Quick Answer: Even with an highly accurate medical test (e.g., 99% accuracy), if you are testing for a very rare disease, a positive test result is often more likely to be a false positive than a true positive. This is called the False Positive Paradox or the Base Rate Fallacy.
The Scenario: A 99% Accurate Test
Imagine a disease affects exactly 1 in 1,000 people (a rare disease). You take a test for this disease that is 99% accurate (it correctly identifies 99% of sick people, and correctly identifies 99% of healthy people). Your test comes back positive. What is the probability that you actually have the disease?
Most people guess 99%. The real answer is less than 10%.
The Math (Using 100,000 People)
Let's run the numbers on a population of 100,000 people to see why.
- Out of 100,000 people, 100 actually have the disease (1 in 1,000).
- The test is 99% accurate, so it correctly identifies 99 of those sick people as positive.
- The remaining 99,900 people are healthy.
- The test is 99% accurate, meaning it is wrong 1% of the time. 1% of 99,900 healthy people is 999 people.
- So, the test will incorrectly flag 999 healthy people as having the disease (False Positives).
The Final Calculation
We now have two groups of people with positive test results:
- True Positives: 99 sick people
- False Positives: 999 healthy people
- Total positive results: 1,098
If you get a positive result, you are one of those 1,098 people. The chance you actually have the disease is 99 ÷ 1,098 = 0.09, or about 9%. Even with a 99% accurate test, a positive result means you probably DON'T have the rare disease.
The Lesson: Base Rate Matters
When estimating probability, the "base rate" (how common the thing is in the general population) matters immensely. When hunting for rare things, even a tiny error rate will produce a mountain of false positives that overwhelm the genuine hits.