Question 529: Zero Evidence Mass Makes This Conditional Probability Undefined

Original question and givens

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

• G001 — Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/11, 1/11, 1/11, 8/11], summing to 1, and canonical event evidence=[]. Event entries are increasing distinct outcome indices from 0 through 3. Every actual conditional probability requires a strictly positive denominator; a dedicated zero-denominator audit records undefined instead of dividing. Priors, likelihoods, intersections and evidence mass all come from this one law. Keep reduced exact fractions with positive denominators, omitting /1 for integers. Bayes conditioning does not imply causation. Hypothesis=[0, 1]. This dedicated boundary requires evidence mass exactly zero. Give (evidence_mass,conditional_value,defined?,conditional_probability_requires_positive_denominator).

Definition and scope

Conditional probability needs a positive denominator. With P(E)=0 it is undefined: no division, and not the value 0.

• mko-bayes-theorem

Step-by-step derivation

P(Ω) := sum(1/11, 1/11, 1/11, 8/11)

P(Ω) = 1

P(E) := sum()

P(E) = 0

P(E)=0

P(H|E)=P(H∩E)/P(E) is not defined here

zero conditioning mass

undefined is not zero, not infinity, and not an empty event

positive-denominator requirement fails

defined=false; value=undefined

Reverse checks

• S005 · premise_and_result_audit — Align events, intersections and conditioning denominators under the same law; do not divide by zero mass.
• S004 · premise_and_result_audit — Align events, intersections and conditioning denominators under the same law; do not divide by zero mass.
• S003 · premise_and_result_audit — Align events, intersections and conditioning denominators under the same law; do not divide by zero mass.
• S002 · premise_and_result_audit — Recompute P(E) with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute P(Ω) with exact rational arithmetic, not rounded decimals.

Result

(0, undefined, No, conditional_probability_requires_positive_denominator)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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.