{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261102-r06-bayes-theorem-0097", "pilot_number": "P1000-0547", "number": 547, "family_id": "qf-mod33-bayes-zero-denominator-boundary", "title": { "zh-Hant": "第 547 題：Bayes定理", "en": "Question 547: Zero Evidence Mass Makes This Conditional Probability Undefined" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261102-r06-bayes-theorem-0097", "batch_id": "batch-20261014-advance-20261102-r06-bayes-theorem-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "997d482972828adc31f421a89c0cc9a31e980ba98c32289b7e423329f6842437", "family_id": "qf-mod33-bayes-zero-denominator-boundary", "seed": 202611026097, "semantic_template": "qf-mod33-bayes-zero-denominator-boundary/{\"evidence\":[],\"hypothesis\":[0,1],\"masses\":[[1,29],[1,29],[1,29],[26,29]]}" }, "fields": { "title": "Zero Evidence Mass Makes This Conditional Probability Undefined", "stem": "Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).", "answer": { "kind": "expression", "value": "(0, undefined, No, conditional_probability_requires_positive_denominator)", "explanation": "Report undefined and No without dividing. This is an intentional audit of the failed positive-denominator premise, not an ordinary Bayes update or a claim of zero posterior probability. The exact result, retaining the requested order, is (0, undefined, No, conditional_probability_requires_positive_denominator)." }, "skills": [ "Zero Evidence Mass Makes This Conditional Probability Undefined" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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。", "en": "Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/29, 1/29, 1/29, 26/29], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "P(Ω) := sum(1/29, 1/29, 1/29, 26/29)", "after": "P(Ω) = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "完整原子機率先加總為1。", "en": "First sum the complete atom masses to1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P(Ω)，不以小數近似替代。", "en": "Recompute P(Ω) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/29", "1/29", "1/29", "26/29" ], "output": "1", "label": "P(Ω)" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "P(E) := sum()", "after": "P(E) = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "只累加條件事件E中的原子。", "en": "Sum only the atoms in the conditioning event E." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 P(E)，不以小數近似替代。", "en": "Recompute P(E) with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [], "output": "0", "label": "P(E)" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "P(E)=0", "after": "P(H|E)=P(H∩E)/P(E) is not defined here", "relation": "implies", "rule_id": "zero_denominator_guard", "conditions": [], "explanation": { "zh-Hant": "條件機率定義需要正分母，本題不進入除法。", "en": "The conditional-probability definition requires a positive denominator, so no division is performed." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "同一 law 的事件、交集與條件分母必須對齊；零質量不作除法。", "en": "Align events, intersections and conditioning denominators under the same law; do not divide by zero mass." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "zero conditioning mass", "after": "undefined is not zero, not infinity, and not an empty event", "relation": "implies", "rule_id": "undefined_not_value", "conditions": [], "explanation": { "zh-Hant": "即使交集亦為零，0/0仍不能指定成數值0。", "en": "Even when the joint mass is zero,0/0 cannot be assigned the value0." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "同一 law 的事件、交集與條件分母必須對齊；零質量不作除法。", "en": "Align events, intersections and conditioning denominators under the same law; do not divide by zero mass." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "positive-denominator requirement fails", "after": "defined=false; value=undefined", "relation": "implies", "rule_id": "contract_boundary", "conditions": [], "explanation": { "zh-Hant": "保留此合約的未定義標記，不改寫機率模型。", "en": "Retain this contract's undefined marker without altering the probability law." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "同一 law 的事件、交集與條件分母必須對齊；零質量不作除法。", "en": "Align events, intersections and conditioning denominators under the same law; do not divide by zero mass." } }, "arithmetic": null } ], "math_model": { "result": [ "0", "undefined", false, "conditional_probability_requires_positive_denominator" ] }, "conclusion": { "canonical_value": "(0, undefined, 否, conditional_probability_requires_positive_denominator)", "explanation": { "zh-Hant": "上述完整代值與條件核對得到原題答案；欄位次序及真假標記保持原始合約。", "en": "The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings." } }, "definition_note": { "zh-Hant": "條件機率需要正分母；P(E)=0 時記為未定義，不除法、不等於 0。", "en": "Conditional probability needs a positive denominator. With P(E)=0 it is undefined: no division, and not the value 0." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261102-r06-bayes-theorem-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r06-bayes-theorem-foundations/questions.jsonl", "batch_sha256": "d42c8d49035f7a60c45c1fb287574d29523621eb200ae86ad41f004ab645ecc1", "original_line": 97, "original_line_sha256": "d00682ce5f96bdf9e9ed44c2689de185245e4b302280c746435126a2d4f5fb27", "question_compact_sha256": "997d482972828adc31f421a89c0cc9a31e980ba98c32289b7e423329f6842437", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r06-bayes-theorem-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r06-bayes-theorem-foundations.en.jsonl", "english_row_compact_sha256": "425f8c3ce1fddf45eb99c0aebbc5899d58c524396f5f61367d15d0e45ece896a" }, "mko_bindings": { "mko-bayes-theorem": { "sha256": "29451b12d334baf07d1c6b72cf87989c5c58d98ccd89384b8b2a6591f52b6179", "english_sha256": "b3f2729c0af9e4e121044352688ea6f3a7a8be149d4a88207d66ada74315e947", "original_path": "public/data/mko/bayes-theorem.json", "english_path": "public/data/i18n/editions/mko-bayes-theorem.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "afa7c0e9f7084cc5929124dcffe9929d5db7def32e40047056d19ea25535873e", "original_candidate_sha256": "e9990c44071ceffe43eef561c5877680815c8a5a74a0c2a5d8372b7c62193500", "original_candidate_line_sha256": "e9990c44071ceffe43eef561c5877680815c8a5a74a0c2a5d8372b7c62193500" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定的教學推導候選；精確算術及有限列舉不是獨立AI審查或一般形式化證明。原題、答案、MKO、公式、AST及歷史保持不變。", "en": "Source-bound pedagogical derivation candidate: exact arithmetic and finite enumeration are not independent AI review or general formal proof. Original questions, answers, MKOs, formulas, ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-003-02", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "Thirty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct sentence shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full answer tuples, model results and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772", "question_first_witness_sha256": "77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 5, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "8481096d6d2895e0db5e676ec5e99d1556627ad5e4e39ff5b7465c6678f8d4c9", "arithmetic_steps": 2 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }