第 21940 題：全期望公式

原題與已知條件

同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={"masses":[[1,24],[2,24],[21,24],[0,1]],"values":[14,15,17,113],"partition":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={"masses":[[1,24],[2,24],[21,24],[0,1]],"values":[14,15,17,113],"partition":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

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。

• mko-law-of-total-expectation

逐步推導

source operation=totalexp-positive-cells-formula;original snapshot and mathematics retained

explicit ordered source output fields=[cell_rows, weighted_sum, mean, equal]

Total expectation complete original mass := sum(1/24, 1/12, 7/8, 0)

Total expectation complete original mass = 1

Total expectation original atom0 weighted X := mul(1/24, 14)

Total expectation original atom0 weighted X = 7/12

Total expectation original atom1 weighted X := mul(1/12, 15)

Total expectation original atom1 weighted X = 5/4

Total expectation original atom2 weighted X := mul(7/8, 17)

Total expectation original atom2 weighted X = 119/8

Total expectation original atom3 weighted X := mul(0, 113)

Total expectation original atom3 weighted X = 0

Total expectation full original mean := sum(7/12, 5/4, 119/8, 0)

Total expectation full original mean = 401/24

ORIGINAL masses=[1/24, 1/12, 7/8, 0],SIGNED values X=[14, 15, 17, 113];ORIGINAL cell order=[[0, 1], [2, 3]]

flattened original atom labels=[0, 1, 2, 3] cover EVERY atom exactly ONCE including null atoms;total mass=1;integrable finite mean=401/24;NO independence premise

Total expectation cell0 mass := sum(1/24, 1/12)

Total expectation cell0 mass = 1/8

Total expectation cell0 numerator := sum(7/12, 5/4)

Total expectation cell0 numerator = 11/6

Total expectation cell0 positive-mass conditional mean := div(11/6, 1/8)

Total expectation cell0 positive-mass conditional mean = 44/3

ORIGINAL cell index=0,atom labels=[0, 1];mass=1/8;weighted numerator=11/6

conditional quotient=44/3;positive denominator qualifies division

Total expectation cell1 mass := sum(7/8, 0)

Total expectation cell1 mass = 7/8

Total expectation cell1 numerator := sum(119/8, 0)

Total expectation cell1 numerator = 119/8

Total expectation cell1 positive-mass conditional mean := div(119/8, 7/8)

Total expectation cell1 positive-mass conditional mean = 17

ORIGINAL cell index=1,atom labels=[2, 3];mass=7/8;weighted numerator=119/8

conditional quotient=17;positive denominator qualifies division

Total expectation positive cell0 weighted mean := mul(1/8, 44/3)

Total expectation positive cell0 weighted mean = 11/6

Total expectation positive cell1 weighted mean := mul(7/8, 17)

Total expectation positive cell1 weighted mean = 119/8

Total expectation sum positive-cell weighted means := sum(11/6, 119/8)

Total expectation sum positive-cell weighted means = 401/24

ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 1/8, 44/3, 11/6], [1, [2, 3], 7/8, 17, 119/8]]

positive-cell weighted sum=401/24;ORIGINAL global mean=401/24;equal=是;disjoint complete cells regroup EVERY original p_i*X_i exactly ONCE,null contributions0

逐步反向核對

• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Total expectation sum positive-cell weighted means，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Total expectation positive cell1 weighted mean，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Total expectation positive cell0 weighted mean，不以小數近似替代。
• S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 以精確有理數重算 Total expectation cell1 positive-mass conditional mean，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Total expectation cell1 numerator，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Total expectation cell1 mass，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Total expectation cell0 positive-mass conditional mean，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Total expectation cell0 numerator，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Total expectation cell0 mass，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 Total expectation full original mean，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom3 weighted X，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom2 weighted X，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom1 weighted X，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Total expectation original atom0 weighted X，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Total expectation complete original mass，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, 是)

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。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。