The terms prior and posterior are relative. Assume that the drug test has been performed, and the outcome is positive, which leads to the posterior probabilities in row 20 of Figure 10.29. Now assume there is a second test, probabilistically independent of the first, that can be used as a follow-up. We assume that its false positive and false negative rates are 0.02 and 0.06.

a. Use the posterior probabilities from row 20 as prior probabilities in a second Bayes’ rule calculation.

(Now prior means prior to the second test.) If the athlete also tests positive in this second test, what is the posterior probability that he is a drug user?

b. We assumed that the two tests are probabilistically independent. Why might this not be realistic? If they are not independent, what kind of additional information would we need about the likelihoods of the test results?