{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r07-probability-0046",
  "pilot_number": "R0055-28396",
  "number": 28396,
  "family_id": "qf-probability-weighted-event",
  "title": {
    "zh-Hant": "第 28396 題：非等可能加權事件機率",
    "en": "Question 28396: Weighted Event Probability without Equiprobability"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r07-probability-0046",
    "batch_id": "batch-20261003-r07-probability-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-probability"
    ],
    "title_zh": "非等可能加權事件機率",
    "stem_zh": "非空有限樣本空間 S={8, 9, 10}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [21/24, 1/24, 2/24]，事件 E=∅。求精確 P(E)，不可假設等可能。",
    "answer": {
      "kind": "expression",
      "value": "0/1",
      "explanation_zh": "P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "非等可能加權事件機率"
      ]
    },
    "variant": {
      "family_id": "qf-probability-weighted-event",
      "seed": 202610037046,
      "parameters": {
        "S": [
          8,
          9,
          10
        ],
        "F": "powerset(S)",
        "weights": [
          21,
          1,
          2
        ],
        "denominator": 24,
        "operation": "weighted_event",
        "E": [],
        "semantic_template": "qf-probability-weighted-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[8,9,10],\"denominator\":\"24\",\"operation\":\"weighted_event\",\"weights\":[[8,\"21\"],[9,\"1\"],[10,\"2\"]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-r07-probability-0046",
    "batch_id": "batch-20261003-r07-probability-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "88eaecac27254336e157a6a76587292002d4bd0dabcd3d42beadd458e856f841",
      "family_id": "qf-probability-weighted-event",
      "seed": 202610037046,
      "semantic_template": "qf-probability-weighted-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[8,9,10],\"denominator\":\"24\",\"operation\":\"weighted_event\",\"weights\":[[8,\"21\"],[9,\"1\"],[10,\"2\"]]}"
    },
    "fields": {
      "title": "Weighted Event Probability without Equiprobability",
      "stem": "The nonempty finite sample space is S={8, 9, 10} and the event family is F=𝒫(S). In the displayed order of S, the weights are [21/(24), 1/(24), 2/(24)] and E=∅. Find P(E) exactly as a reduced fraction with positive denominator; do not assume equiprobability.",
      "answer": {
        "kind": "expression",
        "value": "0/1",
        "explanation": "Check nonnegative weights summing to one, then sum the weights of the event outcomes. The exact probability is 0/1. A nonempty event may have zero probability. Here 𝒫 denotes the full powerset and P denotes the probability law."
      },
      "skills": [
        "Weighted Event Probability without Equiprobability"
      ],
      "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": "非空有限樣本空間 S={8, 9, 10}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [21/24, 1/24, 2/24]，事件 E=∅。求精確 P(E)，不可假設等可能。",
        "en": "The nonempty finite sample space is S={8, 9, 10} and the event family is F=𝒫(S). In the displayed order of S, the weights are [21/(24), 1/(24), 2/(24)] and E=∅. Find P(E) exactly as a reduced fraction with positive denominator; do not assume equiprobability."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-probability-weighted-event, weighted_event]",
      "after": "derived = [[reduced_probability_fraction_retaining_integer_denominator1]]",
      "relation": "implies",
      "rule_id": "source_answer_roles_or_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁明列本題scalar／label／set／point或mode欄序，不強套現代tuple。舊題domain與operation底線格式按原資料，分數整數亦保/1；原中英快照不改。",
        "en": "List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [[8, 9, 10], powerset(S)]",
      "after": "derived = [[8, 9, 10], 3]",
      "relation": "implies",
      "rule_id": "Legacy_probability_NONEMPTY_S_FULLpowerset_exact_declared_event_family",
      "conditions": [],
      "explanation": {
        "zh-Hant": "S為至多100個distinct SAFE整數的非空set；F明示完整powerset，不由單一事件猜event family或省略measurability。",
        "en": "S is a nonempty set of at most100 distinct SAFE integers and F is explicitly the FULL powerset,not an event family inferred from one event."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Legacy probability ORIGINAL ALL numerator weights sum := sum(21, 1, 2)",
      "after": "Legacy probability ORIGINAL ALL numerator weights sum = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy probability ORIGINAL ALL numerator weights sum，不以小數近似替代。",
