Consider a test which detects if a person has a disease. Let R denote the outcome of the test on a person, D denote whether the person actually has the disease and 1 be the probability that the test is positive if the person has it and 0 be the probability that the test is negative if the person doesn't have the disease. That is, the probability that it reports that someone has the disease R=1 when they actually do (D=1) , is 1 , and the probability that it reports that someone doesn't have the disease when they don't is 0 . Formally:

p(R=1D=1)=1 and p(R=0D=0)=0

Finally, the prior probability of a person having this disease is p(D)= .

1. A patient has the test performed. Derive the posterior probability that the person actually has the disease in terms of 1,0 , and given that the test comes back positive or the test comes back negative.
2. After the results of the first test come back positive, the doctor runs it a second time. This time it comes back negative. Derive the posterior probability that the person actually has the disease after this second round of testing in terms of 1,0 and . If 1=0 what did we learn about whether the person has the disease?
3. Suppose 1=0=0.99and =0.001. Suppose 1,000 patients get tested, and they're all negative. On average, how many of these patients actually have the disease? I.E., what is the expected number of false negatives?