{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-advance-20261004-r07-conditional-probability-0064",
  "pilot_number": "R0005-03164",
  "number": 3164,
  "family_id": "qf-conditional-event-validity",
  "title": {
    "zh-Hant": "第 3164 題：條件事件的正機率資格",
    "en": "Question 3164: Positive-Probability Eligibility of a Conditioning Event"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-advance-20261004-r07-conditional-probability-0064",
    "batch_id": "batch-20261003-advance-20261004-r07-conditional-probability-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-conditional-probability"
    ],
    "title_zh": "條件事件的正機率資格",
    "stem_zh": "明示非空 S={1, 2, 3, 4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 2/22, 5/22, 14/22]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "條件質量分子=0。共同分母為正，只有分子>0 才能條件化；非空零質量事件也沒有此資格，不把未定義答案設成零。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "條件事件的正機率資格"
      ]
    },
    "variant": {
      "family_id": "qf-conditional-event-validity",
      "seed": 202610047064,
      "parameters": {
        "S": [
          1,
          2,
          3,
          4
        ],
        "F": "powerset(S)",
        "weights": [
          1,
          2,
          5,
          14
        ],
        "denominator": 22,
        "B": [],
        "operation": "condition_validity",
        "semantic_template": "qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[1,2,3,4],\"denominator\":\"22\",\"operation\":\"condition_validity\",\"weights\":[[1,\"1\"],[2,\"2\"],[3,\"5\"],[4,\"14\"]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent 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-advance-20261004-r07-conditional-probability-0064",
    "batch_id": "batch-20261003-advance-20261004-r07-conditional-probability-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1e5eb5cdd593629ece878598f818f23c2d94c9136b66d70a97573b5f2fabb0b9",
      "family_id": "qf-conditional-event-validity",
      "seed": 202610047064,
      "semantic_template": "qf-conditional-event-validity/{\"B\":[],\"F\":\"powerset(S)\",\"S\":[1,2,3,4],\"denominator\":\"22\",\"operation\":\"condition_validity\",\"weights\":[[1,\"1\"],[2,\"2\"],[3,\"5\"],[4,\"14\"]]}"
    },
    "fields": {
      "title": "Positive-Probability Eligibility of a Conditioning Event",
      "stem": "The nonempty finite sample space is S={1, 2, 3, 4}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [1/(22), 2/(22), 5/(22), 14/(22)], summing exactly to one. Does B=∅ meet the positive-probability requirement for defining P(A|B)?",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "The B mass numerator is 0, with positive common denominator 22. It must be strictly positive, giving No. A nonempty event of zero mass is still ineligible; undefined conditioning is not assigned a zero answer."
      },
      "skills": [
        "Positive-Probability Eligibility of a Conditioning Event"
      ],
      "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": "明示非空 S={1, 2, 3, 4}、事件族 F=𝒫(S)，依 S 順序的非負權重為 [1/22, 2/22, 5/22, 14/22]，總和恰為一。B=∅ 是否具備定義 P(A|B) 所需的正機率條件資格？",
        "en": "The nonempty finite sample space is S={1, 2, 3, 4}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [1/(22), 2/(22), 5/(22), 14/(22)], summing exactly to one. Does B=∅ meet the positive-probability requirement for defining P(A|B)?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "S={1, 2, 3, 4};no particular A supplied;B=∅;F=𝒫(S)",
      "after": "events are declared subsets;conditioning needs P(B)>0",
      "relation": "implies",
      "rule_id": "conditional_model_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "𝒫(S)是事件的全冪集，不是機率值；非空事件也可能機率0。只問B的條件化資格時，不另假設未給定的A。",
        "en": "𝒫(S) is the full event powerset,not a probability value.A nonempty event may have probability0.When checking B's eligibility,do not invent an unspecified A."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "total weight numerator := sum(1, 2, 5, 14)",
      "after": "total weight numerator = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負權重必須和共同正分母相同，才是總機率1。",
        "en": "Nonnegative weights must sum to the common positive denominator to give total probability1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 total weight numerator，不以小數近似替代。",
          "en": "Recompute total weight numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "2",
          "5",
          "14"
        ],
        "output": "22",
        "label": "total weight numerator"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "[[1, 1], [2, 2], [3, 5], [4, 14]]",
      "after": "weights sum to 22;normalization verified",
      "relation": "implies",
      "rule_id": "weight_alignment_and_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "權重依原S順序對齊，不可排序S卻把權重留在原位置。",
        "en": "Weights align with the original S order;do not sort S while leaving weights in their old positions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "B weight numerator := sum()",
      "after": "B weight numerator = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依事件內每個結果的對齊權重相加；零質量結果不變成正質量。",
        "en": "Sum aligned weights over the event;zero-mass outcomes do not become positive mass."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B weight numerator，不以小數近似替代。",
          "en": "Recompute B weight numerator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [],
        "output": "0",
        "label": "B weight numerator"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "B probability mass := div(0, 22)",
      "after": "B probability mass = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除以共同正分母得到精確機率。",
        "en": "Divide by the common positive denominator to obtain exact probability."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B probability mass，不以小數近似替代。",
          "en": "Recompute B probability mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "22"
        ],
        "output": "0",
        "label": "B probability mass"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "P(B)=0",
      "after": "conditioning eligible=否",
      "relation": "implies",
      "rule_id": "positive_conditioning_eligibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有P(B)>0才可用普通條件機率比值；P(B)=0時不把未定義的比值強行設為0。",
        "en": "The ordinary conditional ratio requires P(B)>0;when P(B)=0 do not assign0 to an undefined ratio."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": false,
    "conditioning_mass_B": "0"
  },
  "conclusion": {
    "canonical_value": "否",
    "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."
  },
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      "formalization": "not_formalized",
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      "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": {
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    "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.",
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    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}