4.2 You go to the doctor for a regular medical checkup and the doctor suggests a laboratory test for a particular form of cancer. Assume (unrealistically) that the doctor has available only the one test. We assume that she has not yet discovered any reason to believe, a priori, that you have the disease-it is not in the medical background you have told her, Suppose that the medical test is a very sensitive one in that the chance that the iaboratory test is positive when in fact you do not have thc disease (“false positive”) is 3 percent, and the chance that the laboratory test is negative when in fact you do have the disease (“false negative”) is 1 percent.
Suppose also that the incidence of the disease in the population at large is one in a million (you adopt this value as the prior probability that you have the disease). You take the laboratory test, and to your surprise and horror, the Iaboratoty result comes back positive! What is your posterior probability that you have the disease? (Hint: See Chapter 1, Section 1.1 for a discussion of this problem.)