第 21933 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":[[1,17],[2,17],[14,17],[0,1]],"values":[7,8,10,106],"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":[[1,17],[2,17],[14,17],[0,1]],"values":[7,8,10,106],"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(1/17, 2/17, 14/17, 0)

Total expectation complete original mass = 1

Total expectation original atom0 weighted X := mul(1/17, 7)

Total expectation original atom0 weighted X = 7/17

Total expectation original atom1 weighted X := mul(2/17, 8)

Total expectation original atom1 weighted X = 16/17

Total expectation original atom2 weighted X := mul(14/17, 10)

Total expectation original atom2 weighted X = 140/17

Total expectation original atom3 weighted X := mul(0, 106)

Total expectation original atom3 weighted X = 0

Total expectation full original mean := sum(7/17, 16/17, 140/17, 0)

Total expectation full original mean = 163/17

ORIGINAL masses=[1/17, 2/17, 14/17, 0],SIGNED values X=[7, 8, 10, 106];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=163/17;NO independence premise

Total expectation cell0 mass := sum(1/17, 2/17)

Total expectation cell0 mass = 3/17

Total expectation cell0 numerator := sum(7/17, 16/17)

Total expectation cell0 numerator = 23/17

Total expectation cell0 positive-mass conditional mean := div(23/17, 3/17)

Total expectation cell0 positive-mass conditional mean = 23/3

ORIGINAL cell index=0,atom labels=[0, 1];mass=3/17;weighted numerator=23/17

conditional quotient=23/3;positive denominator qualifies division

Total expectation cell1 mass := sum(14/17, 0)

Total expectation cell1 mass = 14/17

Total expectation cell1 numerator := sum(140/17, 0)

Total expectation cell1 numerator = 140/17

Total expectation cell1 positive-mass conditional mean := div(140/17, 14/17)

Total expectation cell1 positive-mass conditional mean = 10

ORIGINAL cell index=1,atom labels=[2, 3];mass=14/17;weighted numerator=140/17

conditional quotient=10;positive denominator qualifies division

Total expectation positive cell0 weighted mean := mul(3/17, 23/3)

Total expectation positive cell0 weighted mean = 23/17

Total expectation positive cell1 weighted mean := mul(14/17, 10)

Total expectation positive cell1 weighted mean = 140/17

Total expectation sum positive-cell weighted means := sum(23/17, 140/17)

Total expectation sum positive-cell weighted means = 163/17

ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 3/17, 23/3, 23/17], [1, [2, 3], 14/17, 10, 140/17]]

positive-cell weighted sum=163/17;ORIGINAL global mean=163/17;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, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, 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。