Question 12913: Test Independence by the Exact Product Definition

Original question and givens

Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 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. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?).

• G001 — Use one complete finite probability model with aligned exact masses=[16/49, 12/49, 12/49, 9/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[0, 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. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,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(16/49, 12/49, 12/49, 9/49)

event model whole mass total = 1

complete carrier Fin(4);raw aligned masses=[[16, 49], [12, 49], [12, 49], [9, 49]]

normalized masses=[16/49, 12/49, 12/49, 9/49];nonnegative=Yes;total=1

raw carrier events A=[2, 3];B=[0, 3];carrier size=4

normalized A=[2, 3];B=[0, 3];intersection=[3]

A mass := sum(12/49, 9/49)

A mass = 3/7

B mass := sum(16/49, 9/49)

B mass = 25/49

A intersection B mass := sum(9/49)

A intersection B mass = 9/49

event product P_A times P_B := mul(3/7, 25/49)

event product P_A times P_B = 75/343

P_A=3/7;P_B=25/49;P_intersection=9/49;product=75/343

two-event independent=No

Reverse checks

• 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

([2, 3], [0, 3], 3/7, 25/49, 9/49, 75/343, No)

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.