SIMPSON'S PARADOX
(a) There are two crimson jars (labeled C1 and C2) and two mauve jars (labeled M1 and M2). Each jar contains a mixture of green gummi bears and red gummi bears. Show by example that it is possible that C1 has a much higher percentage of green gummi bears than M1, and C2 has a much higher percentage of green gummi bears than M2, yet if the contents of C1 and C2 are merged into a new jar and likewise for M1 and M2, then the combination of C1 and C2 has a lower percentage of green gummi bears than the combination of M1 and M2. (b) Explain how (a) relates to Simpson’s paradox, both intuitively and by explicitly de?ning events A,B,C as in the statement of Simpson’s paradox.
10. (BH 2.54) 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.