For a diagnostic test of a certain disease, π1 denotes the probability that the diagnosis is positive given that a subject has the disease, and π2 denotes the probability that the diagnosis is positive given that a subject does not have it. Let ρ denote the probability that a subject does have the disease.

a. Given that the diagnosis is positive, show that the probability that a subject does have the disease is

b. Suppose that a diagnostic test for HIV+ status has both sensitivity and specificity equal to 0.95, and  ρ=0.005. Find the probability that a subject is truly HIV+, given that the diagnostic test is positive. To better understand this answer, find the joint probabilities relating diagnosis to actual disease status, and discuss their

relative sizes.