Prosecutor's fallacy

The prosecutor's fallacy is a logical 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.

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 (http://www.rss.org.uk/archive/reports/sclark.html).

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.

Before explaining the fallacy mathematically, let's try a thought experiment. I have a big bowl with one thousand balls, some of them made of wood ("guilty" or "creator"), some of them made of plastic ("innocent" or "random origin"). I know that 100% of the wooden balls are white ("DNA match" or "life developed"), 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, we cannot make any statement.

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 is 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 odds of G, the small value of P(E|~G) does therefore not necessesarily imply that Odds(G|E) is large.