{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261027-r06-event-independence-0012",
  "pilot_number": "R0024-12912",
  "number": 12912,
  "family_id": "qf-mod27-eventind-audit",
  "title": {
    "zh-Hant": "第 12912 題：事件獨立",
    "en": "Question 12912: Test Independence by the Exact Product Definition"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261027-r06-event-independence-0012",
    "batch_id": "batch-20261014-advance-20261027-r06-event-independence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-event-independence"
    ],
    "title_zh": "事件獨立",
    "stem_zh": "同一完整有限機率模型中的兩事件。操作=eventind-audit；資料={\"masses\":[[20,49],[15,49],[8,49],[6,49]],\"A\":[2,3],\"B\":[1,3]}。請依 [A,B,P_A,P_B,P_intersection,product,independent] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, 是)",
      "explanation_zh": "Wave4根節點核對=([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, 是)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；主資格是P(A交B)=P(A)P(B)；條件式只在正分母使用且pairwise不冒充mutual。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "事件獨立"
      ]
    },
    "variant": {
      "family_id": "qf-mod27-eventind-audit",
      "seed": 202610276012,
      "parameters": {
        "masses": [
          [
            20,
            49
          ],
          [
            15,
            49
          ],
          [
            8,
            49
          ],
          [
            6,
            49
          ]
        ],
        "A": [
          2,
          3
        ],
        "B": [
          1,
          3
        ],
        "domain": "Exact_finite_event_independence",
        "operation": "eventind-audit",
        "semantic_template": "qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[15,49],[8,49],[6,49]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式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-20261027-r06-event-independence-0012",
    "batch_id": "batch-20261014-advance-20261027-r06-event-independence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "454c300c366ef70afa825b1bfbd213adc9d3b436aa1f2d63537299d92cf280c0",
      "family_id": "qf-mod27-eventind-audit",
      "seed": 202610276012,
      "semantic_template": "qf-mod27-eventind-audit/{\"A\":[2,3],\"B\":[1,3],\"masses\":[[20,49],[15,49],[8,49],[6,49]]}"
    },
    "fields": {
      "title": "Test Independence by the Exact Product Definition",
      "stem": "Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).",
      "answer": {
        "kind": "expression",
        "value": "([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, Yes)",
        "explanation": "Sum the masses of A, B and their intersection separately, then compare the intersection mass with the exact product. Disjointness is not the same as independence. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, Yes)."
      },
      "skills": [
        "Test Independence by the Exact Product Definition"
      ],
      "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": "同一完整有限機率模型中的兩事件。操作=eventind-audit；資料={\"masses\":[[20,49],[15,49],[8,49],[6,49]],\"A\":[2,3],\"B\":[1,3]}。請依 [A,B,P_A,P_B,P_intersection,product,independent] 欄序回答。",
        "en": "Use one complete finite probability model with aligned exact masses=[20/49, 15/49, 8/49, 6/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "event model whole mass total := sum(20/49, 15/49, 8/49, 6/49)",
      "after": "event model whole mass total = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 event model whole mass total，不以小數近似替代。",
          "en": "Recompute event model whole mass total with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20/49",
          "15/49",
          "8/49",
          "6/49"
        ],
        "output": "1",
        "label": "event model whole mass total"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "complete carrier Fin(4);raw aligned masses=[[20, 49], [15, 49], [8, 49], [6, 49]]",
      "after": "normalized masses=[20/49, 15/49, 8/49, 6/49];nonnegative=是;total=1",
      "relation": "implies",
      "rule_id": "event_common_complete_probability_space",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先合格同一完整有限PMF，每個原子非負、總和恰為1；零質量原子仍是載體的一部分，不能只以正質量support取補集。",
        "en": "Qualify one complete finite PMF with nonnegative atoms and exact total1.Zero-mass atoms remain in the carrier;complements are not taken only within positive-mass support."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "raw carrier events A=[2, 3];B=[1, 3];carrier size=4",
      "after": "normalized A=[2, 3];B=[1, 3];intersection=[3]",
      "relation": "implies",
      "rule_id": "event_carrier_label_sets_and_intersection",
      "conditions": [],
      "explanation": {
        "zh-Hant": "事件用載體索引集合，去重排序並保留零質量標籤；交集要同一標籤同時在A與B。這與聯合表的座標值事件不同。",
        "en": "Events are carrier-index sets,normalized without dropping zero-mass labels.Intersection requires the same label in A and B.This differs from coordinate-value events in a joint table."
      },
      "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": "A mass := sum(8/49, 6/49)",
      "after": "A mass = 2/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A mass，不以小數近似替代。",
          "en": "Recompute A mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8/49",
          "6/49"
        ],
        "output": "2/7",
        "label": "A mass"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "B mass := sum(15/49, 6/49)",
      "after": "B mass = 3/7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B mass，不以小數近似替代。",
          "en": "Recompute B mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "15/49",
          "6/49"
        ],
        "output": "3/7",
        "label": "B mass"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "A intersection B mass := sum(6/49)",
      "after": "A intersection B mass = 6/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A intersection B mass，不以小數近似替代。",
          "en": "Recompute A intersection B mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "6/49"
        ],
        "output": "6/49",
        "label": "A intersection B mass"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "event product P_A times P_B := mul(2/7, 3/7)",
      "after": "event product P_A times P_B = 6/49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與完整已知操作數精確計算；不改配對、次序或以近似代替條件。",
        "en": "Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 event product P_A times P_B，不以小數近似替代。",
          "en": "Recompute event product P_A times P_B with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2/7",
          "3/7"
        ],
        "output": "6/49",
        "label": "event product P_A times P_B"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "P_A=2/7;P_B=3/7;P_intersection=6/49;product=6/49",
      "after": "two-event independent=是",
      "relation": "implies",
      "rule_id": "independence_primary_product_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "主定義是交集機率等於邊際乘積，不是互斥，也不先假設獨立再相乘。此處只核兩事件，不是更多事件的互獨立。",
        "en": "The primary definition is intersection probability equal to the marginal product,not disjointness.Do not assume independence before multiplying.This checks two events,not mutual independence of a larger family."
      },
      "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": [
      [
        2,
        3
      ],
      [
        1,
        3
      ],
      "2/7",
      "3/7",
      "6/49",
      "6/49",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([2, 3], [1, 3], 2/7, 3/7, 6/49, 6/49, 是)",
    "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": {
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        "original_path": "public/data/mko/event-independence.json",
        "english_path": "public/data/i18n/editions/mko-event-independence.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "17a80c8482f64498e5a88bd4368d0a11b92d45c259784a0a536a29fbb98662d5",
    "original_candidate_sha256": "9d31e6c7d1b216cd13665fa8e049aed500af36c069b85202487e0759f61c72db",
    "original_candidate_line_sha256": "9d31e6c7d1b216cd13665fa8e049aed500af36c069b85202487e0759f61c72db"
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  "historical_source_state": {
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      "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-20261026-006-p25",
    "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": {
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}