To compute the posterior probabilities P(s1|I), P(s2|I), and P(s3|I), we use Bayes' theorem:
P(s1|I) = P(I|s1) * P(s1) / P(I)
P(s2|I) = P(I|s2) * P(s2) / P(I)
P(s3|I) = P(I|s3) * P(s3) / P(I)
First, let's find the probability of I, P(I):
P(I) = P(I|s1) * P(s1) + P(I|s2) * P(s2) + P(I|s3) * P(s3)
P(I) = (.15)(.1) + (.2)(.6) + (.1)(.3)
P(I) = .015 + .12 + .03
P(I) = .165
Now let's calculate P(s1|I):
P(s1|I) = (.15)(.1) / .165
P(s1|I) ≈ 0.0909
P(s2|I):
P(s2|I) = (.2)(.6) / .165
P(s2|I) ≈ 0.7273
P(s3|I):
P(s3|I) = (.1)(.3) / .165
P(s3|I) ≈ 0.1818
So the computed probabilities are:
P(s1|I) ≈ 0.0909
P(s2|I) ≈ 0.7273
P(s3|I) ≈ 0.1818
Suppose that you are given a decision situation with three possible state of nature: s1, s2, and s3. The prior probabilities are P(s1) =.1, P(s2) = .6, and P(s3) = .3. With sample information I, P(I|s1) =.15, P(I|s2) = .2, and P(I|s3) = .1.
Compute P(s1|I)=
Compute P(s2|I)=
Compute P(s3|I)=
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