Problem 2. Medical test (#Probability, #MathTools) Omer wants to be really certain about a diagnosis so he takes a series of identical medical tests. He hopes multiple tests will reduce his

uncertainty. Events: D: Omer has the disease being tested for. T; : Omer tests positive on the jth test for j = 1, 2, ..., n. Let p = P(D) be the prior probability that he has the disease. Question 4 of 15 2.1 Assume for this part the test results are conditionally independent given Omer's disease status. Let a = P(T;|D) and b₁ = P(T;|D), where ao and bo don't depend on j. Find the posterior probability that Omer has the disease, given that he tests positive on all of the tests. Hint: Since a does not depend on j (the index of a test result), the algebra simplifies a lot since P(T;|D)P(T;|D) = a for any test indices i and j. The same holds for bn. Question 5 of 15 2.2 Suppose some people have a gene that makes them always test positive on this type of medical test. Let G be the event that Omer has the gene. Assume that P(G) and that D and G are independent - that is, the gene does not make you more or less susceptible to the disease. If Omer has the gene, he'll test positive on all tests. If Omer does not have the gene, then the test results are conditionally independent given his disease status. Let a₁ = P(T;|D,G) and b₁ = P(T;|Dº‚Gº), where a₁ and b₁ don't depend on j. Now, suppose that Omer tests positive on all tests and find the posterior probability that Omer has the disease. Question 6 of 15 2.3 Using the same setup as in part (2.), find the posterior probability that Omer has the gene given that he tests positive on all n of the tests.