Question: Fred decides to take a series of n tests, to diagnose whether he has a certain disease (any individual test is not perfectly reliable, so he hopes to reduce his uncertainty by taking multiple tests). Let D be the event that he has the disease, p = P(D) be the prior probability that he has the disease, and q = 1 p. Let Tj be the event that he tests positive on the jth test.
(a) Assume for this part that the test results are conditionally independent given Fred's disease status. Let a = P(Tj|D) and b = P(Tj|Dc), where a and b don't depend on j. Find the posterior probability that Fred has the disease, given that he tests positive on all n of the n tests.
(b) Suppose that Fred tests positive on all n tests. However, some people have a certain gene that makes them always test positive. Let G be the event that Fred has the gene. Assume that P(G) = 1/2 and that D and G are independent. If Fred does not have the gene, then the test results are conditionally independent given his disease status. Let a0 = P(Tj|D, Gc) and b0 = P(Tj|Dc, Gc), where a0 and b0 don't depend on j. Find the posterior probability that Fred has the disease, given that he tests positive on all n of the tests.