2.Bayesian Reasoning II

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 be the likelihood that the test gives the correct result. That is, the probability that it reports that someone has the disease (R= 1) when they actually do ( D= 1), is , and the probability that it reports that someone doesnt have the disease when they dont is also . Formally: p ( R = 1 | D = 1) = p ( R = 0 | D = 0) = Finally, an -fraction of the population actually has this disease, that is, the prior probability of a person having this disease is p ( D ) = .

(a) A patient goes to the doctor, has the test performed and it comes back positive. Derive a general formula for the posterior probability that the person actually has the disease, and simplify it in terms of and . Which value do you get for = 0 . 001 and = .95?