{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-event-independence",
  "semantic_family_id": "sf-event-independence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "事件獨立"
  },
  "summary": {
    "zh-Hant": "兩事件獨立以交集機率等於邊際機率乘積定義；此定義不依賴零機率條件式。"
  },
  "statement": {
    "zh-Hant": "在同一機率空間中，A與B獨立當且僅當P(A交B)=P(A)P(B)。若P(B)>0，這等價於P(A|B)=P(A)；P(B)=0時不得以未定義條件機率取代乘積定義。"
  },
  "formula": {
    "tex": "I(A,B)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>I</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "I",
        "glyph": "I"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        },
        {
          "type": "symbol",
          "name": "B",
          "glyph": "B"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "76e93527fd1ea07416cbdd8c771078accc792da3d83fe169a52c54938e5113a8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "樣本空間、精確有理質量及事件子集完整明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪只處理兩事件；pairwise independence不冒充三事件mutual independence。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "event",
      "role_zh": "第一事件",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "event",
      "role_zh": "第二事件",
      "scope": "representative_formula"
    },
    {
      "token": "P",
      "role": "probability_measure",
      "role_zh": "同一機率測度",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-event",
      "title_zh": "事件",
      "reason_zh": "A、B及交集皆為同一事件空間元素。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "乘積定義依同一機率測度。"
    },
    {
      "id": "mko-conditional-probability",
      "title_zh": "條件機率",
      "reason_zh": "條件式等價只在分母正時使用。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "主資格永遠用交集乘積等式。",
      "正分母時可核條件式等價。",
      "事件與其補集的獨立性可由同一等式推導。"
    ],
    "common_misconception_zh": "把互斥當獨立、在P(B)=0時硬算條件機率，或由pairwise independence推出mutual independence。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核同一完整有限機率模型中的兩事件；不外推事件族mutual independence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入隨機變數獨立。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-event-independence.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-10-27 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}