Question 489: Normalize Partition Hypotheses by the Same Total Evidence Mass

Original question and givens

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. 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. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

• G001 — Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/21, 1/21, 1/21, 18/21], summing to 1, and canonical event evidence=[0, 2]. 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. Ordered hypotheses=[[0, 1], [2, 3]] must be pairwise disjoint and cover every outcome. Each hypothesis has positive prior in this family; target_index=1. Give (target_index,prior,likelihood,evidence_mass,posterior,partition_pairwise_disjoint_exhaustive).

Definition and scope

One complete finite law. Hypotheses are disjoint and exhaustive, priors and evidence mass are positive; posterior = joint mass / P(E).

• mko-bayes-theorem

Step-by-step derivation

P(Ω) := sum(1/21, 1/21, 1/21, 6/7)

P(Ω) = 1

P(E) := sum(1/21, 1/21)

P(E) = 2/21

H_i=[[0, 1], [2, 3]]

pairwise disjoint; union=Ω

P(H_0) := sum(1/21, 1/21)

P(H_0) = 2/21

P(E∩H_0) := sum(1/21)

P(E∩H_0) = 1/21

P(E|H_0) := div(1/21, 2/21)

P(E|H_0) = 1/2

P(E|H_0)P(H_0) := mul(1/2, 2/21)

P(E|H_0)P(H_0) = 1/21

P(H_1) := sum(1/21, 6/7)

P(H_1) = 19/21

P(E∩H_1) := sum(1/21)

P(E∩H_1) = 1/21

P(E|H_1) := div(1/21, 19/21)

P(E|H_1) = 1/19

P(E|H_1)P(H_1) := mul(1/19, 19/21)

P(E|H_1)P(H_1) = 1/21

sum of partition joints := sum(1/21, 1/21)

sum of partition joints = 2/21

partition sum=2/21; direct P(E)=2/21

equal: Yes

P(H_1|E) := div(1/21, 2/21)

P(H_1|E) = 1/2

posterior·P(E) := mul(1/2, 2/21)

posterior·P(E) = 1/21

Reverse checks

• S015 · premise_and_result_audit — Recompute posterior·P(E) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute P(H_1|E) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Align events, intersections and conditioning denominators under the same law; do not divide by zero mass.
• S012 · premise_and_result_audit — Recompute sum of partition joints with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute P(E|H_1)P(H_1) with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P(E|H_1) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute P(E∩H_1) with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute P(H_1) with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute P(E|H_0)P(H_0) with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute P(E|H_0) with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute P(E∩H_0) with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute P(H_0) with exact rational arithmetic, not rounded decimals.
• 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

(1, 19/21, 1/19, 2/21, 1/2, partition_pairwise_disjoint_exhaustive)

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.