Question 3128: Comparing Two Directions of Conditional Probability

Original question and givens

The nonempty finite sample space is S={-10, -9, -8, -7}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(12), 3/(12), 4/(12), 3/(12)], summing exactly to one. A={-9, -8}, B={-9, -8}, and both P(A) and P(B) are positive. Are P(A|B) and P(B|A) equal?

• G001 — The nonempty finite sample space is S={-10, -9, -8, -7}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(12), 3/(12), 4/(12), 3/(12)], summing exactly to one. A={-9, -8}, B={-9, -8}, and both P(A) and P(B) are positive. Are P(A|B) and P(B|A) equal?

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

Step-by-step derivation

S={-10, -9, -8, -7};A={-9, -8};B={-9, -8};F=𝒫(S)

events are declared subsets;conditioning needs P(B)>0

A={-9, -8};B={-9, -8}

A∩B={-9, -8}

total weight numerator := sum(2, 3, 4, 3)

total weight numerator = 12

[[-10, 2], [-9, 3], [-8, 4], [-7, 3]]

weights sum to 12;normalization verified

B weight numerator := sum(3, 4)

B weight numerator = 7

B probability mass := div(7, 12)

B probability mass = 7/12

A∩B weight numerator := sum(3, 4)

A∩B weight numerator = 7

A∩B probability mass := div(7, 12)

A∩B probability mass = 7/12

P(A|B) := div(7/12, 7/12)

P(A|B) = 1

P(A∩B)=7/12;P(B)=7/12>0

P(A|B)=1

A weight numerator := sum(3, 4)

A weight numerator = 7

A probability mass := div(7, 12)

A probability mass = 7/12

P(B|A) := div(7/12, 7/12)

P(B|A) = 1

P(A|B)=1;P(B|A)=1

equal=Yes

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute P(B|A) with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute A probability mass with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute A weight numerator with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute P(A|B) with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute A∩B probability mass with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute A∩B weight numerator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute B probability mass with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute B weight numerator with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute total weight numerator with exact rational arithmetic, not rounded decimals.
• 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

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.