Question

The probability that a person has a certain disease is 0.05 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 test will give a positive result​ (indicating that the disease is​ present) is 0.92 If the disease is not actually​ present, the probability of a positive test result​ (indicating that the disease is​ present) is 0.02 If the medical diagnostic test has given a positive result​ (indicating that the disease is​ present), what is the probability that the disease is actually​ present?

Answer 1

Given that

Probability that the person has a disease is 0.05

P(D) = 0.05

Probability that the person has no disease is

P(D-) = 1-0.05 = 0.95

Disease(D) No Disease(D-)

Positive Result (p) 0.92 0.02

Negetive Result (p-) 1-0.92 = 0.08 1-0.08 = 0.92

P(p|D ) = 0.92

P(p| D-) = 0.02

P(p- | D ) = 0.08

P(p- | D-) = 0.92

If the medical diagnostic test has given a positive result then the probability that the disease actually present is

P( D | p ) = P(p|D) * P(D) / P(p)

= P(p|D) * P(D) / P(p|D) * P(D) + P(p|D-) * P(D-)

= 0.92*0.05 / 0.92*0.05 + 0.02*0.95

= 0.046 / 0.0627

= 0.733