Question 508: The Law of Total Probability Supplies the Bayes Denominator

Original question and givens

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], 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 exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

• G001 — Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/15, 1/15, 1/15, 12/15], 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 exhaustive disjoint hypotheses=[[0, 1], [2, 3]], all with positive prior, and positive evidence mass. Give (priors,likelihoods,joints,total,evidence_mass,equal?), preserving hypothesis order.

Definition and scope

One complete finite law. The prior-times-likelihood joints of the hypotheses must sum to the directly computed P(E).

• mko-bayes-theorem

Step-by-step derivation

P(Ω) := sum(1/15, 1/15, 1/15, 4/5)

P(Ω) = 1

P(E) := sum(1/15, 1/15)

P(E) = 2/15

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

pairwise disjoint; union=Ω

P(H_0) := sum(1/15, 1/15)

P(H_0) = 2/15

P(E∩H_0) := sum(1/15)

P(E∩H_0) = 1/15

P(E|H_0) := div(1/15, 2/15)

P(E|H_0) = 1/2

P(E|H_0)P(H_0) := mul(1/2, 2/15)

P(E|H_0)P(H_0) = 1/15

P(H_1) := sum(1/15, 4/5)

P(H_1) = 13/15

P(E∩H_1) := sum(1/15)

P(E∩H_1) = 1/15

P(E|H_1) := div(1/15, 13/15)

P(E|H_1) = 1/13

P(E|H_1)P(H_1) := mul(1/13, 13/15)

P(E|H_1)P(H_1) = 1/15

sum of partition joints := sum(1/15, 1/15)

sum of partition joints = 2/15

partition sum=2/15; direct P(E)=2/15

equal: Yes

Reverse checks

• 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

([2/15, 13/15], [1/2, 1/13], [1/15, 1/15], 2/15, 2/15, Yes)

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.