Conditional probabilities can be useful in diagnosing disease. Suppose that three different, closely related diseases (A1, A2, and A3) occur in 25%, 15%, and 12% of the population. In addition, suppose that any one of three mutually exclusive symptom states (B1, B2, and B3) may be ­associated with each of these diseases. Experience shows that the likelihood P(B1|A1) of having a given symptom state when the disease is present is as shown in the following table. Find the probability of disease A2 given symptoms B1, B2, B3, and B4, respectively.

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Disease State Ai Symptom State B в, Ba B3 B, (no symptoms) 68 6468 08 17 10 18 12 14 06 07 08