The prosecutor's fallacy: a brief history.

 Prosecutor's fallacy 

The prosecutor's fallacy is a fallacy of statistical reasoning involving a test for an occurrence, such as a DNA match. A positive result in the test may paradoxically be more likely to be an erroneous result than an actual occurrence, even if the test is very accurate. The fallacy is named because it is typically used by a prosecutor to exaggerate the probability of a criminal defendant's guilt. The fallacy can be used to support other claims as well – including the innocence of a defendant. 

A simple example illustrating the prosecutor's fallacy. (The hypothesis is the possibility that the defendant is guilty, whereas the evidence found refers to a positive test result, such as a DNA or blood type match.) Although the probability that evidence is found on the real culprit may be quite high (in this case, two-fifths, or 40%), the probability that a certain individual is guilty given that said evidence was found in him is unrelated to the former, and will often be much smaller (in this example, only two-eighths, or 25%). For instance, if a perpetrator were known to have the same blood type as a given defendant and 10% of the population to share that blood type, then one version of the prosecutor's fallacy would be to claim that, on that basis alone, the probability that the defendant is guilty is 90%. However, this conclusion is only close to correct if the defendant was selected as the main suspect based on robust evidence discovered prior to the blood test and unrelated to it (the blood match may then be an "unexpected coincidence"). Otherwise, the reasoning presented is flawed, as it overlooks the high prior probability (that is, prior to the blood test) that he is a random innocent person. Assume, for instance, that 1000 people live in the town where the murder occurred. This means that 100 people live there who have the perpetrator's blood type; therefore, the true probability that the defendant is guilty – based on the fact that his blood type matches that of the killer – is only 1%, far less than the 90% argued by the prosecutor. 

At its heart, therefore, the fallacy involves assuming that the prior probability of a random match is equal to the probability that the defendant is innocent. When using it, a prosecutor questioning an expert witness may ask: "The odds of finding this evidence on an innocent man are so small that the jury can safely disregard the possibility that this defendant is innocent, correct?"[1] The claim assumes that the probability that evidence is found on an innocent man is the same as the probability that a man is innocent given that evidence was found on him, which is not true. Whilst the former is usually small (approximately 10% in the previous example) due to good forensic evidence procedures, the latter (99% in that example) does not directly relate to it and will often be much higher, since, in fact, it depends on the likely quite high prior odds of the defendant being a random innocent person. 

Mathematically, the fallacy results from misunderstanding the concept of a conditional probability, which is defined as the probability that an event A occurs given that event B is known – or assumed – to have occurred, and it is written as P(A|B). The error is based on assuming that P(A|B) = P(B|A), where A represents the event of finding evidence on the defendant, and B the event that the defendant is innocent. But this equality is not true: in fact, although P(A|B) is usually very small, P(B|A) may still be much higher.