5.116 Bayes’ rule Suppose we know P1A2, P1B•A2, and P1Bc •Ac2, but we want to find P1A•B2.

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a. Using the definition of conditional probability for P(A|B) and for P(B|A), explain why P(A|B) - P(A and B)/P(B) [P(A)P(B|A)]/P(B). b. Splitting the event that B occurs into two parts, according to whether A occurs, explain why P(B) = P(B and A) + P(B and A). c. Using part b and the definition of conditional proba- bility, explain why P(B) = P(A)P(B|A) + P(A)P(B|A). d. Combining what you have shown in parts a-c, reason that P(A|B) = P(A)P(B|A) P(A)P(BA) + P(A)P(B|A)* This formula is called Bayes' rule. It is named after a British clergyman who discovered the formula in 1763.