{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261008-advance-20261010-r04-conditional-probability-0051", "pilot_number": "R0059-30101", "number": 30101, "family_id": "qf-density10-conditional-point-law", "title": { "zh-Hant": "第 30101 題：全部基本結果的條件權重", "en": "Question 30101: Complete Pointwise Conditional Law" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261008-advance-20261010-r04-conditional-probability-0051", "batch_id": "batch-20261008-advance-20261010-r04-conditional-probability-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-conditional-probability" ], "title_zh": "全部基本結果的條件權重", "stem_zh": "有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[0, 1, 2]、D=3，權重為質量/D；B=[1]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。", "answer": { "kind": "expression", "value": "[0/1, 1/1, 0/1]", "explanation_zh": "逐點條件權重=[0/1, 1/1, 0/1]。B內質量除B總質量，B外記0/1；完整向量總和為一，未默認均勻模型。" }, "difficulty": { "band": "upper_secondary", "level": 1, "skills": [ "全部基本結果的條件權重" ] }, "variant": { "family_id": "qf-density10-conditional-point-law", "seed": 202610104051, "parameters": { "masses": [ 0, 1, 2 ], "D": 3, "B": [ 1 ], "operation": "density10_conditional_point_law", "domain": "finite_normalized_probability", "semantic_template": "qf-density10-conditional-point-law/[\"0\",\"1\",\"0\"]" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_integer_rational_arithmetic", "producer_finite_enumeration", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-08", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261008-advance-20261010-r04-conditional-probability-0051", "batch_id": "batch-20261008-advance-20261010-r04-conditional-probability-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "1e8167934a4257b203cab19c375ac979492b76f75bbbba9fff791f88845a34da", "family_id": "qf-density10-conditional-point-law", "seed": 202610104051, "semantic_template": "qf-density10-conditional-point-law/[\"0\",\"1\",\"0\"]" }, "fields": { "title": "Complete Pointwise Conditional Law", "stem": "S={0,…,2}, F=P(S), masses=[0, 1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[1], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.", "answer": { "kind": "expression", "value": "[0/1, 1/1, 0/1]", "explanation": "For a point in B divide its mass by the total B mass; outside B use 0/1. The complete vector sums to one. The exact result, retaining the requested order, is [0/1, 1/1, 0/1]." }, "skills": [ "Complete Pointwise Conditional Law" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[0, 1, 2]、D=3，權重為質量/D；B=[1]，P(B)>0。按索引0起列每個singleton在B條件下的概率，包括B外的零項。", "en": "S={0,…,2}, F=P(S), masses=[0, 1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[1], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-density10-conditional-point-law, density10_conditional_point_law, finite_normalized_probability]", "after": "derived = [[ALL_singleton_conditional_probabilities_index_order]]", "relation": "implies", "rule_id": "R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles", "conditions": [], "explanation": { "zh-Hant": "保原bank的完整operation前綴、domain及輸出欄序；frame只用orthogonal_equal_unit，角度unit只用degrees。分數整數保/1；不增不存在的參數，不改原中英來源。", "en": "Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Conditional ORIGINAL ALL numerator weights sum := sum(0, 1, 2)", "after": "Conditional ORIGINAL ALL numerator weights sum = 3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional ORIGINAL ALL numerator weights sum，不以小數近似替代。", "en": "Recompute Conditional ORIGINAL ALL numerator weights sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1", "2" ], "output": "3", "label": "Conditional ORIGINAL ALL numerator weights sum" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [[0, 1, 2], 3]", "after": "derived = [3, [[0, 0/1], [1, 1/3], [2, 2/3]], 3]", "relation": "implies", "rule_id": "Conditional_FULL_normalized_finite_law_not_candidate_repair", "conditions": [], "explanation": { "zh-Hant": "S=Fin(n)，完整冪集上的每個singleton質量m_i/D，m_i非負且全部和D>0。此四族已要求真law，無效law直接拒絕，不回No後重配權重。", "en": "On S=Fin(n) with its full powerset,each singleton has mass m_i/D,all m_i>=0 and total D>0.These FOUR families REQUIRE an actual normalized law;reject an invalid law rather than return No and repair its weights." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [B, [1], 3]", "after": "derived = [[1]]", "relation": "implies", "rule_id": "Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing", "conditions": [], "explanation": { "zh-Hant": "原list先驗長度<=16且每項為Fin(n)合法SAFE index，再去重數值排序。重複listing不倍加mass；raw17份重複仍超來源界。", "en": "FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "Conditional B ORIGINAL included index1 running0 := sum(0, 1)", "after": "Conditional B ORIGINAL included index1 running0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional B ORIGINAL included index1 running0，不以小數近似替代。", "en": "Recompute Conditional B ORIGINAL included index1 running0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "Conditional B ORIGINAL included index1 running0" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [Conditional B, [1], [0, 1, 2]]", "after": "derived = [1]", "relation": "implies", "rule_id": "Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots", "conditions": [], "explanation": { "zh-Hant": "依規範化事件每個index恰取原slot一次；沒有index時空和為0，不私加均勻權重或換配對。", "en": "For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Conditional ORIGINAL B mass over D := div(1, 3)", "after": "Conditional ORIGINAL B mass over D = 1/3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional ORIGINAL B mass over D，不以小數近似替代。", "en": "Recompute Conditional ORIGINAL B mass over D with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "3" ], "output": "1/3", "label": "Conditional ORIGINAL B mass over D" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "source roles = [[1], [0, 1, 2], 3]", "after": "derived = [1, 1/3, 是]", "relation": "implies", "rule_id": "Conditional_STRICT_positive_B_mass_not_merely_nonempty", "conditions": [], "explanation": { "zh-Hant": "P(B)>0等價b>0，不等於B非空；含有零權重label的非空事件仍可能不可條件化。