Prosecutor's fallacy Guide, Meaning , Facts, Information and Description
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 concludes that the probability of the accused being innocent must be comparably tiny.
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2 Mathematical analysis 3 Defendant's fallacy 4 See also 5 External links |
A concrete example can make it clear why this reasoning is fallacious. Suppose there is a one-in-a-million chance of a match given that the accused is innocent. The prosector says that means there is only a one-in-a-million chance of innocence. But in a community of 10 million people, one expects about 10 matches by pure chance, and the accused is just one of those ten. That would indicate only a one-in-ten chance of guilt, if no other evidence is available.
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 defense claimed that these were two cases of sudden infant death syndrome; neither prosecution nor defense offered any other explanations for the deaths. The prosecution had expert witness Sir Roy Meadow testify that the probability of two children in the same family 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.) A higher court later quashed Sally Clark's conviction, on other grounds, on 29 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
Now consider this case: you win the lottery jackpot. You are then charged with having cheated, for instance with having bribed lottery officials. At the trial, the prosecutor points out that winning the lottery without cheating is extremely unlikely, and that therefore your being innocent must be comparably unlikely. This reasoning is clearly faulty: the prosecutor failed to mention that cheating lottery winners are much more rare than honest winners.
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.
We can view finding a person innocent or guilty in mathematical terms as a form of binary classification.
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 observed evidence, and I stands for "accused is innocent". We know that P(E|I) (the probability that the evidence would be observed if the accused were innocent) is tiny. The prosecutor wrongly concludes that P(I|E) (the probability that the accused is innocent, given the evidence E) is comparatively tiny. However, P(E|I) and P(I|E) are quite different; using Bayes' theorem we see
We can also formulate Bayes' theorem with odds:
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. (In the earlier example of a 10 million city, the presented evidence raises the a priori probability of guilt of 1 in 10 million to an a posteriori probability of guilt of 1 in 10.)
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.
The defendant's fallacy (taking the earlier example) would be to say, "We would expect 10 matches in this city of 10 million people, so this particular piece of evidence suggests there is 90% chance that the accused is innocent. So this evidence cannot be used to point to a conclusion of guilt, and should be excluded."
The problem with the defendant's argument is that there may be other available evidence which on its own is also not conclusive. For example if CCTV cameras surrounding the scene of the crime spotted all one hundred people there at the relevant time, one of which was the accused, then the defendant could claim: "The photograph suggests a 99% chance that the defendant is innocent. The match suggested a 90% chance of innocence. So the conclusion should be a finding of innocence."
When the photographic evidence is combined with the match, the two together point strongly towards guilt, since (assuming the chance of being in the photograph and having the match are independent) the chance that the accused is innocent falls to about 0.01%.
This is an Article on Prosecutor's fallacy. Page Contains Information, Facts Details or Explanation Guide About Prosecutor's fallacy Why this is fallacious: several examples
which is considerably more than 1 in 10,000. (The probability that exactly one of the 20,000 men has a match is about 27%, which is still rather high.)Mathematical analysis
So the a priori probability of innocence P(I) and the overall probability of the observed evidence P(E) need to be taken into account. If P(I) is much larger than P(E), then P(I|E) can be large as well.
Without knowledge of the a priori odds of I, the small value of P(E|I) does not necessarily imply that Odds(I|E) is small. (P(E|~I), the probability that the evidence is observed given the accused is guilty, is assumed to be high.)Defendant's fallacy
See also
External links