[1,17],[1,17],[14,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0011","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,18],[1,18],[1,18],[15,18]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/9, 1/2, 1/9, 1/18, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/9, 1/2, 1/9, 1/18, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026011,"parameters":{"masses":[[1,18],[1,18],[1,18],[15,18]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,18],[1,18],[1,18],[15,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0012","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,19],[1,19],[1,19],[16,19]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/19, 1/2, 2/19, 1/19, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/19, 1/2, 2/19, 1/19, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026012,"parameters":{"masses":[[1,19],[1,19],[1,19],[16,19]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,19],[1,19],[1,19],[16,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0013","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,20],[1,20],[1,20],[17,20]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/10, 1/2, 1/10, 1/20, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/10, 1/2, 1/10, 1/20, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026013,"parameters":{"masses":[[1,20],[1,20],[1,20],[17,20]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,20],[1,20],[1,20],[17,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0014","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,21],[1,21],[1,21],[18,21]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/21, 1/2, 2/21, 1/21, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/21, 1/2, 2/21, 1/21, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026014,"parameters":{"masses":[[1,21],[1,21],[1,21],[18,21]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,21],[1,21],[1,21],[18,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0015","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/11, 1/2, 1/11, 1/22, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/11, 1/2, 1/11, 1/22, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026015,"parameters":{"masses":[[1,22],[1,22],[1,22],[19,22]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,22],[1,22],[1,22],[19,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0016","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,23],[1,23],[1,23],[20,23]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/23, 1/2, 2/23, 1/23, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/23, 1/2, 2/23, 1/23, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026016,"parameters":{"masses":[[1,23],[1,23],[1,23],[20,23]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,23],[1,23],[1,23],[20,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0017","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,24],[1,24],[1,24],[21,24]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/12, 1/2, 1/12, 1/24, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/12, 1/2, 1/12, 1/24, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026017,"parameters":{"masses":[[1,24],[1,24],[1,24],[21,24]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,24],[1,24],[1,24],[21,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0018","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,25],[1,25],[1,25],[22,25]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/25, 1/2, 2/25, 1/25, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/25, 1/2, 2/25, 1/25, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026018,"parameters":{"masses":[[1,25],[1,25],[1,25],[22,25]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,25],[1,25],[1,25],[22,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0019","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,26],[1,26],[1,26],[23,26]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/13, 1/2, 1/13, 1/26, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/13, 1/2, 1/13, 1/26, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026019,"parameters":{"masses":[[1,26],[1,26],[1,26],[23,26]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,26],[1,26],[1,26],[23,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0020","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,27],[1,27],[1,27],[24,27]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/27, 1/2, 2/27, 1/27, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/27, 1/2, 2/27, 1/27, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026020,"parameters":{"masses":[[1,27],[1,27],[1,27],[24,27]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,27],[1,27],[1,27],[24,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0021","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,28],[1,28],[1,28],[25,28]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/14, 1/2, 1/14, 1/28, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/14, 1/2, 1/14, 1/28, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026021,"parameters":{"masses":[[1,28],[1,28],[1,28],[25,28]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,28],[1,28],[1,28],[25,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0022","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,29],[1,29],[1,29],[26,29]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/29, 1/2, 2/29, 1/29, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/29, 1/2, 2/29, 1/29, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026022,"parameters":{"masses":[[1,29],[1,29],[1,29],[26,29]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0023","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,30],[1,30],[1,30],[27,30]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/15, 1/2, 1/15, 1/30, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/15, 1/2, 1/15, 