只在已給正mass下除b。", "en": "P(B)>0 means b>0,not merely B nonempty.A nonempty event containing only null labels cannot be conditioned on.Divide by b ONLY under the given positive-mass premise." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "Conditional ORIGINAL singleton0 over B mass := div(0, 1)", "after": "Conditional ORIGINAL singleton0 over B mass = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional ORIGINAL singleton0 over B mass，不以小數近似替代。", "en": "Recompute Conditional ORIGINAL singleton0 over B mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "Conditional ORIGINAL singleton0 over B mass" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "Conditional ORIGINAL singleton1 over B mass := div(1, 1)", "after": "Conditional ORIGINAL singleton1 over B mass = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional ORIGINAL singleton1 over B mass，不以小數近似替代。", "en": "Recompute Conditional ORIGINAL singleton1 over B mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "1", "1" ], "output": "1", "label": "Conditional ORIGINAL singleton1 over B mass" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "Conditional ORIGINAL singleton2 over B mass := div(0, 1)", "after": "Conditional ORIGINAL singleton2 over B mass = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional ORIGINAL singleton2 over B mass，不以小數近似替代。", "en": "Recompute Conditional ORIGINAL singleton2 over B mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "1" ], "output": "0", "label": "Conditional ORIGINAL singleton2 over B mass" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "Conditional reverse ALL singleton probabilities sum := sum(0, 1, 0)", "after": "Conditional reverse ALL singleton probabilities sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按原角色精確代值，不作浮點近似；SAFE只驗raw輸入，衍生整數／有理數不截斷，不套其他銀行的component cap。", "en": "Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Conditional reverse ALL singleton probabilities sum，不以小數近似替代。", "en": "Recompute Conditional reverse ALL singleton probabilities sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1", "0" ], "output": "1", "label": "Conditional reverse ALL singleton probabilities sum" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "source roles = [[0, 1, 2], 3, [1], 1]", "after": "derived = [[0/1, 1/1, 0/1], 1/1]", "relation": "implies", "rule_id": "Conditional_POINT_law_ALL_indices_NULL_and_outside_ZERO", "conditions": [], "explanation": { "zh-Hant": "每個原index都列P({i}|B)=m_i/b或0/1，包括B外與B內null labels；完整vector和為b/b=1。不能drop零項或改索引。", "en": "For EVERY original index list P({i}|B)=m_i/b or0/1,including outside-B and inside-B null labels.The COMPLETE vector sums to b/b=1.Do not drop zero entries or change indexing." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以原始資料、索引與前提反向代回本步，不聲稱可由結論唯一還原輸入。有限精確核對、獨立審查、形式證明及發布資格分開。", "en": "Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate." } }, "arithmetic": null } ], "math_model": { "result": [ "0/1", "1/1", "0/1" ] }, "conclusion": { "canonical_value": "[0/1, 1/1, 0/1]", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261008-advance-20261010-r04-conditional-probability-foundations", "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r04-conditional-probability-foundations/questions.jsonl", "batch_sha256": "1576d274c739b94db2fbdef2e521cb048f32afed1c6ec141153a6605cf4e6a7b", "original_line": 51, "original_line_sha256": "8b18272166c800c84449ff627b67f8b2f76ba95060b771a80e8af8e3b96412e5", "question_compact_sha256": "1e8167934a4257b203cab19c375ac979492b76f75bbbba9fff791f88845a34da", "english_row_compact_sha256": "37b3f3262f4bc5cf15f89c300aa35eb451ce5181170ae3156906ad1e37733366", "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r04-conditional-probability-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r04-conditional-probability-foundations.en.jsonl" }, "mko_bindings": { "mko-conditional-probability": { "sha256": "7b0f4824aa1f3e4c64ea306c5bed04843566b8e4bbdf96d94dbc1367dbb68419", "english_sha256": "d0ca772e54d88ba2c19452fb83b8dbc37d4f064fe26d89d017fe591f2913621b", "original_path": "public/data/mko/conditional-probability.json", "english_path": "public/data/i18n/editions/mko-conditional-probability.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "17d593a551933064f195bd9277d5085fe2b6c98e11cbda0c8aa27d11b7b07949", "original_candidate_sha256": "94fe86a5f27b09f7f72ce0b3fa1e9339c85d1af46b3b01a39a68e2857d1d2019", "original_candidate_line_sha256": "94fe86a5f27b09f7f72ce0b3fa1e9339c85d1af46b3b01a39a68e2857d1d2019" }, "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": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human 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-20261026-011-p60", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings 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": "4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5", "question_first_witness_sha256": "a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 13, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "11b177064cb3f6f3fa631b1fa41d05cd7f147708baafca29b3f9afb712495559", "arithmetic_steps": 7 }, "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." } }