Suppose that you are given a decision situation with three possible states of nature: S, Sy, and sz. The prior probabilities are P(52) = 0.1, P(s) = 0.5, and P(53) = 0.4. With sample information I, P(I|S,) = 0.1, P(IIS) = 0.05, and P(1|s3) = 0.2. Compute the revised or posterior probabilities: P(s, 11), P(s 11), and P(s3|1). Posterior probabilities are conditional probabilities based on the outcome of the sample information. These can be computed by developing a table using the following process. 1. Enter the states of nature in the first column, the prior probabilities for the states of nature, P(I|S ), in the second column and the conditional probabilities in the third column. 2. In column 4 compute the joint probabilities by multiplying the prior probability values in column 2 by the corresponding conditional probabilities in column 3 3. Sum the joint probabilities in column 4 to obtain the probability of the sample information I, P(I). 4. In column 5, divide each joint probability in column 4 by P(I) to obtain the posterior probabilities, P(5,II). The prior probabilities are given to be P(s) = 0.1, P(s) = 0.5, and P(sz) = 0.4. The conditional probabilities given each state of nature are P(I|S,) = 0.1, P(I|52) = 0.05, and P(I|53) = 0.2. Use the given prior and conditional probabilities to compute the joint probabilities. States of Nature Prior Probabilities P(s;) Conditional Probabilities P(I|s;) Joint Probabilities P(INS;) = S1 Pls,) = 0.1 P(I|92) = 0.1 P(Ins) = P(s)P(I\s) = 0.1(0.1) = S2 P(s) = 0.5 P(I|s ) = 0.05 S3 P(53) = 0.4 = P(I|53) = 0.2 The sum of the Joint Probabilities column gives P(I) =