For a diagnostic test of a certain disease, 1 denotes the probability that the diagnosis is positive given that a subject has the disease,

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 π₁ ρ/[π₁ ρ + π₂(1 – ρ)].

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.