09/02/2026
Imagine you visit a doctor. She runs a test for a rare disease, one that affects one person in a thousand. The test is very accurate. It correctly identifies the disease 99% of the time, and it gives a false positive only 1% of the time. Your test comes back positive. How worried should you be?
I'm sorry to tell you that it's a positive result.
Most people's instinct is very worried. The test is 99% accurate. That sounds nearly certain. But the mathematics tells a different story, and it is one of the most counterintuitive results in all of probability. Here's why.
The test was accurate in two different ways. Among the small group of people who had the disease, it caught 99% of them. Those are true positives. Among the much larger group who did not have the disease, it correctly cleared 99% of them true negatives. But look at the 1% it got wrong. Those are healthy people the test flagged as sick. False positives. Let me show you. Start with 10,000 people. About 10 have the disease. 9,990 do not. Among the 10, roughly 10 true positives. Among the 9,990, about 100 false positives. 1% of a very large group is bigger than 99% of a very small one. So, put the two side by side. About 10 true positives, about 100 false positives. 110 positive results in total, and only 10 of them are real. 10 divided by 110 is roughly 9%. A positive result on a 99% accurate test does not mean a 99% chance of disease. In this example, there is only about a 9% chance that a positive result means the person actually has the disease.
That result is shocking the first time you encounter it.
Yes, it certainly is.
It was also for most of Western history incalculable. No general systematic method was widely available for combining prior knowledge, the rarity of the disease with new evidence, the test result, to arrive at an updated probability until a minister in Tundridge, Wales, England, worked it out in private sometime in the middle of the 18th century. Welcome to the pioneers of probability with me, Mark Hebner. https://www.ifa.com/videos/pioneers-of-probability-thomas-bayes
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