          "en": "Recompute Legacy probability ORIGINAL ALL numerator weights sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "1",
          "2"
        ],
        "output": "24",
        "label": "Legacy probability ORIGINAL ALL numerator weights sum"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [[8, 9, 10], [21, 1, 2], 24]",
      "after": "derived = [[[8, 21], [9, 1], [10, 2]], 24, 是, 是, 是, 是]",
      "relation": "implies",
      "rule_id": "Legacy_probability_ORIGINAL_rawS_weight_alignment_and_model_qualification",
      "conditions": [],
      "explanation": {
        "zh-Hant": "weights先綁原始S slots，set排序不移動這個關聯。valid須正分母、全部非負、BigInt總分子等分母；model-validity題可合法回No，不除0也不改號／重正規化成別的model。",
        "en": "Bind weights to ORIGINAL raw S slots;set sorting does not change these associations.Validity needs a positive denominator,all nonnegative weights and the BigInt total equal to the denominator.A model-validity question may legally answerNo,without0/0,sign repair or renormalizing another model."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Legacy probability ORIGINAL normalized event or complement fraction := div(0, 24)",
      "after": "Legacy probability ORIGINAL normalized event or complement fraction = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原始角色精確代值，不作浮點近似或換操作數；額外教學等式不另縮原題域。",
        "en": "Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Legacy probability ORIGINAL normalized event or complement fraction，不以小數近似替代。",
          "en": "Recompute Legacy probability ORIGINAL normalized event or complement fraction with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "24"
        ],
        "output": "0",
        "label": "Legacy probability ORIGINAL normalized event or complement fraction"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [[8, 9, 10], [], [[8, 21], [9, 1], [10, 2]], 24]",
      "after": "derived = [0, [8, 9, 10], 0, 0/1]",
      "relation": "implies",
      "rule_id": "Legacy_complete_finite_probability_axioms_and_same_carrier_complement",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在完整powerset上對任何A以原weights求和，非負、全集1，disjoint事件的分子可重排合併，故有限（亦countable）additivity。E可非空但mass0；complement在同一S內，不把weight來源換成sorted slots或uniform counts。",
        "en": "On the full powerset define every event by its ORIGINAL weighted sum.Nonnegativity,total1 and regrouping disjoint events give finite(and countable,for finite S)additivity.A nonempty E can have mass0.The complement is in the SAME S;do not substitute sorted-slot weights or uniform counts."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查完整來源資料、原操作數及本步條件；有限核對、條件式理由、形式證明與發布資格分開。",
          "en": "Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct."
        }
      },
      "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-20261003-r07-probability-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-r07-probability-foundations/questions.jsonl",
      "batch_sha256": "abc51053d58d9d3d2507fbc398af5d154233f38157394df4a2b5b1d480826a61",
      "original_line": 46,
      "original_line_sha256": "cfec9a462c8b20e504068611cf4d95132311fa6c6999e95be5780e0c7aa2d5ca",
      "question_compact_sha256": "88eaecac27254336e157a6a76587292002d4bd0dabcd3d42beadd458e856f841",
      "english_row_compact_sha256": "43f7df989ee8c5e01d5263db9a5b4e5d6c9c140b5f5ed66e91942c94cdaa9b8a",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-r07-probability-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-r07-probability-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-probability": {
        "sha256": "07f2b780895b31a69725da388129f9624510427b4a2937b15a61418f94e267ca",
        "english_sha256": "916cfce506d3d89365ae56463d792e878efa37eeafd9f269c15a87f152202f75",
        "original_path": "public/data/mko/probability.json",
        "english_path": "public/data/i18n/editions/mko-probability.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b5763df142e86ef7f9787b6ea86050afc60bd3c5c53538d4839606827cc44686",
    "original_candidate_sha256": "1a91b1beb538f93316e2ce65248d6464f29ab98899e6fdf37648d34df4f901db",
    "original_candidate_line_sha256": "1a91b1beb538f93316e2ce65248d6464f29ab98899e6fdf37648d34df4f901db"
  },
  "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-p56",
    "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": "6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683",
      "question_first_witness_sha256": "611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e"
    },
    "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."
  }
}