Prosecutor’s Fallacy

Lucia de Berk

damned lies

The prosecutor’s fallacy is a fallacy of statistical reasoning, typically used by the prosecution to argue for the guilt of a defendant during a criminal trial (though some variants are utilized by defense lawyers arguing for the innocence of their client). The fallacy involves assuming that the prior probability of a random match is equal to the probability that the defendant is innocent. For instance, if a perpetrator is known to have the same blood type as a defendant and 10% of the population share that blood type, then to argue on that basis alone that the probability of the defendant being guilty is 90% makes the prosecutors’s fallacy.

Consider the case of a lottery winner accused of cheating based on the improbability of winning. At the trial, the prosecutor calculates the (very small) probability of winning the lottery without cheating and argues that this is the chance of innocence. The logical flaw is that the prosecutor has failed to account for the large number of people who play the lottery.

The fallacy results from misunderstanding conditional probability (the probability of an event given that another event has occurred) and neglecting the prior odds of a defendant being guilty before that evidence was introduced. When 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 occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. This is a fallacious argument because the odds in this scenario do not relate to the odds of being guilty,but merely to the odds of being picked at random.

The term was originated by lawyer and psychologist William C. Thompson and statistician Edward Schumann in 1987. The fallacy can arise from multiple testing, such as when evidence is compared against a large database. The size of the database elevates the likelihood of finding a match by pure chance alone; e.g., DNA evidence is soundest when a match is found after a single directed comparison because the existence of matches against a large database where the test sample is of poor quality may be less unlikely by mere chance. ‘Cold hits’ like this on DNA databanks are now understood to require careful presentation as trial evidence.

Berkson’s paradox, a subtype of the prosecutor’s fallacy involves mistaking conditional probability for unconditional. It led to several wrongful convictions of British mothers, accused of murdering two of their children in infancy, where the primary evidence against them was the statistical improbability of two children dying accidentally in the same household. Though multiple accidental sudden infant deaths are rare, so are multiple murders; with only the facts of the deaths as evidence, it is the ratio of these (prior) improbabilities that gives the correct ‘posterior probability’ of murder.

Some authors have cited defence arguments in the O. J. Simpson murder trial as an example of this fallacy regarding the context in which the accused had been brought to court: crime scene blood matched Simpson’s with characteristics shared by 1 in 400 people. The defense argued that a football stadium could be filled with Angelenos matching the sample and that the figure of 1 in 400 was useless. Also at the trial, the prosecution presented evidence that Simpson had been violent toward his wife, while the defense argued that there was only one woman murdered for every 2500 women who were subjected to spousal abuse, and that any history of Simpson being violent toward his wife was irrelevant to the trial. However, some regard the reasoning behind the defense’s calculation as fallacious. According to author Gerd Gigerenzer, the correct probability requires the context—that Simpson’s wife had not only been subjected to domestic violence, but subjected to domestic violence and murdered—to be taken into account. Gigerenzer writes ‘the chances that a batterer actually murdered his partner, given that she has been killed, is about 8 in 9 or approximately 90%.’

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