Question

The probability that a person has a certain disease is 0.03

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

a. 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?

b. If the medical diagnostic test has given a negative result​ (indicating that the disease is not​ present), what is the probability that the disease is not​ present?

Answer 1

Answer:

a)

Given,

P(T+) = P(D)*P(T+|D) + P(D')*P(T+|D')

substitute values

= 0.03*0.92 + 0.97*0.02

= 0.0276 + 0.0194

= 0.047

P(D|T+) = P(D)*P(T+|D) / [P(D)*P(T+|D) + P(D')*P(T+|D')

substitute values

= 0.03*0.92 / (0.03*0.92 + 0.97*0.02)

= 0.0276 / (0.0276 + 0.0194)

= 0.0276 / 0.047

= 0.5872

b)

P(T-) = P(D')*P(T-|D') + P(D)*P(T-|D)

substitute values

= 0.97*0.98 + 0.03*0.08

= 0.953

P(D'|T-) = P(D')*P(T-|D') / P(D')*P(T-|D') + P(D)*P(T-|D)

substitute values

= 0.97*0.98 / (0.97*0.98 + 0.03*0.08)

= 0.9506 / 0.9506 + 0.0024)

= 0.9975