Question: Fred has just tested positive for a certain disease.

(a) Given this information, find the posterior odds that he has the disease, in terms of the prior odds, the sensitivity of the test, and the specificity of the test.

(b) Not surprisingly, Fred is much more interested in P(have disease|test positive), known as the positive predictive value, than in the sensitivity P(test positive|have disease). A handy rule of thumb in biostatistics and epidemiology is as follows: For a rare disease and a reasonably good test, specificity matters much more than sensitivity in determining the positive predictive value. Explain intuitively why this rule of thumb works. For this part you can make up some specific numbers and interpret probabilities in a frequentist way as proportions in a large population, e.g., assume the disease of a population of 10000 people and then consider various possibilities for the sensitivity and specificity.