The Prosecutor's Fallacy
When rare evidence doesn't mean guilt
The Fallacy
A DNA sample matches the defendant. The probability of a random match is 1 in a million. Therefore, the defendant is almost certainly guilty—right?
Wrong. This reasoning has sent innocent people to prison. The probability that evidence matches an innocent person is NOT the same as the probability that a matching person is innocent.
The prosecutor's fallacy confuses two very different probabilities:
- P(Evidence | Innocent) — The probability of finding this evidence if the person is innocent
- P(Innocent | Evidence) — The probability that the person is innocent given the evidence
These are NOT the same. In fact, they can be dramatically different—and confusing them has led to some of the greatest miscarriages of justice in modern history.
⚖️ The Probability Calculator
See how rare evidence doesn't always mean guilt
The Sally Clark Tragedy
Sally Clark, a British solicitor, lost two infant sons to what appeared to be Sudden Infant Death Syndrome (SIDS). She was charged with murdering both children.
At trial, pediatrician Roy Meadow testified that the probability of two SIDS deaths in one family was 1 in 73 million. He arrived at this by squaring the probability of a single SIDS death (1 in 8,543).
The jury convicted her. She spent three years in prison.
But the statistics were catastrophically wrong:
- SIDS deaths are NOT independent—they run in families
- Even if rare, the probability of guilt ≠ probability of rare event
- Double murder of one's own children is ALSO extremely rare
- When compared properly, SIDS was 4-9x MORE likely than murder
Her conviction was overturned in 2003 after hidden evidence emerged. Tragically, Sally Clark never recovered from the trauma and died in 2007 from alcohol poisoning.
The Royal Statistical Society issued a public statement condemning the misuse of statistics in her case. It became a landmark example of how statistical illiteracy can destroy lives.
The Mathematics
Bayes' Theorem
The probability of guilt given evidence depends on the base rate of guilt in the population—not just how rare the evidence is.
A Worked Example
Imagine a city of 1 million people. One person committed a crime. DNA evidence has a 1 in 1 million random match rate.
- The guilty person will match (probability = 1)
- Among 999,999 innocent people, about 1 will randomly match
- So 2 people match the evidence: 1 guilty, 1 innocent
- If you pick one of the matching people, there's a 50% chance they're guilty
The prosecutor who claims "1 in a million chance of innocence" is off by a factor of 500,000.
Real-World Implications
The False Positive Paradox
This problem gets worse with mass surveillance. If you test 1 million airline passengers with a 99% accurate terrorist detector:
- If there are 10 actual terrorists, you catch ~10
- But 1% of 999,990 innocent people = ~10,000 false positives
- So 99.9% of the people flagged are innocent!
Medical Testing
The same logic applies to rare disease screening. A 99% accurate test for a disease affecting 1 in 10,000 people will mostly produce false positives.
How to Avoid It
- Always consider base rates — How common is guilt/disease in the population?
- Use Bayes' Theorem — Combine evidence probability with prior probability
- Beware of "1 in X" claims — Ask: 1 in X of what population?
- Require corroborating evidence — One rare match isn't proof
Other Famous Cases
A nurse convicted of murdering patients based on the statistical "impossibility" that she was present at so many deaths. The court found a 1 in 342 million chance. She served 6 years before exoneration when the statistical reasoning was debunked.
Blood evidence matched Simpson at 1 in 400 odds. Defense attorney Alan Dershowitz argued that in Los Angeles, this matched thousands of people—filling a football stadium. The 1 in 400 figure alone was not proof of guilt.