# Question 3171: Positive-Probability Eligibility of a Conditioning Event [Return to catalogue page 16](https://ocme.evemisslab.com/en/questions/page/16/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r07-conditional-probability-foundations/#q-20261003-advance-20261004-r07-conditional-probability-0071) 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. R0005-03171 · q-20261003-advance-20261004-r07-conditional-probability-0071 ## Original question and givens The nonempty finite sample space is S={8, 9, 10, 11}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(26), 1/(26), 2/(26), 21/(26)], summing exactly to one. Does B={8} meet the positive-probability requirement for defining P(A|B)? - G001 — The nonempty finite sample space is S={8, 9, 10, 11}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(26), 1/(26), 2/(26), 21/(26)], summing exactly to one. Does B={8} meet the positive-probability requirement for defining P(A|B)? ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-conditional-probability](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/) ## Step-by-step derivation S001 · impliesBefore S={8, 9, 10, 11};no particular A supplied;B={8};F=𝒫(S) After events are declared subsets;conditioning needs P(B)>0 Conditions: none 𝒫(S) is the full event powerset,not a probability value.A nonempty event may have probability0.When checking B's eligibility,do not invent an unspecified A. Step identifiers and dependenciesRule: conditional_model_domain · Uses: G001 S002 · calculationBefore total weight numerator := sum(2, 1, 2, 21) After total weight numerator = 26 Conditions: none Nonnegative weights must sum to the common positive denominator to give total probability1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore [[8, 2], [9, 1], [10, 2], [11, 21]] After weights sum to 26;normalization verified Conditions: none Weights align with the original S order;do not sort S while leaving weights in their old positions. Step identifiers and dependenciesRule: weight_alignment_and_normalization · Uses: G001, S002 S004 · calculationBefore B weight numerator := sum(2) After B weight numerator = 2 Conditions: none Sum aligned weights over the event;zero-mass outcomes do not become positive mass. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore B probability mass := div(2, 26) After B probability mass = 1/13 Conditions: none Divide by the common positive denominator to obtain exact probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore P(B)=1/13 After conditioning eligible=Yes Conditions: none The ordinary conditional ratio requires P(B)>0;when P(B)=0 do not assign0 to an undefined ratio. Step identifiers and dependenciesRule: positive_conditioning_eligibility · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute B probability mass with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute B weight numerator with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute total weight numerator with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result Yes The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## 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: 404b0f28cc7cbe1446d94cf78390b6c6292c223744cac1a7b6437353ea41fd8f - candidate record SHA-256: 7eec6314bd1b0fecf79f90283e0932d432d5afc245c6e8eaf971757f340a9589 - semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2 - question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r07-conditional-probability-0071.json)