The Prosecutor's Fallacy

A crime has been committed. DNA found at the scene is tested against a database of 100,000 people. One person's DNA matches. The test is 99.9% accurate, with a false positive rate of 0.1%.

The prosecutor tells the jury: "There is only a 0.1% chance this match is wrong. The probability of the defendant's innocence is one in a thousand."

This sounds compelling. It is also completely wrong, and understanding why is one of the most useful things you can learn about probability.

Two Very Different Questions

The prosecutor is conflating two probabilities that feel similar but are vastly different:

The first probability is about the test's accuracy. The second is about the defendant's guilt. They are not interchangeable, and the gap between them depends on something the prosecutor didn't mention: how many people were tested.

The Math That Changes Everything

If you search a database of 100,000 people with a test that has a 0.1% false positive rate, you should expect about 100 false matches (100,000 × 0.001). Plus the one true match (the actual perpetrator, if they're in the database).

So you have ~101 total matches, and only 1 is the real perpetrator. If the defendant was found through this database search, the probability they're guilty isn't 99.9%. It's closer to 1 in 101, or about 1%.

The test is extremely accurate. And the defendant is almost certainly innocent.

See It

Adjust the database size and test accuracy below, and watch how the probability of guilt changes. Pay attention to what happens when you search larger populations:

Simulate the Database Search

Click below to simulate actual database searches. Each run tests every person in the database and reports how many matches are found. You'll see that most matches are false positives: innocent people who happened to trigger the test.

Why This Happens

The core issue is base rates. In a large database, the vast majority of people are innocent. Even a very accurate test, applied to a mostly-innocent population, will produce more false positives than true positives, because there are so many more innocent people to falsely match.

Think of it this way: if 1 person in 100,000 is guilty and the test is 99.9% accurate, the test correctly identifies the guilty person almost every time. But it also incorrectly flags ~100 innocent people. The true match is buried in a haystack of false ones.

This is Bayes' theorem in action. The probability of guilt given a match depends not just on the test's accuracy, but on how rare guilt is in the tested population. The rarer the thing you're looking for, the more false positives dominate.

Real Cases

• The UK National DNA Database: In the early 2000s, cold case searches against the database produced matches that were presented as near-certain evidence. Defense attorneys began challenging these on prosecutor's fallacy grounds, leading to acquittals and revised forensic guidelines.
• Sally Clark case (1999): A mother was convicted of murdering her two children partly based on an expert's claim that the odds of two crib deaths in one family were 1 in 73 million. This ignored base rates and assumed independence between the events. She was eventually exonerated.
• COVID testing (2020-2021): When rapid tests with ~1% false positive rates were used for mass screening of low-prevalence populations, a significant fraction of positive results were false positives, leading to unnecessary quarantines and public confusion about test reliability.

The Transferable Insight

The prosecutor's fallacy isn't just a courtroom problem. It appears whenever a rare condition is tested for in a large population:

• Medical screening: A mammogram with 90% sensitivity and a 5% false positive rate sounds reliable. But if only 1% of women screened have cancer, most positive results are false positives. This is why follow-up tests exist. The first screen is a filter, not a diagnosis.
• Security and fraud detection: An airport security system that's 99% accurate sounds great until you consider that it's screening millions of innocent travelers. The false alarm rate overwhelms the true detection rate.
• Hiring and filtering: Any filter applied to a large pool where the "hit" is rare will produce mostly false positives. The more candidates you screen, the less a single signal means.

The habit worth building: whenever a test result seems definitive, ask how many people were tested and how common the thing being tested for actually is. A match from a test is the beginning of an investigation, not the end of one. The rarer the condition and the larger the population, the more skeptical you should be of any single positive result.