第 21903 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-partition-audit；資料={"masses":[[1,12],[2,12],[9,12],[0,1]],"values":[2,3,5,101],"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-partition-audit；資料={"masses":[[1,12],[2,12],[9,12],[0,1]],"values":[2,3,5,101],"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-partition-audit;original snapshot and mathematics retained

explicit ordered source output fields=[partition, cell_masses, complete_disjoint_certificate, mean]

Total expectation complete original mass := sum(1/12, 1/6, 3/4, 0)

Total expectation complete original mass = 1

Total expectation original atom0 weighted X := mul(1/12, 2)

Total expectation original atom0 weighted X = 1/6

Total expectation original atom1 weighted X := mul(1/6, 3)

Total expectation original atom1 weighted X = 1/2

Total expectation original atom2 weighted X := mul(3/4, 5)

Total expectation original atom2 weighted X = 15/4

Total expectation original atom3 weighted X := mul(0, 101)

Total expectation original atom3 weighted X = 0

Total expectation full original mean := sum(1/6, 1/2, 15/4, 0)

Total expectation full original mean = 53/12

ORIGINAL masses=[1/12, 1/6, 3/4, 0],SIGNED values X=[2, 3, 5, 101];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=53/12;NO independence premise

Total expectation cell0 mass := sum(1/12, 1/6)

Total expectation cell0 mass = 1/4

Total expectation cell0 numerator := sum(1/6, 1/2)

Total expectation cell0 numerator = 2/3

Total expectation cell0 positive-mass conditional mean := div(2/3, 1/4)

Total expectation cell0 positive-mass conditional mean = 8/3

ORIGINAL cell index=0,atom labels=[0, 1];mass=1/4;weighted numerator=2/3

conditional quotient=8/3;positive denominator qualifies division

Total expectation cell1 mass := sum(3/4, 0)

Total expectation cell1 mass = 3/4

Total expectation cell1 numerator := sum(15/4, 0)

Total expectation cell1 numerator = 15/4

Total expectation cell1 positive-mass conditional mean := div(15/4, 3/4)

Total expectation cell1 positive-mass conditional mean = 5

ORIGINAL cell index=1,atom labels=[2, 3];mass=3/4;weighted numerator=15/4

conditional quotient=5;positive denominator qualifies division

逐步反向核對

• 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], [2, 3]], [[0, 1/4], [1, 3/4]], 是, 53/12)

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。