Question

The probability that a person has a certain disease is 2%. Medical diagnostic tests are available to determine whether the person actually has the disease. If the disease is actually present, the probability that the medical diagnostic tests will give a positive result (correct diagnosis) is 95%. If the disease is not actually present, the probability of a positive test result (incorrect diagnosis) is 0.5%. Suppose that the medical diagnostic test shows a positive result, (a) What is the probability that the disease is actually present? (b) Suppose that the medical diagnostic test shows a negative test result, what is the probability that the disease is actually absent?

Answer 1

Probability of disease, P(D) = 0.02

P(D') = 1- P(D) = 0.98

P(+ | D) = 0.95

P(+ | D') = 0.005

a)

probability that the disease is actually present given that test is positive, P(D | +) =

= [P(D)*P(+ | D)] / [P(D)*P(+ | D) + P(D')*P(+ | D')] = [0.02*0.95] / [ 0.02*0.95 + 0.98*0.005] = 0.7950

b)

P(- | D) = 1-P(+ | D) = 1-0.95 = 0.05

P(- | D') = 1-P(+ | D') = 1-0.005 = 0.995

probability that the disease is actually absent given that result is negative, P(D' | -) =

= [P(D')*P(- | D')] / [P(D)*P(- | D) + P(D')*P(- | D')] = [0.98*0.995] / [ 0.02*0.05 + 0.98*0.995] = 0.9990