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Start with a consistent stated model
The model supplies P(A|B) = 65%, P(B) = 40% and P(A) = 50%. Multiply 65% by 40% to obtain joint probability P(A∩B) = 26%. That overlap is no greater than either marginal probability.
Divide by the new conditioning probability
For P(B|A), divide the 26% overlap by P(A) = 50%. The result is 52%. For P(A|B), the denominator was 40%, giving 65%.
| Quantity | Numerator | Denominator | Result |
|---|---|---|---|
| P(A|B) | 26% | 40% | 65% |
| P(B|A) | 26% | 50% | 52% |
| P(A ∩ B) | 26% | Full model | 26% |
Check the remaining model cells
A only is 24%, B only is 14%, both is 26% and neither is 36%. The cells sum to 100% and reproduce the supplied marginal probabilities. This check supports the consistency of the example.
Avoid stopping at the joint probability
Reporting 26% as P(B|A) stops one step too early. Reporting 65% simply reuses the conditional in the other direction. Identify the condition in words before choosing its denominator.