Question

The probability that a person actually has the disease is 0.3, implying that out of 1000 people, 30 people have the disease. A new test has a false positive probability of 0.01 and a false negative probability of 0.02. A person is given the test and the result is positive, what is the probability they actually have the disease?

Answer 1

Let D shows the event that person has disease. Let P shows the event that test gives positive result and N shows the event that test gives negative result. So we have

And we have

By the complement rule we have

(a)

By the law of total probability, we have

If the test reports a positive result for a person selected at random from the population, the probability that this person actually has the disease is

Answer: 0.7519