1/30, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026023,"parameters":{"masses":[[1,30],[1,30],[1,30],[27,30]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,30],[1,30],[1,30],[27,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0024","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,31],[1,31],[1,31],[28,31]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(2/31, 1/2, 2/31, 1/31, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(2/31, 1/2, 2/31, 1/31, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026024,"parameters":{"masses":[[1,31],[1,31],[1,31],[28,31]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,31],[1,31],[1,31],[28,31]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0025","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-two-event-reversal；資料={\"masses\":[[1,32],[1,32],[1,32],[29,32]],\"hypothesis\":[0,1],\"evidence\":[0,2]}。請依 [prior,likelihood,evidence_mass,joint,posterior,boundary] 欄序回答。","answer":{"kind":"expression","value":"(1/16, 1/2, 1/16, 1/32, 1/2, not_causal)","explanation_zh":"Wave5第二層核對=(1/16, 1/2, 1/16, 1/32, 1/2, not_causal)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-two-event-reversal","seed":202611026025,"parameters":{"masses":[[1,32],[1,32],[1,32],[29,32]],"hypothesis":[0,1],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-two-event-reversal","semantic_template":"qf-mod33-bayes-two-event-reversal/{\"evidence\":[0,2],\"hypothesis\":[0,1],\"masses\":[[1,32],[1,32],[1,32],[29,32]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0026","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,8],[1,8],[1,8],[5,8]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/4, 1/2, 1/4, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/4, 1/2, 1/4, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026026,"parameters":{"masses":[[1,8],[1,8],[1,8],[5,8]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,8],[1,8],[1,8],[5,8]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0027","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,9],[1,9],[1,9],[6,9]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 7/9, 1/7, 2/9, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 7/9, 1/7, 2/9, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026027,"parameters":{"masses":[[1,9],[1,9],[1,9],[6,9]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,9],[1,9],[1,9],[6,9]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0028","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,10],[1,10],[1,10],[7,10]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/5, 1/2, 1/5, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/5, 1/2, 1/5, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026028,"parameters":{"masses":[[1,10],[1,10],[1,10],[7,10]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,10],[1,10],[1,10],[7,10]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0029","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,11],[1,11],[1,11],[8,11]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 9/11, 1/9, 2/11, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 9/11, 1/9, 2/11, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026029,"parameters":{"masses":[[1,11],[1,11],[1,11],[8,11]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,11],[1,11],[1,11],[8,11]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0030","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,12],[1,12],[1,12],[9,12]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/6, 1/2, 1/6, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/6, 1/2, 1/6, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026030,"parameters":{"masses":[[1,12],[1,12],[1,12],[9,12]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,12],[1,12],[1,12],[9,12]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0031","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,13],[1,13],[1,13],[10,13]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 11/13, 1/11, 2/13, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 11/13, 1/11, 2/13, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026031,"parameters":{"masses":[[1,13],[1,13],[1,13],[10,13]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,13],[1,13],[1,13],[10,13]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0032","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,14],[1,14],[1,14],[11,14]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/7, 1/2, 1/7, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/7, 1/2, 1/7, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026032,"parameters":{"masses":[[1,14],[1,14],[1,14],[11,14]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,14],[1,14],[1,14],[11,14]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0033","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,15],[1,15],[1,15],[12,15]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 13/15, 1/13, 2/15, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 13/15, 1/13, 2/15, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026033,"parameters":{"masses":[[1,15],[1,15],[1,15],[12,15]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,15],[1,15],[1,15],[12,15]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0034","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,16],[1,16],[1,16],[13,16]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/8, 1/2, 1/8, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/8, 1/2, 1/8, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026034,"parameters":{"masses":[[1,16],[1,16],[1,16],[13,16]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,16],[1,16],[1,16],[13,16]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0035","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,17],[1,17],[1,17],[14,17]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 