Prosecutor's Fallacy

Transposing the conditional

The heart of the prosecutor's fallacy is treating P(evidence | innocent) as if it were P(innocent | evidence). The two are linked by Bayes' theorem, and between them sits the prior probability, how likely the defendant was to be guilty before the evidence was seen. If a bloodstain with a one-in-a-million random match probability is found in a city of 10 million, about 10 people would be expected to match by chance. For someone arrested on the match alone, with no other evidence, the probability of guilt is not 999,999 in a million but closer to about 1 in 11. Thompson and Schumann (1987) showed in mock jury experiments that many people commit this error. Because it ignores prior probabilities, it is also a form of the base rate fallacy.

Courtroom cases and DNA evidence

In the 1968 California Supreme Court case People v. Collins, the prosecution multiplied the probabilities of the culprits' characteristics to argue that the chance an innocent couple would share them all was 1 in 12 million, but the court reversed the conviction, faulting both the assumption that the characteristics were independent and the interpretation of the probability. Finkelstein and Fairley (1970) proposed Bayesian evaluation of evidence, while Tribe (1971) warned of the dangers of mathematical reasoning in the courtroom. Koehler (1993) and Balding and Donnelly (1994) analyzed cases in which random match probabilities for DNA evidence were misrepresented through the prosecutor's fallacy. In the UK's Sally Clark case (1999), testimony that the chance of two sudden infant deaths in one family was 1 in 73 million helped secure a conviction. That figure wrongly assumed the two deaths were independent and was also presented as if it were the probability of innocence. Hill (2004) and Nobles and Schiff (2005) analyzed this misuse of statistics, and the conviction was overturned on appeal in 2003.

The defense attorney's fallacy and Bayesian evaluation

Thompson and Schumann (1987) also identified the opposite error, the defense attorney's fallacy. It argues that since dozens of people would match, the defendant is just one of them and the evidence is meaningless, ignoring the fact that the evidence has greatly narrowed the pool of suspects. Robertson and Vignaux (1995) and Aitken and Taroni (2004) established as the forensic standard an approach that expresses the strength of evidence as a likelihood ratio, the ratio of the probability of the evidence under the guilty and innocent hypotheses. Dawid (2002) analyzed jurors' weighing of evidence with Bayes' theorem, and Fenton and Neil (2011) and Fenton, Neil and Lagnado (2013) showed how Bayesian networks can model dependencies among complex pieces of evidence. Gigerenzer (2002) showed that expressing such probabilistic information as natural frequencies greatly improves understanding among lawyers and jurors.

FAQ

What is the prosecutor's fallacy in simple terms?

It's the mistake of reading "an innocent person is very unlikely to match this evidence by chance" as "this person is very unlikely to be innocent." Those two probabilities can be wildly different.

Why is it called the prosecutor's fallacy?

Because this reasoning usually works against the defendant, making the evidence seem to point strongly toward guilt. In practice, expert witnesses, judges, jurors and the media can make the same mistake, not just prosecutors.

Sources reviewed: 2026-10-26

References

1. Finkelstein, M. O., & Fairley, W. B. (1970). A Bayesian approach to identification evidence. Harvard Law Review, 83(3), 489–517.
2. Tribe, L. H. (1971). Trial by mathematics: Precision and ritual in the legal process. Harvard Law Review, 84(6), 1329–1393.
3. Thompson, W. C., & Schumann, E. L. (1987). Interpretation of statistical evidence in criminal trials: The prosecutor's fallacy and the defense attorney's fallacy. Law and Human Behavior, 11(3), 167–187.
4. Koehler, J. J. (1993). Error and exaggeration in the presentation of DNA evidence at trial. Jurimetrics, 34(1), 21–39.
5. Balding, D. J., & Donnelly, P. (1994). The prosecutor's fallacy and DNA evidence. Criminal Law Review, 711–721.
6. Robertson, B., & Vignaux, G. A. (1995). Interpreting Evidence: Evaluating Forensic Science in the Courtroom. Chichester: Wiley.
7. Gigerenzer, G. (2002). Calculated Risks: How to Know When Numbers Deceive You. New York: Simon & Schuster.
8. Dawid, A. P. (2002). Bayes's theorem and weighing evidence by juries. In R. Swinburne (Ed.), Bayes's Theorem (Proceedings of the British Academy, 113, pp. 71–90). Oxford: Oxford University Press.
9. Aitken, C. G. G., & Taroni, F. (2004). Statistics and the Evaluation of Evidence for Forensic Scientists (2nd ed.). Chichester: Wiley.
10. Hill, R. (2004). Multiple sudden infant deaths—coincidence or beyond coincidence? Paediatric and Perinatal Epidemiology, 18(5), 320–326.
11. Nobles, R., & Schiff, D. (2005). Misleading statistics within criminal trials: The Sally Clark case. Significance, 2(1), 17–19.
12. Fenton, N., & Neil, M. (2011). Avoiding probabilistic reasoning fallacies in legal practice using Bayesian networks. Australian Journal of Legal Philosophy, 36, 114–151.
13. Fenton, N., Neil, M., & Lagnado, D. A. (2013). A general structure for legal arguments about evidence using Bayesian networks. Cognitive Science, 37(1), 61–102.