1. The prior probabilities for events A1 and A2 are P(A1)  .40 and P(A2)  .60. It is also known...

Question

1. The prior probabilities for events A1 and A2 are P(A1)  .40 and P(A2)  .60. It is also known that P(A1  A2)  0. Suppose P(B  A1)  .20 and P(B  A2)  .05.
a. Are A1 and A2 mutually exclusive? Explain.
b. Compute P(A1  B) and P(A2  B).
c. Compute P(B).
d. Apply Bayes’ theorem to compute P(A1  B) and P(A2  B).
2. The prior probabilities for events A1, A2, and A3 are P(A1)  .20, P(A2)  .50, and P(A3) 
.30. The conditional probabilities of event B given A1, A2, and A3 are P(B  A1)  .50,
P(B  A2)  .40, and P(B  A3)  .30.
a. Compute P(B  A1), P(B  A2), and P(B  A3).
b. Apply Bayes’theorem, equation (4.19), to compute the posterior probability P(A2  B).
c. Use the tabular approach to applying Bayes’ theorem to compute P(A1  B), P(A2  B),
and P(A3  B).

Robert · Accepted Answer

Solution 
1) P (A1) = 0.40 
     P (A2) = 0.60 
     P (A1 A2) = 0 
     P (B| A1) = 0.20 
     P (B| A2) = 0.05 
a) Yes A1 and A2 are mutually exclusive because their intersection contains no element 
and therefore the probability is zero. 
b) P (A1and B) = P (B|A1)* P (A1) 
                          = 0.20*0.40 
                          = 0.08 
    P (A2 and B) = P (B|A2)* P (A2) 
                           = 0.03 
 c) P (B) = [P (A1)*P (B|A1) + P (A2)* P (B|A2)] 
              = 0.08 + 0.03 
              = 0.

1. The prior probabilities for events A1 and A2 are P(A1) .40 and P(A2) .60. It is also known that P(A1 A2) 0. Suppose P(B A1) .20 and P(B A2) .05. a. Are A1 and A2 mutually exclusive?...

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