15/17, 1/15, 2/17, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 15/17, 1/15, 2/17, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026035,"parameters":{"masses":[[1,17],[1,17],[1,17],[14,17]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,17],[1,17],[1,17],[14,17]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0036","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,18],[1,18],[1,18],[15,18]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/9, 1/2, 1/9, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/9, 1/2, 1/9, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026036,"parameters":{"masses":[[1,18],[1,18],[1,18],[15,18]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,18],[1,18],[1,18],[15,18]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0037","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,19],[1,19],[1,19],[16,19]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 17/19, 1/17, 2/19, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 17/19, 1/17, 2/19, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026037,"parameters":{"masses":[[1,19],[1,19],[1,19],[16,19]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,19],[1,19],[1,19],[16,19]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0038","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,20],[1,20],[1,20],[17,20]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/10, 1/2, 1/10, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/10, 1/2, 1/10, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026038,"parameters":{"masses":[[1,20],[1,20],[1,20],[17,20]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,20],[1,20],[1,20],[17,20]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0039","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,21],[1,21],[1,21],[18,21]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026039,"parameters":{"masses":[[1,21],[1,21],[1,21],[18,21]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,21],[1,21],[1,21],[18,21]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0040","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026040,"parameters":{"masses":[[1,22],[1,22],[1,22],[19,22]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0041","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,23],[1,23],[1,23],[20,23]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 21/23, 1/21, 2/23, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 21/23, 1/21, 2/23, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026041,"parameters":{"masses":[[1,23],[1,23],[1,23],[20,23]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,23],[1,23],[1,23],[20,23]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0042","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,24],[1,24],[1,24],[21,24]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/12, 1/2, 1/12, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/12, 1/2, 1/12, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026042,"parameters":{"masses":[[1,24],[1,24],[1,24],[21,24]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,24],[1,24],[1,24],[21,24]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0043","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,25],[1,25],[1,25],[22,25]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 23/25, 1/23, 2/25, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 23/25, 1/23, 2/25, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026043,"parameters":{"masses":[[1,25],[1,25],[1,25],[22,25]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,25],[1,25],[1,25],[22,25]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0044","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,26],[1,26],[1,26],[23,26]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/13, 1/2, 1/13, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/13, 1/2, 1/13, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026044,"parameters":{"masses":[[1,26],[1,26],[1,26],[23,26]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,26],[1,26],[1,26],[23,26]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0045","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,27],[1,27],[1,27],[24,27]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 25/27, 1/25, 2/27, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 25/27, 1/25, 2/27, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026045,"parameters":{"masses":[[1,27],[1,27],[1,27],[24,27]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,27],[1,27],[1,27],[24,27]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0046","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,28],[1,28],[1,28],[25,28]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/14, 1/2, 1/14, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/14, 1/2, 1/14, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026046,"parameters":{"masses":[[1,28],[1,28],[1,28],[25,28]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,28],[1,28],[1,28],[25,28]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0047","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,29],[1,29],[1,29],[26,29]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 27/29, 1/27, 2/29, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 27/29, 1/27, 2/29, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026047,"parameters":{"masses":[[1,29],[1,29],[1,29],[26,29]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,29],[1,29],[1,29],[26,29]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0048","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,30],[1,30],[1,30],[27,30]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/15, 1/2, 1/15, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/15, 1/2, 