Question 14014: A Selected Pair of Value Events under the Same Joint Law

Original question and givens

Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[26], y_event=[3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

• G001 — Use the complete joint grid with original x_values=[1, 26], y_values=[3, 33], and aligned row-by-column exact masses=[[12/49, 9/49], [16/49, 12/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[26], y_event=[3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).

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-random-variable-independence

Step-by-step derivation

joint grid model whole mass sum := sum(12/49, 9/49, 16/49, 12/49)

joint grid model whole mass sum = 1

joint grid model:raw aligned masses=[[12, 49], [9, 49], [16, 49], [12, 49]]

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

original x row labels=[1, 26];original y column labels=[3, 33]

normalized aligned row-by-column joint grid=[[12/49, 9/49], [16/49, 12/49]]

X row0 marginal := sum(12/49, 9/49)

X row0 marginal = 3/7

X row1 marginal := sum(16/49, 12/49)

X row1 marginal = 4/7

Y column0 marginal := sum(12/49, 16/49)

Y column0 marginal = 4/7

Y column1 marginal := sum(9/49, 12/49)

Y column1 marginal = 3/7

raw x event=[26];raw y event=[3]

normalized events=[[26], [3]];original matching rows=[1];columns=[0]

selected X event mass := sum(4/7)

selected X event mass = 4/7

selected Y event mass := sum(4/7)

selected Y event mass = 4/7

selected joint event rectangle mass := sum(16/49)

selected joint event rectangle mass = 16/49

selected event marginal product := mul(4/7, 4/7)

selected event marginal product = 16/49

P_X_event=4/7;P_Y_event=4/7;P_joint=16/49;product=16/49

selected event pair factorizes=Yes;not an all-event independence certificate

Reverse checks

• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute selected event marginal product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute selected joint event rectangle mass with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute selected Y event mass with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute selected X event 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 Y column1 marginal with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Y column0 marginal with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute X row1 marginal with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute X row0 marginal 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 joint grid model whole mass sum with exact rational arithmetic, not rounded decimals.

Result

([26], [3], 4/7, 4/7, 16/49, 16/49, 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.