Probability in the courtroom

Probabilities are often used in daily life to make decisions. For example, should I go swimming, if the probability for sunshine is $80\%$? In court, such decisions can have tragic consequences, if not applied correctly. Here is a famous example. The text is taken and adapted from Wikipedia.

Sally Clark (August 1964 - 15 March 2007) was an English solicitor who, in November 1999, became the victim of a miscarriage of justice when she was found guilty of the murder of two of her sons. Although the conviction was overturned and she was freed from prison in 2003, the experience caused her to develop serious psychiatric problems and she died in her home in March 2007 from alcohol poisoning.

The case

Clark's first son died suddenly within a few weeks of his birth in December 1996, and in January 1998 her second died in a similar manner. A month later, she was arrested and subsequently tried for the murder of both children. The prosecution case relied on significantly flawed statistical evidence presented by paediatrician Professor Sir Roy Meadow, who testified that the chance of two children from an affluent family suffering sudden infant death syndrome was 1 in 73 million. He had arrived at this figure by squaring 1 in 8500, as being the likelihood of a cot death in similar circumstances.

The Royal Statistical Society later issued a statement arguing that there was "no statistical basis" for Meadow's claim, and expressing its concern at the "misuse of statistics in the courts".

So, what was wrong with Meadow's argument? Let's define the event $D_1$="cot death of child 1" and "cot death of child 2". Meadow assumed that the two events are unrelated, so that the probability of cot death for both children is $p(D_1\cap D_2)=p(D_1)\cdot p(D_2)$ and because Meadow found that $1$ in $8500$ families experienced a cot death, we get $p(D_1\cap D_2)=(1/8500)^2$, that is, this would happen once in $73$ Mio families. Indeed, this sounds improbable, Meadow argued, so there is a big likelihood for murder.

However, it is not clear at all that the two events are independent. In fact, a genetic predisposition or environmental influences could make the events highly dependent: Given that there is one cot death (indicating that there a genetical predisposition), it may by highly likely that the second child has a similar predisposition, and thus there will a high probability for a second cot death. For example, suppose that $p(D_2|D_1)=0.6$, then $p(D_1\cap D_2)=p(D_2|D_1)\cdot p(D_1)=0.6\cdot (1/8500)$, which reduces the probability for two cot deaths to $1$ in $14 000$ families - much less dramatic, especially if this number is compared to 1 cot death in 8500 families.

In addition, if should be noted that the relevant probability to look at is not $p(D_1\cap D_2)$. What we actually should consider are the families where two deaths occurred, and compare how many of those were due to cot death or accidental death, and how many were due to double homicide. In other words, we should compare the probabilities

and

Only if the first probability is much smaller than the second one, suspicion should arise. In fact, a study shows that the first probability is about 10 times higher than the second probability, making accidental death such as cot death a more likely explaining for two deaths in a family.

Clark was convicted in November 1999. The convictions were upheld at appeal in October 2000, but overturned in a second appeal in January 2003, after it emerged that Dr Alan Williams, the prosecution forensic pathologist who examined both of her babies, had incompetently failed to disclose microbiological reports that suggested the second of her sons had died of natural causes. She was released from prison having served more than three years of her sentence. The journalist Geoffrey Wansell called Clark's experience "one of the great miscarriages of justice in modern British legal history". As a result of her case, the Attorney-General ordered a review of hundreds of other cases, and two other women had their convictions overturned.