1/15, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026048,"parameters":{"masses":[[1,30],[1,30],[1,30],[27,30]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,30],[1,30],[1,30],[27,30]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0049","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,31],[1,31],[1,31],[28,31]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":1}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(1, 29/31, 1/29, 2/31, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(1, 29/31, 1/29, 2/31, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026049,"parameters":{"masses":[[1,31],[1,31],[1,31],[28,31]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":1,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,31],[1,31],[1,31],[28,31]],\"target_index\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0050","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-partition-posterior；資料={\"masses\":[[1,32],[1,32],[1,32],[29,32]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。","answer":{"kind":"expression","value":"(0, 1/16, 1/2, 1/16, 1/2, partition_pairwise_disjoint_exhaustive)","explanation_zh":"Wave5第二層核對=(0, 1/16, 1/2, 1/16, 1/2, partition_pairwise_disjoint_exhaustive)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-partition-posterior","seed":202611026050,"parameters":{"masses":[[1,32],[1,32],[1,32],[29,32]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"target_index":0,"domain":"Exact_finite_atomic_Bayes","operation":"bayes-partition-posterior","semantic_template":"qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,32],[1,32],[1,32],[29,32]],\"target_index\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0051","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,8],[1,8],[1,8],[5,8]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/4, 3/4], [1/2, 1/6], [1/8, 1/8], 1/4, 1/4, 是)","explanation_zh":"Wave5第二層核對=([1/4, 3/4], [1/2, 1/6], [1/8, 1/8], 1/4, 1/4, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026051,"parameters":{"masses":[[1,8],[1,8],[1,8],[5,8]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,8],[1,8],[1,8],[5,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0052","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,9],[1,9],[1,9],[6,9]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/9, 7/9], [1/2, 1/7], [1/9, 1/9], 2/9, 2/9, 是)","explanation_zh":"Wave5第二層核對=([2/9, 7/9], [1/2, 1/7], [1/9, 1/9], 2/9, 2/9, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026052,"parameters":{"masses":[[1,9],[1,9],[1,9],[6,9]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,9],[1,9],[1,9],[6,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0053","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,10],[1,10],[1,10],[7,10]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/5, 4/5], [1/2, 1/8], [1/10, 1/10], 1/5, 1/5, 是)","explanation_zh":"Wave5第二層核對=([1/5, 4/5], [1/2, 1/8], [1/10, 1/10], 1/5, 1/5, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026053,"parameters":{"masses":[[1,10],[1,10],[1,10],[7,10]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,10],[1,10],[1,10],[7,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0054","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,11],[1,11],[1,11],[8,11]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/11, 9/11], [1/2, 1/9], [1/11, 1/11], 2/11, 2/11, 是)","explanation_zh":"Wave5第二層核對=([2/11, 9/11], [1/2, 1/9], [1/11, 1/11], 2/11, 2/11, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026054,"parameters":{"masses":[[1,11],[1,11],[1,11],[8,11]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,11],[1,11],[1,11],[8,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0055","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,12],[1,12],[1,12],[9,12]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/6, 5/6], [1/2, 1/10], [1/12, 1/12], 1/6, 1/6, 是)","explanation_zh":"Wave5第二層核對=([1/6, 5/6], [1/2, 1/10], [1/12, 1/12], 1/6, 1/6, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026055,"parameters":{"masses":[[1,12],[1,12],[1,12],[9,12]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,12],[1,12],[1,12],[9,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0056","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,13],[1,13],[1,13],[10,13]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/13, 11/13], [1/2, 1/11], [1/13, 1/13], 2/13, 2/13, 是)","explanation_zh":"Wave5第二層核對=([2/13, 11/13], [1/2, 1/11], [1/13, 1/13], 2/13, 2/13, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026056,"parameters":{"masses":[[1,13],[1,13],[1,13],[10,13]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,13],[1,13],[1,13],[10,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0057","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,14],[1,14],[1,14],[11,14]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/7, 6/7], [1/2, 1/12], [1/14, 1/14], 1/7, 1/7, 是)","explanation_zh":"Wave5第二層核對=([1/7, 6/7], [1/2, 1/12], [1/14, 1/14], 1/7, 1/7, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026057,"parameters":{"masses":[[1,14],[1,14],[1,14],[11,14]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,14],[1,14],[1,14],[11,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0058","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,15],[1,15],[1,15],[12,15]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, 是)","explanation_zh":"Wave5第二層核對=([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026058,"parameters":{"masses":[[1,15],[1,15],[1,15],[12,15]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,15],[1,15],[1,15],[12,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0059","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,16],[1,16],[1,16],[13,16]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/8, 7/8], [1/2, 1/14], [1/16, 1/16], 1/8, 1/8, 是)","explanation_zh":"Wave5第二層核對=([1/8, 7/8], [1/2, 1/14], [1/16, 1/16], 1/8, 