{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261102-r06-bayes-theorem-0040",
  "pilot_number": "P1000-0490",
  "number": 490,
  "family_id": "qf-mod33-bayes-partition-posterior",
  "title": {
    "zh-Hant": "第 490 題：Bayes定理",
    "en": "Question 490: Normalize Partition Hypotheses by the Same Total Evidence Mass"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261102-r06-bayes-theorem-0040",
    "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-partition-posterior；資料={\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)",
      "explanation_zh": "Wave5第二層核對=(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)；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": 2,
      "skills": [
        "Bayes定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod33-bayes-partition-posterior",
      "seed": 202611026040,
      "parameters": {
        "masses": [
          [
            1,
            22
          ],
          [
            1,
            22
          ],
          [
            1,
            22
          ],
          [
            19,
            22
          ]
        ],
        "hypotheses": [
          [
            0,
            1
          ],
          [
            2,
            3
          ]
        ],
        "evidence": [
          0,
          2
        ],
        "target_index": 0,
        "domain": "Exact_finite_atomic_Bayes",
        "operation": "bayes-partition-posterior",
        "semantic_template": "qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"target_index\":0}"
      }
    },
    "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-0040",
    "batch_id": "batch-20261014-advance-20261102-r06-bayes-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "73ba62de3f5a137cf745dc6a1fb3c0df4e7af7059f690cde0f8ca5fe31117aad",
      "family_id": "qf-mod33-bayes-partition-posterior",
      "seed": 202611026040,
      "semantic_template": "qf-mod33-bayes-partition-posterior/{\"evidence\":[0,2],\"hypotheses\":[[0,1],[2,3]],\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"target_index\":0}"
    },
    "fields": {
      "title": "Normalize Partition Hypotheses by the Same Total Evidence Mass",
      "stem": "Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. 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. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).",
      "answer": {
        "kind": "expression",
        "value": "(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)",
        "explanation": "Compute each prior-times-likelihood joint from the common atomic law. Their sum equals the positive evidence mass. Divide the selected joint by that total; do not omit hypotheses, priors or the partition obligations. The exact result, retaining the requested order, is (0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)."
      },
      "skills": [
        "Normalize Partition Hypotheses by the Same Total Evidence Mass"
      ],
      "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-partition-posterior；資料={\"masses\":[[1,22],[1,22],[1,22],[19,22]],\"hypotheses\":[[0,1],[2,3]],\"evidence\":[0,2],\"target_index\":0}。請依 [target_index,prior,likelihood,evidence_mass,posterior,partition_certificate] 欄序回答。",
        "en": "Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/22, 1/22, 1/22, 19/22], summing to 1, and canonical event evidence=[0, 2]. 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. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=0. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "P(Ω) := sum(1/22, 1/22, 1/22, 19/22)",
      "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": [
          "1/22",
          "1/22",
          "1/22",
          "19/22"
        ],
        "output": "1",
        "label": "P(Ω)"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "P(E) := sum(1/22, 1/22)",
      "after": "P(E) = 1/11",
      "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": [
          "1/22",
          "1/22"
        ],
        "output": "1/11",
        "label": "P(E)"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "H_i=[[0, 1], [2, 3]]",
      "after": "pairwise disjoint; union=Ω",
      "relation": "implies",
      "rule_id": "complete_partition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整partition使每個原子恰落入一個假說，分解E時不重複也不遺漏。",
        "en": "A complete partition assigns each atom to exactly one hypothesis, so decomposing E neither duplicates nor omits mass."
