In 1999, a British woman named Sally Clark lost two infant sons within two years, both to what appeared to be sudden infant death syndrome (SIDS). A disease that is so rare that she was consequently tried for the murders of her kids. At the trial the evidence was presented that the probability of two deaths by SIDS happening in the same family was 1 in 73 million, something so improbable that the jurors had no other choice but to convict her for murder. She sat in prison for three years until statisticians proved that this probability was anything but correct. This trial caused her to have huge psychiatric problems and she died of alcohol intoxication not too much longer.
The flaw in reasoning used in Sally Clark’s trial was already widely known and studied in statistics, it is called the prosecutor’s fallacy. You fall into this trap by confusing the probability of A given B, with the probability of B given A. Mathematically this looks very easy, however humans are very bad at interpreting the difference. This difference in the courtroom can destroy someone’s life.
An easy example is as follows. Imagine that a woman’s purse is stolen by a man. She reports to the police that the man has the following descriptive features: a thick moustache, a scar on his neck and red curly hair. The police find a man that fits the description and instantly arrest him. In court it is presented that only one in a million innocent people share these features. The jury, hence, thinks that he only has a one in a million chance of being innocent and reaches the verdict that the man is guilty. This, however, is false. Let’s say that the man comes out of a city with a population of 20 million, then 20 people would match his description. Hence he would have a 19 out of 20 chance of being innocent. This is a probability of 95%! The prosecutor calculated the probability of having those features given innocence. The jury needed the probability of innocence given those features.
The most infamous case of the abuse of this fallacy was in the OJ Simpson case. The prosecution spent its opening ten days documenting OJ's history of abusing Nicole Brown Simpson, arguing that a man with that pattern of violence had the motive to kill. Alan Dershowitz, the Harvard law professor advising the defence, hit back with a statistic: fewer than 1 in 2,500 men who abuse their partners ever go on to murder them. The jury heard it as a demolition of the prosecution's evidence. It was nothing of the sort.
Dershowitz's number answered the question: of all men who abuse their partners, what fraction eventually kill them? That number is low, and it is correct. But Nicole was already dead. The relevant question was what probability to assign to OJ being the killer, given both the history of abuse and the fact that she had been murdered. Statisticians worked through that version and got roughly 90%. The abuse history was not a distraction from the evidence, it should have been the most powerful evidence in the trial. OJ was set free in October 1995, after the jury deliberated for less than four hours. What makes the prosecutor's fallacy so effective is that it does not require anyone to lie. The numbers are real. Dershowitz's 1 in 2,500 is accurate.
To get back to Sally Clark, the argument that convinced the jury sounded deceptively simple. The prosecutor claims that SIDS occurs in roughly 1 out of every 8,500 babies in families like the Clark’s. If that is true, then two SIDS deaths in the same household would occur with a probability of 1 in 8500 squared or about 1 in 73 million. To many people in the courtroom, that number sounded so astronomically small that innocence itself began to seem implausible. The jury heard that number and drew the obvious conclusion: Sally Clark must be guilty. This is precisely the fallacy. What was calculated was the probability of two SIDS deaths given innocence, P(evidence | innocent). What the jury needed was the probability of innocence given the evidence, P(innocent | evidence), and those are again completely different questions. To answer the second, you cannot just look at how unlikely SIDS is. You also have to ask how likely double murder is, because that is the only alternative explanation on the table. When statisticians actually ran that comparison, the numbers flipped entirely. Double infant murder is rarer than double SIDS, making it roughly 4 to 9 times more probable that both deaths were natural. A later analysis put the odds of Sally Clark's guilt at around 4%. She was convicted on a statistic that, correctly interpreted, pointed toward her innocence.
The lesson is not that statistics do not belong in court. Numbers are some of the most powerful evidence we have, but only if they answer the right question. Sally Clark’s 1 in 73 million was real arithmetic. Dershowitz's 1 in 2,500 was a real statistic. Both were deployed to answer something other than what the jury needed to know, and both worked. That is the uncomfortable truth the prosecutor's fallacy leaves you with; you do not need to fabricate anything to mislead a jury. You just need to pick the right number and let human intuition do the rest.