1/8, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026059,"parameters":{"masses":[[1,16],[1,16],[1,16],[13,16]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,16],[1,16],[1,16],[13,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0060","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,17],[1,17],[1,17],[14,17]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/17, 15/17], [1/2, 1/15], [1/17, 1/17], 2/17, 2/17, 是)","explanation_zh":"Wave5第二層核對=([2/17, 15/17], [1/2, 1/15], [1/17, 1/17], 2/17, 2/17, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026060,"parameters":{"masses":[[1,17],[1,17],[1,17],[14,17]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,17],[1,17],[1,17],[14,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0061","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,18],[1,18],[1,18],[15,18]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/9, 8/9], [1/2, 1/16], [1/18, 1/18], 1/9, 1/9, 是)","explanation_zh":"Wave5第二層核對=([1/9, 8/9], [1/2, 1/16], [1/18, 1/18], 1/9, 1/9, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026061,"parameters":{"masses":[[1,18],[1,18],[1,18],[15,18]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,18],[1,18],[1,18],[15,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0062","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,19],[1,19],[1,19],[16,19]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/19, 17/19], [1/2, 1/17], [1/19, 1/19], 2/19, 2/19, 是)","explanation_zh":"Wave5第二層核對=([2/19, 17/19], [1/2, 1/17], [1/19, 1/19], 2/19, 2/19, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026062,"parameters":{"masses":[[1,19],[1,19],[1,19],[16,19]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,19],[1,19],[1,19],[16,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0063","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,20],[1,20],[1,20],[17,20]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/10, 9/10], [1/2, 1/18], [1/20, 1/20], 1/10, 1/10, 是)","explanation_zh":"Wave5第二層核對=([1/10, 9/10], [1/2, 1/18], [1/20, 1/20], 1/10, 1/10, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026063,"parameters":{"masses":[[1,20],[1,20],[1,20],[17,20]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,20],[1,20],[1,20],[17,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0064","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,21],[1,21],[1,21],[18,21]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/21, 19/21], [1/2, 1/19], [1/21, 1/21], 2/21, 2/21, 是)","explanation_zh":"Wave5第二層核對=([2/21, 19/21], [1/2, 1/19], [1/21, 1/21], 2/21, 2/21, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026064,"parameters":{"masses":[[1,21],[1,21],[1,21],[18,21]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,21],[1,21],[1,21],[18,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0065","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/11, 10/11], [1/2, 1/20], [1/22, 1/22], 1/11, 1/11, 是)","explanation_zh":"Wave5第二層核對=([1/11, 10/11], [1/2, 1/20], [1/22, 1/22], 1/11, 1/11, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026065,"parameters":{"masses":[[1,22],[1,22],[1,22],[19,22]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,22],[1,22],[1,22],[19,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0066","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,23],[1,23],[1,23],[20,23]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/23, 21/23], [1/2, 1/21], [1/23, 1/23], 2/23, 2/23, 是)","explanation_zh":"Wave5第二層核對=([2/23, 21/23], [1/2, 1/21], [1/23, 1/23], 2/23, 2/23, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026066,"parameters":{"masses":[[1,23],[1,23],[1,23],[20,23]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,23],[1,23],[1,23],[20,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0067","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,24],[1,24],[1,24],[21,24]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/12, 11/12], [1/2, 1/22], [1/24, 1/24], 1/12, 1/12, 是)","explanation_zh":"Wave5第二層核對=([1/12, 11/12], [1/2, 1/22], [1/24, 1/24], 1/12, 1/12, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026067,"parameters":{"masses":[[1,24],[1,24],[1,24],[21,24]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,24],[1,24],[1,24],[21,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0068","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,25],[1,25],[1,25],[22,25]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/25, 23/25], [1/2, 1/23], [1/25, 1/25], 2/25, 2/25, 是)","explanation_zh":"Wave5第二層核對=([2/25, 23/25], [1/2, 1/23], [1/25, 1/25], 2/25, 2/25, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026068,"parameters":{"masses":[[1,25],[1,25],[1,25],[22,25]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,25],[1,25],[1,25],[22,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0069","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,26],[1,26],[1,26],[23,26]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/13, 12/13], [1/2, 1/24], [1/26, 1/26], 1/13, 1/13, 是)","explanation_zh":"Wave5第二層核對=([1/13, 12/13], [1/2, 1/24], [1/26, 1/26], 1/13, 1/13, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026069,"parameters":{"masses":[[1,26],[1,26],[1,26],[23,26]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,26],[1,26],[1,26],[23,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0070","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,27],[1,27],[1,27],[24,27]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/27, 25/27], [1/2, 1/25], [1/27, 1/27], 2/27, 2/27, 是)","explanation_zh":"Wave5第二層核對=([2/27, 25/27], [1/2, 1/25], [1/27, 1/27], 2/27, 2/27, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026070,"parameters":{"masses":[[1,27],[1,27],[1,27],[24,27]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,27],[1,27],[1,27],[24,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0071","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,28],[1,28],[1,28],[25,28]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/14, 