{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-probability",
  "semantic_family_id": "sf-probability",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限樣本空間上的機率"
  },
  "summary": {
    "zh-Hant": "在非空有限樣本空間上指定非負且總和為一的權重，事件機率為其成員權重之和。"
  },
  "statement": {
    "zh-Hant": "令 S 為非空有限樣本空間，事件族 F=𝒫(S)，其中 𝒫 表示全冪集、P 表示機率律。指定各 s∈S 的有理權重 w_s≥0 且 Σ_(s∈S)w_s=1，則事件 E⊆S 的機率 P(E)=Σ_(s∈E)w_s。只有另明示所有結果等可能，才可用 P(E)=|E|/|S|。"
  },
  "formula": {
    "tex": "P(E)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "E",
          "glyph": "E"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "73cc253edb074a732a19d68da0db4f6eb0133f1a4dacf15ee31104156708ef3e"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "S 有限、非空、基本結果互異；事件族選定全冪集。"
    },
    {
      "id": "a2",
      "zh-Hant": "權重為精確有理數，均非負且總和恰為一；事件 E 必須為 S 的子集合。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "finite_probability_law",
      "role_zh": "有限機率律",
      "scope": "representative_formula"
    },
    {
      "token": "E",
      "role": "event_subset",
      "role_zh": "樣本空間的事件子集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-event",
      "title_zh": "事件",
      "reason_zh": "先固定樣本空間與事件子集合，不把結果和事件混同。"
    },
    {
      "id": "mko-ratio-proportion",
      "title_zh": "比與比例",
      "reason_zh": "等可能與有理權重計算需要合法非零比值。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限模型必須先驗證全部權重及其總和，再相加事件成員的權重。事件基數不是一般機率；例如不同基本結果可具有不同權重。",
      "P(∅)=0、P(S)=1，補事件相對同一個 S 定義，P(S∖E)=1−P(E)。非空事件也可能機率為零，只要其中各基本結果權重均為零。",
      "本輪用明示整數結果標籤及共分母有理權重作精確計算與模型合法性檢查。不推論真實實驗的機率、不建立獨立性或條件機率，也不擴張為一般測度論定義。"
    ],
    "common_misconception_zh": "未聲明等可能就套有利結果數除以總結果數，或以 P(E)=0 推出 E 為空。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示有限有理權重可完整重算；一般機率測度與可數可加性形式 Evidence 尚未建立。",
    "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": "綁定有限 PMF、有理權重歸一化、事件和與相對補事件；測度層另開語義橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-probability.md 整理；2026-10-03 脈衝 R07，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}