The Deadly Disease Puzzler

This is one of the world's most significant puzzlers. This puzzler dramatically shows how easy it is to jump to a terribly wrong conclusion if you don't understand statistics. Furthermore, this is a realistic situation. Thousands of people every year face similar situations.

Your doctor gives you the bad news that you have tested positive for a deadly disease. The test is 99% reliable. Of people who actually have the disease, 99% test positive and only 1% test negative. Of people who don't have the disease, 99% test negative and only 1% test positive.

Unfortunately, you have tested positive. What is the probability that you have the disease? Or answer this simpler question: Are you likely to have the disease or not?

Think about it before looking at the answer.


The answer is: You cannot answer the question! You don't have enough information. Your chance of having the disease may be very low or very high, depending on whether the disease is rare or common in the general population.

For example, say that in a population of 1 million people, only 100 people have the disease. Of the people who test positive, some will be true positives (they actually have the disease) and some will be false positives (they don't have the disease). Let's determine approximately how many people out of a population of 1 million are true positives and how many are false positives. If you were to test the 100 people who actually have the disease, 99%, or about 100 people, would test positive (these are the true positives). If you were to test the approximatey 1 million people who do not have the disease, 1%, or about 10,000 people, would test positive (these are the false positives). So there are about 100 true positives and about 10,000 false positives. So, in the case of this rare disease, if you test positive, there is only about a 1% chance that you are a true positive and actually have the disease.