Question

A diagnostic test has a 95% probability of giving a positive result when given to a person who has a certain disease. It has a 10% probability of giving a (false) positive result when given to a person who doesn’t have the disease. It is estimated that 15% of the population suffers from this disease.

(a) What is the probability that a test result is positive?

(b) A person recieves a positive test result. What is the probability that this person actually has the disease? (probability of a true positive)

(c) A person recieves a positive test result. What is the probability that this person doesn’t actually have the disease? (probability of a false negative)

Answer 1

P(having the disease) = 0.15

P(positive test | have the disease) = 0.95

P(positive test | don't have the disease) = 0.1

a) P(positive test result) = P(positive test | have the disease) * P(have the disease) + P(positive test | don't have the disease) * P(don't have the disease)

                                      = 0.95 * 0.15 + 0.1 * (1 - 0.15)

                                      = 0.2275

b) P(has the disease | positive test) = P(positive test | have the disease) * P(have the disease) / P(positive test result)

                                                         = 0.95 * 0.15 / 0.2275

                                                         = 0.6264

c) P(doesn't have the disease | positive test) = P(positive test | don't have the disease) * P(don't have the disease) / P(positive test result)

                                                                       = 0.1 * (1 - 0.15) / 0.2275

                                                                       = 0.3736