# Question 545: Zero Evidence Mass Makes This Conditional Probability Undefined [Return to catalogue page 3](https://ocme.evemisslab.com/en/questions/page/3/) [← 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-0095) 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. P1000-0545 · q-20261014-advance-20261102-r06-bayes-theorem-0095 ## Original question and givens Use the same complete finite atomic probability law with aligned nonnegative exact masses=[1/27, 1/27, 1/27, 24/27], 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=[1/27, 1/27, 1/27, 24/27], 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](https://ocme.evemisslab.com/en/mko/mko-bayes-theorem/) ## Step-by-step derivation S001 · calculationBefore P(Ω) := sum(1/27, 1/27, 1/27, 8/9) 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() After P(E) = 0 Conditions: none Sum only the atoms in the conditioning event E. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore P(E)=0 After P(H|E)=P(H∩E)/P(E) is not defined here Conditions: none The conditional-probability definition requires a positive denominator, so no division is performed. Step identifiers and dependenciesRule: zero_denominator_guard · Uses: G001, S002 S004 · impliesBefore zero conditioning mass After undefined is not zero, not infinity, and not an empty event Conditions: none Even when the joint mass is zero,0/0 cannot be assigned the value0. Step identifiers and dependenciesRule: undefined_not_value · Uses: G001, S003 S005 · impliesBefore positive-denominator requirement fails After defined=false; value=undefined Conditions: none Retain this contract's undefined marker without altering the probability law. Step identifiers and dependenciesRule: contract_boundary · Uses: G001, S004 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: f2edad4ec50eb5e7e22baf8ef83c41b38e03116c5450c61c7a6f90901abdc4f1 - candidate record SHA-256: 2d3881081833a5c94ffd54b3e325ec837cf739db43bf63df5b744304b130adde - semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772 - question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b - 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-0095.json)