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 6.28. Now assume there is a second test, independent of the first that can be used as a follow-up. 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. 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 independent. Why might this not be realistic? If they are not independent, what kind of additional information would you need about the likelihoods of the test results?

Transcribed Image Text:

B C 1 llustration of Bayes' rule using drug example A D. Figure 6.28 Bayes' Rule for Drug-Testing Example 2 3 Prior probabilities of drug user status User Non-user 0.95 0.05 4 7 Likelihoods of test results, given drug user status User Non-user 0.93 0.07 8 9 Test positive 10 Test negative 11 0.03 0.97 1 12 13 Unconditional probabilities of test results (denominators of Bayes' rule) 14 Test positive 15 Test negative 16 17 18 Posterior probabilities of drug user status (Bayes' rule) 0.075 0.925 User Non-user 0.620 0.380 0.004 0.996 19 20 Test positive 21 Test negative