The mistake in plain words
Suppose a piece of evidence is very rare among innocent people. Say that only 1 in 10,000 innocent people would match it. A person matches. It is tempting to say: “there is only a 1 in 10,000 chance this person is innocent.” That sentence is the prosecutors fallacy.
The error is a swap of two different questions:
- How likely is this evidence if the person is innocent? Here, 1 in 10,000.
- How likely is it that the person is innocent, given this evidence? This is a different number, and it can be far larger.
These two are conditional probabilities running in opposite directions. As the article on conditional probability explains, P(A | B) and P(B | A) are generally not equal. The fallacy treats them as if they were.
This article is an explanation of a reasoning error, not legal advice, and it does not describe how any particular court or legal system works. The example is invented.
A worked example with counts
Imagine a crime for which the guilty person must be one of 1,000,000 people in some population, and assume for the example that the offender is in that group. A forensic feature is found at the scene. It is shared by 1 in 10,000 people, and a suspect is found to have it.
The match probability is 1 in 10,000. What is the chance the suspect is the source, if the match is the only evidence?
Count the people in the population:
- The offender has the feature (assume it is detected reliably). That is 1 person.
- Of the other 999,999 people, about 1 in 10,000 also have it: roughly 100 people. (999,999 / 10,000 is just under 100, so about 100.)
So around 101 people match, and exactly one of them is the offender. If the suspect was picked only because they match, the chance they are the offender is about 1/101, which is roughly 1%.
Compare that to the fallacious reading, “99.99% chance of guilt”. The gap between the two is enormous, and it comes from the same place as base rate neglect: with a large pool of possible people, even a rare feature is shared by many.
Where the numbers come from
The correct approach is Bayes theorem. Start with a prior: before the evidence, any particular person in the pool is the offender with chance 1 in 1,000,000. Then the evidence updates it. The likelihood of the match is 1 if they are the offender and 1/10,000 if they are not. After updating, the odds that the suspect is the offender are:
(1 / 999,999) × (1 / (1/10,000)) = 10,000 / 999,999
That is about 1 to 100, which gives roughly 1/101 as the chance. The match did raise the chance dramatically, from about one in a million to about one in a hundred, a gain of around ten thousandfold. It simply did not raise it to near certainty.
Why real cases differ
The simple example deliberately includes no other evidence. In real situations the picture is rarely that bare:
- There is often other information that narrows the pool or raises the prior, such as where the person was.
- The suspect may be identified for reasons unrelated to the match, so they are not just a random draw from the matching group.
- The match probability itself can be uncertain, and it may be calculated under assumptions that are not quite right.
These points can push the answer either way. The lesson is not that matches are weak. It is that the number “1 in 10,000” describes the evidence, and the question of guilt needs the prior and the whole of the evidence together.
The opposite error
A related mistake sometimes called the defence attorney’s fallacy goes the other way: “100 other people match, so this evidence is worthless.” That also misleads. Going from one in a million to about one in a hundred is a big shift. The evidence is informative; it just needs to be combined sensibly with everything else. Treating it as either proof or nothing at all ignores the arithmetic.
Spotting the fallacy elsewhere
The same pattern appears far from courtrooms:
- A screening or detection system says a result is “99% unlikely under normal conditions”, so people assume the result is 99% likely to be a real signal. This is the structure behind false positives.
- A seemingly extraordinary streak is described as having “a one in a million chance of happening by luck”, so someone concludes it was not luck. But the relevant question is how many people and streaks are being looked at, a point also made in discussions of coincidence myths.
- A player on a hot streak “could only happen 1% of the time by chance”, ignoring that many players are observed.
In each case, the rare probability describes how often the pattern occurs under one assumption. It does not directly give the chance that the assumption is true.
A checklist
Before drawing a conclusion from a rare coincidence, ask:
- What exactly is the probability describing: the evidence given a hypothesis, or the hypothesis given the evidence?
- How large is the pool of people or events in which a match could turn up?
- How many matches would we expect by chance in that pool?
- What other information bears on the prior?
Converting to counts of people, as in the example, answers the second and third questions almost automatically. And as a general reminder in the context of games of chance: patterns that look striking are common in random data, past results do not alter future odds, and nobody should read a streak as a signal to bet more. If gambling is causing harm, a local support service can help.