Refer to Examples 1 and 2. Suppose that a black ball was selected. What is the probability that the black ball came from urn I?

Approach We know that

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P(U) = 1 P(U) = 3 4 P(B|U) = P(B|U2) 23 4 = P(B) = 13 36 91 We wish to know P(U|B). Using the Conditional Probability Rule, we write P(U|B) as P(U|B) = P(UB) P(B) But, P(UB) = P(U) P(B|U) and P(B) = P(U) P(B|U) + P(U) P(B|U) So we have P(UB) P(U|B) = = P(U) P(B\U). P(B) P(U) P(B U) + P(U2) P(B|U) (4)