Question 12948: Audit Complement Independence without Assuming the Original Flag

Original question and givens

Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

• G001 — Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Take A_complement in the same full carrier. Give (original_independent?,A_complement,P_complement,P_complement_intersection_B,product,complement_independent?).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-event-independence

Step-by-step derivation

event model whole mass total := sum(8/49, 20/49, 6/49, 15/49)

event model whole mass total = 1

complete carrier Fin(4);raw aligned masses=[[8, 49], [20, 49], [6, 49], [15, 49]]

normalized masses=[8/49, 20/49, 6/49, 15/49];nonnegative=Yes;total=1

raw carrier events A=[2, 3];B=[1, 3];carrier size=4

normalized A=[2, 3];B=[1, 3];intersection=[3]

A mass := sum(6/49, 15/49)

A mass = 3/7

B mass := sum(20/49, 15/49)

B mass = 5/7

A intersection B mass := sum(15/49)

A intersection B mass = 15/49

event product P_A times P_B := mul(3/7, 5/7)

event product P_A times P_B = 15/49

P_A=3/7;P_B=5/7;P_intersection=15/49;product=15/49

two-event independent=Yes

A complement mass := sum(8/49, 20/49)

A complement mass = 4/7

A complement intersection B mass := sum(20/49)

A complement intersection B mass = 20/49

complement product P_complement times P_B := mul(4/7, 5/7)

complement product P_complement times P_B = 20/49

original independence residual := sub(15/49, 15/49)

original independence residual = 0

complement independence residual := sub(20/49, 20/49)

complement independence residual = 0

full carrier=[0, 1, 2, 3];A complement=[0, 1];P_Ac=4/7;P_AcB=20/49

P_Ac=1-P_A;P_AcB=P_B-P_AB;complement residual=0=-(original residual=0)

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute complement independence residual with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute original independence residual with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute complement product P_complement times P_B with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute A complement intersection B mass with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute A complement mass with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute event product P_A times P_B with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute A intersection B mass with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute B mass with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute A mass with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute event model whole mass total with exact rational arithmetic, not rounded decimals.

Result

(Yes, [0, 1], 4/7, 20/49, 20/49, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.