The following problem is designed to encourage your consideration of Bayes’ theorem. It shows how unaided judgment about probabilities can often be far off the mark. The problem is adapted from one appearing in an article in The Economist, “Getting the goat,” February 20, 1999, p.72. This article discusses how people who guess at probabilities can frequently be wrong:
A disease is present in the population at the rate of one person per thousand. A test for the disease becomes available. The drug company that is marketing the test randomly selects you to take the test. You agree, and the test results are positive. If the disease is present, the test always shows a positive result. However, the test has a 5% probability of showing a positive result when in fact the disease is not present. What is the probability that you have the disease?