# Question 119: Prior and Likelihood Must Both Enter a Bayes Reversal [Return to catalogue page 1](https://ocme.evemisslab.com/en/questions/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r06-bayes-theorem-foundations/#q-20261014-advance-20261102-r06-bayes-theorem-0019) 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. P250-0119 · q-20261014-advance-20261102-r06-bayes-theorem-0019 ## Original question and givens Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], 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. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal). - G001 — Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/26, 1/26, 1/26, 23/26], 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. Hypothesis=[0, 1] must have positive prior, and evidence must have positive mass. Give (prior,likelihood,evidence_mass,joint,posterior,not_causal). ## Definition and scope Use one complete finite probability law; both conditioning events have strictly positive mass. - [mko-bayes-theorem](https://ocme.evemisslab.com/en/mko/mko-bayes-theorem/) ## Step-by-step derivation S001 · calculationBefore P(Ω) := sum(1/26, 1/26, 1/26, 23/26) After P(Ω) = 1 Conditions: none Sum all atom masses first to verify normalization. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore H=[0, 1]; E=[0, 2] After H∩E=[0] Conditions: none Only sample points in both events contribute to their joint probability. Step identifiers and dependenciesRule: event_intersection · Uses: G001, S001 S003 · calculationBefore P(H) := sum(1/26, 1/26) After P(H) = 1/13 Conditions: none Add the masses of the points in H. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore P(E) := sum(1/26, 1/26) After P(E) = 1/13 Conditions: none Add the masses in E; this is the posterior's conditioning denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore P(H∩E) := sum(1/26) After P(H∩E) = 1/26 Conditions: none Sum the intersection once, without double-counting. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore P(E|H) := div(1/26, 1/13) After P(E|H) = 1/2 Conditions: none Since P(H)=1/13>0, divide the joint probability by P(H). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore P(E|H)·P(H) := mul(1/2, 1/13) After P(E|H)·P(H) = 1/26 Conditions: none Multiply the conditional probability by the conditioning event's probability to recover the joint. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore P(H|E) := div(1/26, 1/13) After P(H|E) = 1/2 Conditions: none When reversing the condition, the denominator is P(E)=1/13>0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore P(H|E)=1/2; P(H∩E)=P(E|H)P(H) After P(H|E)=P(E|H)P(H)/P(E)=1/2 Conditions: P(E)>0; P(H)>0 Substitute the established joint-product identity into the posterior definition to obtain Bayes' formula for this instance. Step identifiers and dependenciesRule: bayes_substitution · Uses: G001, S008 S010 · calculationBefore P(H|E)·P(E) := mul(1/2, 1/13) After P(H|E)·P(E) = 1/26 Conditions: none Multiply back by P(E) to recover the same joint probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore probability identity After no causal direction has been established Conditions: none This is a conditional-probability identity in one law, not evidence of a causal direction. Step identifiers and dependenciesRule: probability_not_causality · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — With zero conditioning mass the quotient is undefined, not a numerical 0/0 answer. - S010 · premise_and_result_audit — Recompute P(H|E)·P(E) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Reversing a conditional does not mean swapping labels while keeping the old denominator. - S008 · premise_and_result_audit — Recompute P(H|E) with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute P(E|H)·P(H) with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute P(E|H) with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute P(H∩E) with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute P(E) with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute P(H) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Do not multiply the event probabilities: independence has not been assumed. - S001 · premise_and_result_audit — Recompute P(Ω) with exact rational arithmetic, not rounded decimals. ## Result (1/13, 1/2, 1/13, 1/26, 1/2, not_causal) P(H|E)=1/2; the complete answer also retains the prior, likelihood, evidence and joint. ## This question review and source binding Review date: 2026-10-24 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: d96bb1f3a311d1695ba395f3f827520f93ac35e2e726700860d98c2066d26f21 - candidate record SHA-256: 2080a585408670bcd148ff9209ce22220c556c12103fe3b156cc1a84231d91b9 - semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b - question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r06-bayes-theorem-0019.json)