Lesson 6

Let's consider a canonical example of using this formula, which will once again turn out to be counterintuitive. Suppose there is a certain disease that, on average, affects one person out of a hundred. And there is a test that detects this disease fairly well. It is reliable but not 100 percent accurate:

• If a person is healthy, the test will indicate illness with a probability of \(5\) percent (false positive error).


• If a person is ill, the test will indicate no disease with a probability of \(10\) percent (false negative error).

The test seems quite reliable (it errs with a probability of only \(5\)–\(10\) percent). Suppose that for a randomly chosen person, the test was positive. Is the person really ill? We already know that this may not be the case since the test sometimes makes mistakes. Therefore, a more reasonable question is: what is the probability that this person is actually ill?

Problem. Check your intuition! What is the approximate probability that a randomly chosen person with a positive test result is actually ill?