The Prosecutor's Fallacy

 This is a new one to me by name but I recognize the statistical issue.  The Prosecutor's Fallacy:

The prosecutor's fallacy is a fallacy of statistical reasoning typically used by a prosecutor to exaggerate the likelihood of a criminal defendant's guilt. The fallacy can be used to support other claims as well – including the innocence of a defendant.

The following claim demonstrates the fallacy in the context of a prosecutor questioning an expert witness: "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".[1] The claim obscures that the likelihood of the defendant's innocence, given the evidence found on him, in fact depends on the likely quite high prior odds of the defendant being a random innocent person – as well as the stated low odds of finding the evidence on such a random innocent person, not to mention the underlying high odds that the evidence is indeed indicative of guilt.

At its heart, the fallacy involves assuming that the prior likelihood of a random match is equal to the likelihood that the defendant is innocent. For instance, if a perpetrator were known to have the same blood type as a 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 likelihood of the defendant's guilt is 90%. The actual likelihood would depend on the size of the population with a matching blood type and would likely be much lower.

Which seems pretty esoteric.  An actual (tragic) case is a better illustration.  Also from the Wikipedia. 

The Sally Clark case

Sally Clark, a British woman, was accused in 1998 of having killed her first child at 11 weeks of age and then her second child at 8 weeks of age. The prosecution had expert witness Sir Roy Meadow, a professor and consultant paediatrician, testify that the probability of two children in the same family dying from SIDS is about 1 in 73 million. That was much less frequent than the actual rate measured in historical data – Meadow estimated it from single-SIDS death data, and the assumption that the probability of such deaths should be uncorrelated between infants.

Meadow acknowledged that 1-in-73 million is not an impossibility, but argued that such accidents would happen "once every hundred years" and that, in a country of 15 million 2-child families, it is vastly more likely that the double-deaths are due to Münchausen syndrome by proxy than to such a rare accident. However, there is good reason to suppose that the likelihood of a death from SIDS in a family is significantly greater if a previous child has already died in these circumstances (a genetic predisposition to SIDS is likely to invalidate that assumed statistical independence) making some families more susceptible to SIDS and the error an outcome of the ecological fallacy. The likelihood of two SIDS deaths in the same family cannot be soundly estimated by squaring the likelihood of a single such death in all otherwise similar families.

1-in-73 million greatly underestimated the chance of two successive accidents, but, even if that assessment were accurate, the court seems to have missed the fact that the 1-in-73 million number meant nothing on its own. As an a priori probability, it should have been weighed against the a priori probabilities of the alternatives. Given that two deaths had occurred, one of the following explanations must be true, and all of them are a priori extremely improbable:

Two successive deaths in the same family, both by SIDS

Double homicide (the prosecution's case)

Other possibilities (including one homicide and one case of SIDS)

It's unclear whether an estimate of the probability for the second possibility was ever proposed during the trial, or whether the comparison of the first two probabilities was understood to be the key estimate to make in the statistical analysis assessing the prosecution's case against the case for innocence.

Mrs. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistakes.

In 2002, Ray Hill (Mathematics professor at Salford) attempted to accurately compare the chances of these two possible explanations; he concluded that successive accidents are between 4.5 and 9 times more likely than are successive murders, so that the '"a priori"' odds of Clark's guilt were between 4.5 to 1 and 9 to 1 against.

After it was found that the forensic pathologist who had examined both babies had withheld exculpatory evidence, a higher court later quashed Sally Clark's conviction, on 29 January 2003.

Sally Clark was a practising solicitor before the conviction. After her three-year imprisonment she developed a number of serious psychiatric problems including serious alcohol dependency and died in 2007 from acute alcohol poisoning.

Statistics is a marvelously useful tool set but at the theoretical margin, you have to be extremely careful in its application.