# Question 3181: Conditional Probability with Explicit Equiprobability

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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.

R0005-03181 · q-20261003-advance-20261004-r07-conditional-probability-0031



## Original question and givens

The nonempty finite sample space is S={-7, -6, -5, -4, -3, -2}, with event family F=𝒫(S). All elementary outcomes are explicitly equiprobable. A={-7, -5, -3} and B={-3, -2} is nonempty. Compute P(A|B) as a reduced fraction with positive denominator.


- G001 — The nonempty finite sample space is S={-7, -6, -5, -4, -3, -2}, with event family F=𝒫(S). All elementary outcomes are explicitly equiprobable. A={-7, -5, -3} and B={-3, -2} is nonempty. Compute P(A|B) as a reduced fraction with positive denominator.





## 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={-7, -6, -5, -4, -3, -2};A={-7, -5, -3};B={-3, -2};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 · impliesBefore

A={-7, -5, -3};B={-3, -2}

After

A∩B={-3}

Conditions: none

The intersection keeps outcomes in both events;neither a union nor independence is assumed.

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





S003 · impliesBefore

all elementary outcomes explicitly equiprobable

After

P(A∩B)=|A∩B|/6;P(B)=|B|/6

Conditions: none

Only explicit equiprobability permits the cardinality ratio;nonempty B then has positive probability.

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





S004 · calculationBefore

uniform conditional probability := div(1, 2)

After

uniform conditional probability = 1/2

Conditions: none

Cancel the common positive denominator |S|,leaving intersection count divided by conditioning count.

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









## Reverse checks


- S004 · premise_and_result_audit — Recompute uniform conditional probability 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

1/2

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: a441a07f708bcec69ce744143f4b263320d5f0176ea1cac6a7df150ec68b6a20
- candidate record SHA-256: b3e5cf281408fcb7083ae640619de4a0288e09ccf807ac2c388f0942ea51908d
- 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-0031.json)