{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261004-advance-20261005-r05-random-variable-0095", "pilot_number": "R0056-28545", "number": 28545, "family_id": "qf-random-variable-point-mass", "title": { "zh-Hant": "第 28545 題：重複值的有限點機率", "en": "Question 28545: Finite Point Probability of a Repeated Value" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261004-advance-20261005-r05-random-variable-0095", "batch_id": "batch-20261004-advance-20261005-r05-random-variable-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-random-variable" ], "title_zh": "重複值的有限點機率", "stem_zh": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -20), (1, -20), (2, 21), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/25, 21/25, 3/25, 0/25]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。", "answer": { "kind": "expression", "value": "0/1", "explanation_zh": "P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "重複值的有限點機率" ] }, "variant": { "family_id": "qf-random-variable-point-mass", "seed": 202610055095, "parameters": { "S": [ 0, 1, 2, 3 ], "F": "powerset(S)", "values": [ -20, -20, 21, 1018 ], "weights": [ 1, 21, 3, 0 ], "denominator": 25, "t": 0, "operation": "rv_point_mass", "semantic_template": "qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-20\",\"22/25\"],[\"21\",\"3/25\"]],\"operation\":\"rv_point_mass\",\"t\":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": "Codex independent producer (inherited model; exact runtime model not exposed)", "generated_at": "2026-10-04", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261004-advance-20261005-r05-random-variable-0095", "batch_id": "batch-20261004-advance-20261005-r05-random-variable-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "869b4dba3f1fe262e3ab2a2265c936602040e5bf383dd54b13c8d2bb0cb5faa5", "family_id": "qf-random-variable-point-mass", "seed": 202610055095, "semantic_template": "qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-20\",\"22/25\"],[\"21\",\"3/25\"]],\"operation\":\"rv_point_mass\",\"t\":0}" }, "fields": { "title": "Finite Point Probability of a Repeated Value", "stem": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 21), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(25), 21/(25), 3/(25), 0/(25)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.", "answer": { "kind": "expression", "value": "0/1", "explanation": "Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator." }, "skills": [ "Finite Point Probability of a Repeated Value" ], "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": "非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -20), (1, -20), (2, 21), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/25, 21/25, 3/25, 0/25]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。", "en": "The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 21), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(25), 21/(25), 3/(25), 0/(25)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-random-variable-point-mass, rv_point_mass]", "after": "derived = [probability_of_complete_value_fibre]", "relation": "implies", "rule_id": "R56_source_answer_role_and_EXACT_family_interface", "conditions": [], "explanation": { "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。", "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[0, 1, 2, 3], powerset(S), [-20, -20, 21, 1018]]", "after": "derived = [[[0, -20], [1, -20], [2, 21], [3, 1018]]]", "relation": "implies", "rule_id": "RV_full_powerset_measurability_ORIGINAL_outcome_value_alignment", "conditions": [], "explanation": { "zh-Hant": "values逐slot對齊原S，不能把兩列分開排序。有限full powerset包含每個X的逆像，故這個實值函數可測；求value／range／event不須另造均勻權重。", "en": "Align each value with the ORIGINAL S slot,not independently sorted columns.The full powerset contains every preimage of this real-valued function,hence measurability.No invented uniform weights are needed for a value,range or event." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "RV ORIGINAL all aligned numerator weights := sum(1, 21, 3, 0)", "after": "RV ORIGINAL all aligned numerator weights = 25", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 RV ORIGINAL all aligned numerator weights，不以小數近似替代。", "en": "Recompute RV ORIGINAL all aligned numerator weights with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "21", "3", "0" ], "output": "25", "label": "RV ORIGINAL all aligned numerator weights" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "source roles = [[[0, -20, 1], [1, -20, 21], [2, 21, 3], [3, 1018, 0]], 25, 0]", "after": "derived = [25, [[0, -20, 1, 否], [1, -20, 21, 否], [2, 21, 3, 否], [3, 1018, 0, 否]]]", "relation": "implies", "rule_id": "RV_point_mass_normalized_law_ON_S_and_ALL_fibre_terms", "conditions": [], "explanation": { "zh-Hant": "同一S的非負權重總和須等den>0；P(X=t)加完整fiber的權重，重複X值不可只取首項，零權重也保留。不以range大小作分母。", "en": "Nonnegative weights on the SAME S must sum to den>0.P(X=t) adds the COMPLETE fibre,including repeated X values and zero weights;range size is not the denominator." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "RV ORIGINAL complete fibre mass over denominator := div(0, 25)", "after": "RV ORIGINAL complete fibre mass over denominator = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。", "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 RV ORIGINAL complete fibre mass over denominator，不以小數近似替代。", "en": "Recompute RV ORIGINAL complete fibre mass over denominator with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "div", "inputs": [ "0", "25" ], "output": "0", "label": "RV ORIGINAL complete fibre mass over denominator" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "source roles = [[[0, -20, 1], [1, -20, 21], [2, 21, 3], [3, 1018, 0]], 0, 0, 25]", "after": "derived = [0/1]", "relation": "implies", "rule_id": "RV_point_mass_unattained_value_zero_and_reduced_fraction", "conditions": [], "explanation": { "zh-Hant": "空fiber依空和0得到0/1；否則逐項加總再除原den。約分與正分母不改事件，整數機率仍顯示/1。", "en": "An unattained value has empty-fibre sum0 and probability0/1.Otherwise add all matching terms and divide by the ORIGINAL denominator.Reduction preserves the event;integer probabilities retain/1." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。", "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance." } }, "arithmetic": null } ], "math_model": { "result": "0/1" }, "conclusion": { "canonical_value": "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-20261004-advance-20261005-r05-random-variable-foundations", "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r05-random-variable-foundations/questions.jsonl", "batch_sha256": "e65fbafe078025619c0c64f1e8c66b8d0c0386bc721517cbf25b94c3d3ba5e1f", "original_line": 95, "original_line_sha256": "00c4f6ed2296ba4fd8370b2e3aeb09cfd7b97dba56f521a5c7bdaa1d2aa1f916", "question_compact_sha256": "869b4dba3f1fe262e3ab2a2265c936602040e5bf383dd54b13c8d2bb0cb5faa5", "english_row_compact_sha256": "948d3cc68c0b2fe7585cdfb2040130401ab77d8b88c4b99cf4c6f854a49a5e9d", "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r05-random-variable-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r05-random-variable-foundations.en.jsonl" }, "mko_bindings": { "mko-random-variable": { "sha256": "bba00300a619b37481007904b211059750f93a4a2c4ba2b8209eac472ad33e06", "english_sha256": "456778ef562a2c57a14293522fad0ac209632686518a816ad9409094a4c350e1", "original_path": "public/data/mko/random-variable.json", "english_path": "public/data/i18n/editions/mko-random-variable.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5", "original_candidate_sha256": "691ee0c44b0355b1231b9d16d1cd8127b943e0c7bfb9d28451fd64100d1030e5", "original_candidate_line_sha256": "691ee0c44b0355b1231b9d16d1cd8127b943e0c7bfb9d28451fd64100d1030e5" }, "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-20261026-010-p57", "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8", "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 6, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "50b4fd74186db67771e77408f10c852f795cc877be527ae88ed40fd9065d90aa", "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." } }