In using Bayes’ rule for the presence of a disease (see Figure 9.17 and the file Bayes Rule for Disease.xlsx), we assumed that there are only two test results, positive or negative. Suppose there is another possible test result, “maybe.” The 2 x 2 range B9:C10 in the file should now be replaced by a 2 x 3 range, B9:D10, for positive, maybe, and negative (in that order). Let the corresponding probabilities in row 9 be 0.85, 0.10, and 0.05, and let those in row 10 be 0.05, 0.15, and 0.80. Redo the Bayes’ rule calculations with both the frequency approach and the probability approach.

Figure 9.17

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10,000 middle-aged men 99% -1% 100 with 9,900 without disease disease 90% 95% 5 test negative 95 test 990 test 8910 test positive 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% -5%