第 21921 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":[[1,30],[2,30],[27,30],[0,1]],"values":[20,21,23,119],"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":[[1,30],[2,30],[27,30],[0,1]],"values":[20,21,23,119],"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(1/30, 1/15, 9/10, 0)

Total expectation complete original mass = 1

Total expectation original atom0 weighted X := mul(1/30, 20)

Total expectation original atom0 weighted X = 2/3

Total expectation original atom1 weighted X := mul(1/15, 21)

Total expectation original atom1 weighted X = 7/5

Total expectation original atom2 weighted X := mul(9/10, 23)

Total expectation original atom2 weighted X = 207/10

Total expectation original atom3 weighted X := mul(0, 119)

Total expectation original atom3 weighted X = 0

Total expectation full original mean := sum(2/3, 7/5, 207/10, 0)

Total expectation full original mean = 683/30

ORIGINAL masses=[1/30, 1/15, 9/10, 0],SIGNED values X=[20, 21, 23, 119];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=683/30;NO independence premise

Total expectation cell0 mass := sum(1/30, 1/15)

Total expectation cell0 mass = 1/10

Total expectation cell0 numerator := sum(2/3, 7/5)

Total expectation cell0 numerator = 31/15

Total expectation cell0 positive-mass conditional mean := div(31/15, 1/10)

Total expectation cell0 positive-mass conditional mean = 62/3

ORIGINAL cell index=0,atom labels=[0, 1];mass=1/10;weighted numerator=31/15

conditional quotient=62/3;positive denominator qualifies division

Total expectation cell1 mass := sum(9/10, 0)

Total expectation cell1 mass = 9/10

Total expectation cell1 numerator := sum(207/10, 0)

Total expectation cell1 numerator = 207/10

Total expectation cell1 positive-mass conditional mean := div(207/10, 9/10)

Total expectation cell1 positive-mass conditional mean = 23

ORIGINAL cell index=1,atom labels=[2, 3];mass=9/10;weighted numerator=207/10

conditional quotient=23;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, 1/10], [1, 9/10]], 是, 683/30)

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。