第 21942 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":[[1,26],[2,26],[23,26],[0,1]],"values":[16,17,19,115],"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":[[1,26],[2,26],[23,26],[0,1]],"values":[16,17,19,115],"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(1/26, 1/13, 23/26, 0)

Total expectation complete original mass = 1

Total expectation original atom0 weighted X := mul(1/26, 16)

Total expectation original atom0 weighted X = 8/13

Total expectation original atom1 weighted X := mul(1/13, 17)

Total expectation original atom1 weighted X = 17/13

Total expectation original atom2 weighted X := mul(23/26, 19)

Total expectation original atom2 weighted X = 437/26

Total expectation original atom3 weighted X := mul(0, 115)

Total expectation original atom3 weighted X = 0

Total expectation full original mean := sum(8/13, 17/13, 437/26, 0)

Total expectation full original mean = 487/26

ORIGINAL masses=[1/26, 1/13, 23/26, 0],SIGNED values X=[16, 17, 19, 115];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=487/26;NO independence premise

Total expectation cell0 mass := sum(1/26, 1/13)

Total expectation cell0 mass = 3/26

Total expectation cell0 numerator := sum(8/13, 17/13)

Total expectation cell0 numerator = 25/13

Total expectation cell0 positive-mass conditional mean := div(25/13, 3/26)

Total expectation cell0 positive-mass conditional mean = 50/3

ORIGINAL cell index=0,atom labels=[0, 1];mass=3/26;weighted numerator=25/13

conditional quotient=50/3;positive denominator qualifies division

Total expectation cell1 mass := sum(23/26, 0)

Total expectation cell1 mass = 23/26

Total expectation cell1 numerator := sum(437/26, 0)

Total expectation cell1 numerator = 437/26

Total expectation cell1 positive-mass conditional mean := div(437/26, 23/26)

Total expectation cell1 positive-mass conditional mean = 19

ORIGINAL cell index=1,atom labels=[2, 3];mass=23/26;weighted numerator=437/26

conditional quotient=19;positive denominator qualifies division

Total expectation positive cell0 weighted mean := mul(3/26, 50/3)

Total expectation positive cell0 weighted mean = 25/13

Total expectation positive cell1 weighted mean := mul(23/26, 19)

Total expectation positive cell1 weighted mean = 437/26

Total expectation sum positive-cell weighted means := sum(25/13, 437/26)

Total expectation sum positive-cell weighted means = 487/26

ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 3/26, 50/3, 25/13], [1, [2, 3], 23/26, 19, 437/26]]

positive-cell weighted sum=487/26;ORIGINAL global mean=487/26;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, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, 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。