Question 12991: A Zero-Mass Conditioning Event Still Has a Product Definition

Original question and givens

Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] 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. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

• G001 — Use one complete finite probability model with aligned exact masses=[18/49, 24/49, 3/49, 4/49, 0/1] on Fin(5)={0,…,4}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[4] 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. Require P(B)=0, allowing a nonempty B of zero-mass outcomes. Give (P_A,P_B,P_intersection,product,independent?,conditional_defined?,conditional_probability_undefined_use_product_definition).

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(18/49, 24/49, 3/49, 4/49, 0)

event model whole mass total = 1

complete carrier Fin(5);raw aligned masses=[[18, 49], [24, 49], [3, 49], [4, 49], [0, 1]]

normalized masses=[18/49, 24/49, 3/49, 4/49, 0];nonnegative=Yes;total=1

raw carrier events A=[2, 3];B=[4];carrier size=5

normalized A=[2, 3];B=[4];intersection=[]

A mass := sum(3/49, 4/49)

A mass = 1/7

B mass := sum(0)

B mass = 0

A intersection B mass := sum()

A intersection B mass = 0

event product P_A times P_B := mul(1/7, 0)

event product P_A times P_B = 0

P_A=1/7;P_B=0;P_intersection=0;product=0

two-event independent=Yes

P_B=0;A intersection B subset B;all atom masses nonnegative;P_A=1/7

0<=P_intersection<=P_B=0;product=0;independent=Yes;conditional ratio undefined

Reverse checks

• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(1/7, 0, 0, 0, Yes, No, conditional_probability_undefined_use_product_definition)

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.