The Sally Clark case – how bad maths can cause a miscarriage of justice

*This is something that I wrote ages ago, but never hit ‘publish’. So here you go …*

Photo from the Guardian

Photo from the Guardian

The case of Sally Clark is an instructive one when looking at how courts (mis)handle statistics. Wikipedia has a useful summary with good links to some of the problems that come up from it.

The essence of the (statistical) problem was that one of the Prosecution expert witnesses (Roy Meadow) said that the chance of a SIDS (cot death) was 1 in 8,543. Sally Clark’s two children had died from cot death. He then took from that that the chances of both children suffering a cot death was 1 in (8543*8543), in other words 1 in 72,982,849 and on that basis the inference was 1 in 73 million is so unlikely that you can discount cot death and conclude that this was murder.

This is completely and embarrassingly wrong. There are two major problems.

The first is that to find the probability of two events happening by multiplying the individual probabilities is only valid if the two events are ‘independent‘. For example, the Probability of getting a head if you toss a fair coin [P(H)] is ½. The Probability of getting a one if you roll a fair dice [P(1)] is 1/6.

The Probability of getting a head and rolling a one if you toss a coin and roll a dice is ½ x 1/6= 1/12. In other words, for two independent events P (A and B) = P(A) x P(B).

So multiplying the odds is only valid if there are no environmental or genetic factors present in cot deaths. This seems on the face of it unlikely (and the full report that the stats came from indicate that there are environmental factors).

Roy Meadow was made the subject of a complaint to the GMC. This ended up in the Court of Appeal were it was concluded (by a majority) that he was not guilty of Serious Professional Misconduct.

Roy Meadow is obviously an educated man. He is a man of science. Whilst he is not a statistician, this is not complicated in any way. It’s basic maths. One would have thought that someone who is a doctor should have seen this coming a long way off.

But, whilst he got rightly criticised for this, one wonders why nobody in court from amongst the lawyers picked up how obviously wrong this was? Anyone with a GCSE in maths should have spotted the error and asked some questions as to how it was right to assume independence.

The second problem is that is another example of the Prosecutor’s Fallacy (most often seen in DNA evidence). 73 million sounds a large number. 1 in 73 million sounds like very long odds indeed, doesn’t it? But even if that figure is correct (which it almost certainly isn’t) what does that tell us?

Because we are very bad at reasoning with numbers, it’s tempting to conclude that the chance of Sally Clark being guilty is 1 in 73 million. It needed to be explained carefully why this wasn’t the case.

What it actually told us is the chance of a person, chosen at random in the UK, suffering two cot deaths is 1 in 73 million. But, the jury was looking at one person where her two children had died and deciding whether it was murder. Yes, two cot deaths are very unlikely, but so are double murders. If anything, a double murder is even less likely.

I think that the best way of looking at it is by looking at what information we have. In assessing the odds of Sally Clark being guilty, we shouldn’t ask what the odds of two children dying from SIDS is, but,given that Sally Clark’s two children have died, what is the probability that they were SIDS? These two questions can sound similar, but the answer is very different and can be very misleading.

The second point is a bit more subtle, but in a murder case really should have been understood. But reading the Court of Appeal judgments (certainly the first one, but also the second one to some extent) frankly they didn’t seem to get the first point, let alone the second. It’s pretty depressing that people can get locked up on this basis.

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5 thoughts on “The Sally Clark case – how bad maths can cause a miscarriage of justice

  1. I cite this case in my next book.

    “I was drawn to the case of Sally Clark, a solicitor who, in November 1999, was convicted of murder after the deaths of her two sons. Her defence counsel had said from the outset that they had both died from cot death syndrome. It was, they said, a tragic coincidence that almost defied belief. The prosecution convinced the jury that she had in fact murdered them. Despite the Court of Appeal’s Law Lords’ acceptance that key expert evidence advanced by the prosecution at her original trial had been flawed, her first appeal failed. It was only the resolve of her husband, Stephen Clark, a finance partner in a top ten City law firm, which meant she would get a second chance. With the steadfast belief in his wife’s innocence, he spent years researching her case, eventually proving that their second son’s death was consequent to the presence of staphylococcus aureus, a bacteria that can cause a quickly spreading and lethal infection in infants. Whilst it remains unclear why the prosecution failed to disclose this evidence, its acquisition resulted in Clark mounting a formidable challenge on the safety of his wife’s conviction. After serving more than three years in prison, Sally Clark was freed after the Law Lords finally agreed that her convictions were unsafe. It had taken a monumental effort to the point of a miracle to see his wife walk free, but, after exhausting all judicial avenues, he had taken it upon himself to get her case to the Court of Appeal. Despite the intervention of the Royal Statistical Society and evidence from the Foundation for the Study of Infant Deaths (used in the failed appeal), it was evidence uncovered by the hard work of Sally Clark’s husband that proved critical in overturning her conviction. Determination to prove his wife’s innocence had proved a more formidable weapon than the best legal brains in the land. Stephen Clark became my inspiration and, with the same belief in my innocence, I began investigating the case against me.”

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