The terms prior and posterior are relative. Assume that the test in Figure 9.17 (and in the file Bayes Rule for Disease.xlsx) has been performed, and the outcome is positive, which leads to the posterior probabilities shown. Now assume there is a second test, independent of the first, that can be used as a followup. Assume that its false-positive and false-negative rates are 0.02 and 0.06.

a. Use the posterior probabilities as prior probabilities in a second Bayes’ rule calculation. (Now prior means prior to the second test.) If Joe also tests positive in this second test, what is the posterior probability that he has the disease?

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?

Figure 9.17

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10,000 middle-aged men -1% 99% 100 with 9,900 without disease disease 95% 95 test 5 test 990 test positive negative 8910 test positive negative Joe's chance of having the disease, given a positive test = 95/(95+990) = 8.75% Joe's chance of having the disease, given a negative test = 5/(5+8910) = 0.056% Joe's chance of testing positive = (95+990)/10000 = 10.9% 90% %0- -5%