Suppose a test can determine with 99 percent accuracy whether someone has a certain disease, which only affects one out of every 10,000 people in the population. If a given person’s test comes back positive, what are the approximate odds that the person has the disease?

Many people assume that because the test is 99 percent accurate, a person with positive results has about a 99 percent chance of having the disease. However, the correct answer is about one percent. To understand why, imagine the test is given to 10,000 people. We know that on average only one of these people will have the disease. A test that is 99 percent accurate will be wrong one out of every 100 times the test is given. So if 10,000 people are tested, the one person who has the disease will likely test positive, but there will likely be 100 false positives. Therefore, for any person who tests positive, the odds of them having the disease are one out of 101, or about one percent.

The false positive paradox is a type of base rate fallacy, where prior probability information, in this case the incidence of the disease in the population, is ignored. This is a real problem in the field of medicine, as surveys indicate that many patients and even physicians simply do not understand how to interpret information such as cancer screening statistics.