Question

A diagnostic test is being developed to diagnose a new virus that has infected 10% of the world's population. If a person is infected with the virus, the probability that the diagnostic test comes back positive is 0.9 and if a person is not infected with the virus, the probability that the diagnostic test comes back positive is 0.25. Suppose a randomly seleted person is given the diagnostic test. Given that the test came back negative, find the probability that the person is infected with the virus.

Answer 1

P(person is infected) = 0.1

P(test comes back positive | person is infected) = 0.9

P(test comes back positive | person is not infected) = 0.25

P(test comes back positive) = P(test comes back positive | person is infected) * P(person is infected) + P(test comes back positive | person is not infected) * P(person is not infected)

                                             = 0.9 * 0.1 + 0.25 * (1 - 0.1)

                                             = 0.315

P(test comes back negative) = 1 - P(test comes back positive) = 1 - 0.315 = 0.685

P(test comes back negative | person is infected) = 1 - P(test comes back positive | person is infected) = 1 - 0.9 = 0.1

P(person is infected | test comes back negative) = P(test comes back negative | person is infected) * P( person is infected) / P(test comes back negative)

                                                                             = 0.1 * 0.1 / 0.685

                                                                             = 0.0146