{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-r07-probability-0020",
  "pilot_number": "R0055-28370",
  "number": 28370,
  "family_id": "qf-probability-uniform-event",
  "title": {
    "zh-Hant": "第 28370 題：明示等可能的事件機率",
    "en": "Question 28370: Event Probability with Explicit Equiprobability"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-r07-probability-0020",
    "batch_id": "batch-20261003-r07-probability-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-probability"
    ],
    "title_zh": "明示等可能的事件機率",
    "stem_zh": "非空有限樣本空間 S={7, 8, 9, 10, 11, 12}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={7, 8, 9, 10, 11, 12} 的精確 P(E) 是多少？",
    "answer": {
      "kind": "expression",
      "value": "1/1",
      "explanation_zh": "P(E)=1/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "明示等可能的事件機率"
      ]
    },
    "variant": {
      "family_id": "qf-probability-uniform-event",
      "seed": 202610037020,
      "parameters": {
        "S": [
          7,
          8,
          9,
          10,
          11,
          12
        ],
        "E": [
          7,
          8,
          9,
          10,
          11,
          12
        ],
        "F": "powerset(S)",
        "equiprobable": true,
        "operation": "uniform_event",
        "semantic_template": "qf-probability-uniform-event/{\"E\":[7,8,9,10,11,12],\"F\":\"powerset(S)\",\"S\":[7,8,9,10,11,12],\"equiprobable\":true,\"operation\":\"uniform_event\"}"
      }
    },
    "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-0020",
    "batch_id": "batch-20261003-r07-probability-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4cfe5921a6c4b04bcfc52882534fe1aaab77526400a86230f7030c363e5f2821",
      "family_id": "qf-probability-uniform-event",
      "seed": 202610037020,
      "semantic_template": "qf-probability-uniform-event/{\"E\":[7,8,9,10,11,12],\"F\":\"powerset(S)\",\"S\":[7,8,9,10,11,12],\"equiprobable\":true,\"operation\":\"uniform_event\"}"
    },
    "fields": {
      "title": "Event Probability with Explicit Equiprobability",
      "stem": "The nonempty finite sample space is S={7, 8, 9, 10, 11, 12} and the event family is F=𝒫(S). All elementary outcomes are explicitly equiprobable. For E={7, 8, 9, 10, 11, 12}, find P(E) exactly as a reduced fraction with positive denominator.",
      "answer": {
        "kind": "expression",
        "value": "1/1",
        "explanation": "Explicit equiprobability permits |E|/|S|=6/6=1/1. Here 𝒫 denotes the full powerset and P denotes the probability law."
      },
      "skills": [
        "Event Probability with Explicit 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={7, 8, 9, 10, 11, 12}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={7, 8, 9, 10, 11, 12} 的精確 P(E) 是多少？",
        "en": "The nonempty finite sample space is S={7, 8, 9, 10, 11, 12} and the event family is F=𝒫(S). All elementary outcomes are explicitly equiprobable. For E={7, 8, 9, 10, 11, 12}, find P(E) exactly as a reduced fraction with positive denominator."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-probability-uniform-event, uniform_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 = [[7, 8, 9, 10, 11, 12], powerset(S)]",
      "after": "derived = [[7, 8, 9, 10, 11, 12], 6]",
      "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 uniform ORIGINAL event count over carrier count := div(6, 6)",
      "after": "Legacy uniform ORIGINAL event count over carrier count = 1",
      "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 uniform ORIGINAL event count over carrier count，不以小數近似替代。",
          "en": "Recompute Legacy uniform ORIGINAL event count over carrier count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "6"
        ],
        "output": "1",
        "label": "Legacy uniform ORIGINAL event count over carrier count"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [[7, 8, 9, 10, 11, 12], [7, 8, 9, 10, 11, 12], 是]",
      "after": "derived = [1/1]",
      "relation": "implies",
      "rule_id": "Legacy_uniform_requires_EXACT_equiprobability_not_cardinality_guess",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有exactBoolean true的等可能前提才可用|E|/|S|；空事件合法0/1，全集1/1，整數分數保留/1。",
        "en": "Only the exactBoolean true equiprobability premise permits |E|/|S|.An empty event gives0/1,the whole space1/1,and integer fractions retain/1."
      },
      "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": "1/1"
  },
  "conclusion": {
    "canonical_value": "1/1",
    "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": 20,
      "original_line_sha256": "fda114553bb74a2e829c4b39243c2cb102bb0c07f8a37659abd138c48f955d38",
      "question_compact_sha256": "4cfe5921a6c4b04bcfc52882534fe1aaab77526400a86230f7030c363e5f2821",
      "english_row_compact_sha256": "43440e42230959b3e795d1ec304ce3a501514601893a48e2431aff106fb7d091",
      "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": "7881682e83365e2e76dbe369ac45cd948e5850065de07e0feef2ddb48fe00962",
    "original_candidate_line_sha256": "7881682e83365e2e76dbe369ac45cd948e5850065de07e0feef2ddb48fe00962"
  },
  "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": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "77ad907044c6a347ebfe2a7f6048fef59f02952971417b21f490e2c1cbc98636",
      "arithmetic_steps": 1
    },
    "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."
  }
}