13/14], [1/2, 1/26], [1/28, 1/28], 1/14, 1/14, 是)","explanation_zh":"Wave5第二層核對=([1/14, 13/14], [1/2, 1/26], [1/28, 1/28], 1/14, 1/14, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026071,"parameters":{"masses":[[1,28],[1,28],[1,28],[25,28]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,28],[1,28],[1,28],[25,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0072","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,29],[1,29],[1,29],[26,29]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/29, 27/29], [1/2, 1/27], [1/29, 1/29], 2/29, 2/29, 是)","explanation_zh":"Wave5第二層核對=([2/29, 27/29], [1/2, 1/27], [1/29, 1/29], 2/29, 2/29, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026072,"parameters":{"masses":[[1,29],[1,29],[1,29],[26,29]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0073","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,30],[1,30],[1,30],[27,30]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, 是)","explanation_zh":"Wave5第二層核對=([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026073,"parameters":{"masses":[[1,30],[1,30],[1,30],[27,30]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,30],[1,30],[1,30],[27,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0074","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,31],[1,31],[1,31],[28,31]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([2/31, 29/31], [1/2, 1/29], [1/31, 1/31], 2/31, 2/31, 是)","explanation_zh":"Wave5第二層核對=([2/31, 29/31], [1/2, 1/29], [1/31, 1/31], 2/31, 2/31, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026074,"parameters":{"masses":[[1,31],[1,31],[1,31],[28,31]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,31],[1,31],[1,31],[28,31]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0075","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-total-evidence；資料={\"masses\":[[1,32],[1,32],[1,32],[29,32]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2]}。請依 [priors,likelihoods,joints,total,evidence_mass,equal] 欄序回答。","answer":{"kind":"expression","value":"([1/16, 15/16], [1/2, 1/30], [1/32, 1/32], 1/16, 1/16, 是)","explanation_zh":"Wave5第二層核對=([1/16, 15/16], [1/2, 1/30], [1/32, 1/32], 1/16, 1/16, 是)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-total-evidence","seed":202611026075,"parameters":{"masses":[[1,32],[1,32],[1,32],[29,32]],"hypotheses":[[0,1],[2,3]],"evidence":[0,2],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-total-evidence","semantic_template":"qf-mod33-bayes-total-evidence/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,32],[1,32],[1,32],[29,32]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0076","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,8],[1,8],[1,8],[5,8]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026076,"parameters":{"masses":[[1,8],[1,8],[1,8],[5,8]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,8],[1,8],[1,8],[5,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0077","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,9],[1,9],[1,9],[6,9]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026077,"parameters":{"masses":[[1,9],[1,9],[1,9],[6,9]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,9],[1,9],[1,9],[6,9]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0078","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,10],[1,10],[1,10],[7,10]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026078,"parameters":{"masses":[[1,10],[1,10],[1,10],[7,10]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,10],[1,10],[1,10],[7,10]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0079","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,11],[1,11],[1,11],[8,11]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026079,"parameters":{"masses":[[1,11],[1,11],[1,11],[8,11]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,11],[1,11],[1,11],[8,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0080","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,12],[1,12],[1,12],[9,12]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026080,"parameters":{"masses":[[1,12],[1,12],[1,12],[9,12]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,12],[1,12],[1,12],[9,12]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0081","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,13],[1,13],[1,13],[10,13]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026081,"parameters":{"masses":[[1,13],[1,13],[1,13],[10,13]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,13],[1,13],[1,13],[10,13]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0082","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,14],[1,14],[1,14],[11,14]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026082,"parameters":{"masses":[[1,14],[1,14],[1,14],[11,14]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,14],[1,14],[1,14],[11,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0083","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,15],[1,15],[1,15],[12,15]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026083,"parameters":{"masses":[[1,15],[1,15],[1,15],[12,15]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,15],[1,15],[1,15],[12,15]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0084","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,16],[1,16],[1,16],[13,16]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026084,"parameters":{"masses":[[1,16],[1,16],[1,16],[13,16]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,16],[1,16],[1,16],[13,16]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0085","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,17],[1,17],[1,17],[14,17]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