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# Prosecutor's fallacy

The prosecutor's fallacy is a fallacy commonly occurring in criminal trials and elsewhere. A prosecutor has collected some evidence (for instance a DNA match) and has an expert testify that the probability of finding this evidence if the accused were innocent is tiny. The fallacy is committed if one then proceeds to claim that the probability of the accused being innocent is comparably tiny.

The fallacy lies in the fact that the a priori probability of guilt is not taken into account. If this probability is small, then the only effect of the presented evidence is to increase that probability somewhat, but not necessarily dramatically.

## Examples

Consider for instance the case of Sally Clark, who was accused in 1998 of having killed her first child at 11 weeks of age, then conceived another child and killed it at 8 weeks of age. The prosecution had an expert witness testify that the probability of two children dying from sudden infant death syndrome is about 1 in 73 million. To provide proper context for this number, the probability of a mother killing one child, conceiving another and killing that one too, should have been estimated and compared to the 1 in 73 million figure, but it wasn't. Ms. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistake. (See link at end of article.) Sally Clark's conviction was eventually quashed on other grounds on appeal on 29th January 2003.

In another scenario, assume a rape has been committed in a town, and 20,000 men in the town have their DNA compared to a sample from the crime. One of these men has matching DNA, and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000. This does not mean the probability that the suspect is innocent is 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance; the probability that there was at least one DNA match is 1-(1-1/10000)^20000, about 86% -- considerably more than 1 in 10,000. (The probability that exactly one of the 20,000 men has a match is 20000*(1/10000)*(1-1/10000)^19999, or about 27%, which is still rather high.)

Another instance of the prosecutor's fallacy is sometimes encountered when discussing the origins of life: the probability of life arising at random out of the physical laws is estimated to be tiny, and this is presented as evidence for a creator, without regard for the possibility that the probability of such a creator could be even tinier.

## Mathematical explanation of the general prosecutor's fallacy

We start with a thought experiment. I have a big bowl with one thousand balls, some of them made of wood, some of them made of plastic. I know that 100% of the wooden balls are white, and only 1% of the plastic balls are white, the others being red. Now I pull a ball out at random, and observe that it is actually white. Given this information, how likely is it that the ball I pulled out is made of wood? Is it 99%? No! Maybe the bowl contains only 10 wooden and 990 plastic balls. Without that information (the a priori probability), we cannot make any statement. In this thought experiment, you should think of the wooden balls as "accused is guilty" or "life originated from a creator", the plastic balls as "accused is innocent" or "life emerged without a creator", and the white balls as "the evidence is observed" or "life developed".

The fallacy can be analyzed using conditional probability: Suppose E is the evidence, and G stands for "guilt". We are interested in Odds(G|E) (the odds that the accused is guilty, given the evidence) and we know that P(E|~G) (the probability that the evidence would be observed if the accused were innocent) is tiny. One formulation of Bayes' theorem then states:

Odds(G|E) = Odds(G) · P(E|G)/P(E|~G)
Without knowledge of the a priori odds of G, the small value of P(E|~G) does not necessarily imply that Odds(G|E) is large.

The prosecutor's fallacy is therefore no fallacy if the a priori odds of guilt are assumed to be 1:1. In an Bayesian approach to personal probabilities, where probabilities represent degrees of belief of reasonable persons, this assumption can be justified as follows: a completely unbiased person, without having been shown any evidence and without any prior knowledge, will estimate the a priori odds of guilt as 1:1.

In this picture then, the fallacy consists in the fact that the prosecutor claims an absolutely low probability of innocence, without mentioning that the information he conveniently omitted would have led to a different estimate.

In legal terms, the prosecutor is operating in terms of a presumption of guilt, something which is contrary to the normal presumption of innocence where a person is assumed to be innocent unless found guilty. A more reasonable value for the prior odds of guilt might be a value estimated from the overall frequency of the given crime in the general population.