# Question 523: The Law of Total Probability Supplies the Bayes Denominator

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

P1000-0523 · q-20261014-advance-20261102-r06-bayes-theorem-0073



## Original question and givens

Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/30, 1/30, 1/30, 27/30], 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=[1/30, 1/30, 1/30, 27/30], 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](https://ocme.evemisslab.com/en/mko/mko-bayes-theorem/)





## Step-by-step derivation

S001 · calculationBefore

P(Ω) := sum(1/30, 1/30, 1/30, 9/10)

After

P(Ω) = 1

Conditions: none

First sum the complete atom masses to1.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · calculationBefore

P(E) := sum(1/30, 1/30)

After

P(E) = 1/15

Conditions: none

Sum only the atoms in the conditioning event E.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

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

After

pairwise disjoint; union=Ω

Conditions: none

A complete partition assigns each atom to exactly one hypothesis, so decomposing E neither duplicates nor omits mass.

Step identifiers and dependenciesRule: complete_partition · Uses: G001, S002





S004 · calculationBefore

P(H_0) := sum(1/30, 1/30)

After

P(H_0) = 1/15

Conditions: none

Calculate this hypothesis's prior.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

P(E∩H_0) := sum(1/30)

After

P(E∩H_0) = 1/30

Conditions: none

Calculate the joint mass from the event intersection.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

P(E|H_0) := div(1/30, 1/15)

After

P(E|H_0) = 1/2

Conditions: none

This quotient is licensed by the positive prior.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

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

After

P(E|H_0)P(H_0) = 1/30

Conditions: none

Multiplying by the prior must recover the same joint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

P(H_1) := sum(1/30, 9/10)

After

P(H_1) = 14/15

Conditions: none

Calculate this hypothesis's prior.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

P(E∩H_1) := sum(1/30)

After

P(E∩H_1) = 1/30

Conditions: none

Calculate the joint mass from the event intersection.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

P(E|H_1) := div(1/30, 14/15)

After

P(E|H_1) = 1/28

Conditions: none

This quotient is licensed by the positive prior.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

P(E|H_1)P(H_1) := mul(1/28, 14/15)

After

P(E|H_1)P(H_1) = 1/30

Conditions: none

Multiplying by the prior must recover the same joint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

sum of partition joints := sum(1/30, 1/30)

After

sum of partition joints = 1/15

Conditions: none

The disjoint, exhaustive intersections decompose E completely.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · impliesBefore

partition sum=1/15; direct P(E)=1/15

After

equal: Yes

Conditions: none

Verify the denominator by total probability before using a posterior answer.

Step identifiers and dependenciesRule: total_probability · Uses: G001, S012









## 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

([1/15, 14/15], [1/2, 1/28], [1/30, 1/30], 1/15, 1/15, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 5d4d4fdb97003ea24f82865d34ce723785954e4af5ea66cebefaae4a7f0e700d
- candidate record SHA-256: 49968a96e9f318ef01e0e621041fbc08f911d7c0aee0c7aaa9210c076531f0d3
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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