026085,"parameters":{"masses":[[1,17],[1,17],[1,17],[14,17]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,17],[1,17],[1,17],[14,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0086","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,18],[1,18],[1,18],[15,18]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026086,"parameters":{"masses":[[1,18],[1,18],[1,18],[15,18]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,18],[1,18],[1,18],[15,18]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0087","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,19],[1,19],[1,19],[16,19]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026087,"parameters":{"masses":[[1,19],[1,19],[1,19],[16,19]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,19],[1,19],[1,19],[16,19]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0088","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,20],[1,20],[1,20],[17,20]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026088,"parameters":{"masses":[[1,20],[1,20],[1,20],[17,20]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,20],[1,20],[1,20],[17,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0089","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,21],[1,21],[1,21],[18,21]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026089,"parameters":{"masses":[[1,21],[1,21],[1,21],[18,21]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,21],[1,21],[1,21],[18,21]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0090","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026090,"parameters":{"masses":[[1,22],[1,22],[1,22],[19,22]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,22],[1,22],[1,22],[19,22]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0091","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,23],[1,23],[1,23],[20,23]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026091,"parameters":{"masses":[[1,23],[1,23],[1,23],[20,23]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,23],[1,23],[1,23],[20,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0092","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,24],[1,24],[1,24],[21,24]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026092,"parameters":{"masses":[[1,24],[1,24],[1,24],[21,24]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,24],[1,24],[1,24],[21,24]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0093","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,25],[1,25],[1,25],[22,25]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026093,"parameters":{"masses":[[1,25],[1,25],[1,25],[22,25]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,25],[1,25],[1,25],[22,25]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0094","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,26],[1,26],[1,26],[23,26]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026094,"parameters":{"masses":[[1,26],[1,26],[1,26],[23,26]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,26],[1,26],[1,26],[23,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0095","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,27],[1,27],[1,27],[24,27]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026095,"parameters":{"masses":[[1,27],[1,27],[1,27],[24,27]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,27],[1,27],[1,27],[24,27]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0096","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,28],[1,28],[1,28],[25,28]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026096,"parameters":{"masses":[[1,28],[1,28],[1,28],[25,28]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,28],[1,28],[1,28],[25,28]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0097","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,29],[1,29],[1,29],[26,29]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026097,"parameters":{"masses":[[1,29],[1,29],[1,29],[26,29]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0098","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,30],[1,30],[1,30],[27,30]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026098,"parameters":{"masses":[[1,30],[1,30],[1,30],[27,30]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,30],[1,30],[1,30],[27,30]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0099","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,31],[1,31],[1,31],[28,31]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026099,"parameters":{"masses":[[1,31],[1,31],[1,31],[28,31]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,31],[1,31],[1,31],[28,31]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261102-r06-bayes-theorem-0100","batch_id":"batch-20261014-advance-20261102-r06-bayes-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-bayes-theorem"],"title_zh":"Bayes定理","stem_zh":"完整有限原子law、事件與正conditioning denominator。操作=bayes-zero-denominator-boundary；資料={\"masses\":[[1,32],[1,32],[1,32],[29,32]],\"hypothesis\":[0,1],\"evidence\":[]}。請依 [evidence_mass,conditional_value,defined,boundary] 欄序回答。","answer":{"kind":"expression","value":"(0, undefined, 否, conditional_probability_requires_positive_denominator)","explanation_zh":"Wave5第二層核對=(0, undefined, 否, conditional_probability_requires_positive_denominator)；依完整明示資料重算；partition須pairwise disjoint且exhaustive並由total probability正規化；不可漏prior、反轉零分母或把Bayes當causation。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Bayes定理"]},"variant":{"family_id":"qf-mod33-bayes-zero-denominator-boundary","seed":202611026100,"parameters":{"masses":[[1,32],[1,32],[1,32],[29,32]],"hypothesis":[0,1],"evidence":[],"domain":"Exact_finite_atomic_Bayes","operation":"bayes-zero-denominator-boundary","semantic_template":"qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,32],[1,32],[1,32],[29,32]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}