      },
      "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": "P(H_0) := sum(1/22, 1/22)",
      "after": "P(H_0) = 1/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "算出這個假說的prior。",
        "en": "Calculate this hypothesis's prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(H_0)，不以小數近似替代。",
          "en": "Recompute P(H_0) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/22",
          "1/22"
        ],
        "output": "1/11",
        "label": "P(H_0)"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "P(E∩H_0) := sum(1/22)",
      "after": "P(E∩H_0) = 1/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以交集事件算共同質量。",
        "en": "Calculate the joint mass from the event intersection."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(E∩H_0)，不以小數近似替代。",
          "en": "Recompute P(E∩H_0) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/22"
        ],
        "output": "1/22",
        "label": "P(E∩H_0)"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "P(E|H_0) := div(1/22, 1/11)",
      "after": "P(E|H_0) = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正prior才允許此條件商。",
        "en": "This quotient is licensed by the positive prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(E|H_0)，不以小數近似替代。",
          "en": "Recompute P(E|H_0) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1/22",
          "1/11"
        ],
        "output": "1/2",
        "label": "P(E|H_0)"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "P(E|H_0)P(H_0) := mul(1/2, 1/11)",
      "after": "P(E|H_0)P(H_0) = 1/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘回prior應恢復同一joint。",
        "en": "Multiplying by the prior must recover the same joint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(E|H_0)P(H_0)，不以小數近似替代。",
          "en": "Recompute P(E|H_0)P(H_0) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/11"
        ],
        "output": "1/22",
        "label": "P(E|H_0)P(H_0)"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "P(H_1) := sum(1/22, 19/22)",
      "after": "P(H_1) = 10/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "算出這個假說的prior。",
        "en": "Calculate this hypothesis's prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(H_1)，不以小數近似替代。",
          "en": "Recompute P(H_1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/22",
          "19/22"
        ],
        "output": "10/11",
        "label": "P(H_1)"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "P(E∩H_1) := sum(1/22)",
      "after": "P(E∩H_1) = 1/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以交集事件算共同質量。",
        "en": "Calculate the joint mass from the event intersection."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(E∩H_1)，不以小數近似替代。",
          "en": "Recompute P(E∩H_1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/22"
        ],
        "output": "1/22",
        "label": "P(E∩H_1)"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "P(E|H_1) := div(1/22, 10/11)",
      "after": "P(E|H_1) = 1/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正prior才允許此條件商。",
        "en": "This quotient is licensed by the positive prior."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(E|H_1)，不以小數近似替代。",
          "en": "Recompute P(E|H_1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1/22",
          "10/11"
        ],
        "output": "1/20",
        "label": "P(E|H_1)"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "P(E|H_1)P(H_1) := mul(1/20, 10/11)",
      "after": "P(E|H_1)P(H_1) = 1/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘回prior應恢復同一joint。",
        "en": "Multiplying by the prior must recover the same joint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(E|H_1)P(H_1)，不以小數近似替代。",
          "en": "Recompute P(E|H_1)P(H_1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/20",
          "10/11"
        ],
        "output": "1/22",
        "label": "P(E|H_1)P(H_1)"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "sum of partition joints := sum(1/22, 1/22)",
      "after": "sum of partition joints = 1/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "互斥且完備的交集把E完整分解。",
        "en": "The disjoint, exhaustive intersections decompose E completely."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum of partition joints，不以小數近似替代。",
          "en": "Recompute sum of partition joints with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1/22",
          "1/22"
        ],
        "output": "1/11",
        "label": "sum of partition joints"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "partition sum=1/11; direct P(E)=1/11",
      "after": "equal: 是",
      "relation": "implies",
      "rule_id": "total_probability",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先用全機率確認分母，不從後驗答案反推。",
        "en": "Verify the denominator by total probability before using a posterior answer."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "P(H_0|E) := div(1/22, 1/11)",
      "after": "P(H_0|E) = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "取指定假說的joint除以完整evidence質量。",
        "en": "Divide the target joint by the full evidence mass."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 P(H_0|E)，不以小數近似替代。",
          "en": "Recompute P(H_0|E) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1/22",
          "1/11"
        ],
        "output": "1/2",
        "label": "P(H_0|E)"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "posterior·P(E) := mul(1/2, 1/11)",
      "after": "posterior·P(E) = 1/22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "反向乘回核對指定假說的joint。",
        "en": "Multiply back to verify the target joint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 posterior·P(E)，不以小數近似替代。",
          "en": "Recompute posterior·P(E) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1/2",
          "1/11"
        ],
        "output": "1/22",
        "label": "posterior·P(E)"
      }
    }
  ],
  "math_model": {
    "result": [
      0,
      "1/11",
      "1/2",
      "1/11",
      "1/2",
      "partition_pairwise_disjoint_exhaustive"
    ]
  },
  "conclusion": {
    "canonical_value": "(0, 1/11, 1/2, 1/11, 1/2, partition_pairwise_disjoint_exhaustive)",
    "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": "同一完整有限機率律；假說須互斥且完備、先驗與證據質量為正，後驗＝聯合質量除以 P(E)。",
    "en": "One complete finite law. Hypotheses are disjoint and exhaustive, priors and evidence mass are positive; posterior = joint mass / P(E)."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261102-r06-bayes-theorem-foundations",
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