Ignoring probability theory is dangerous

In 1999, Sally Clark, a young British lawyer, was convicted of murdering her two newborn babies over a two-year period and given a life sentence. A pediatrician had testified for the prosecution that the chance that the two boys had died of sudden infant death syndrome (SIDS) was about 1 in 73,000,000. This was the only real evidence of the crime.

But the probability assessment, which convinced the jury, was flawed. The two deaths were assumed to be statistically independent events, which justified multiplying their respective probabilities of both events occurring: 1/8543 × 1/8543 is approximately equal to 1/73,000,000. In reality, however, two cot death deaths in the same family are not independent events: one such death increases the medical probability that a second death will occur by 10. Furthermore, Ray Hill, professor of mathematics at the University of Salford, later calculated that the probability of two cot deaths in the same family is between 4.5 and 9 times greater than the chance of two little brothers and sisters being murdered. (See Ray Hill, ‘Multiple Sudden Infant Deaths – Coincidence or More Than Coincidence?’ [Paediatric and Perinatal Epidemiology 2004, 18, 320–326].)

Ms Clark’s conviction was overturned on appeal after she served three years in prison. Statistician David Hand notes that ‘she never recovered from her ordeal’ and was found dead in 2007 ‘of acute alcohol poisoning’. If Tim Harford, the Financial times“undercover economist,” it said“she drank herself to death at the age of 42.” A very tragic and disturbing story.

There are other documented cases of murder convictions resulting from a similar ignorance of probabilities.

On probabilities and coincidences I recommend Professor Hand’s book The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day (Scientific American and Farrar, Straus and Giroux, 2014). I think this book is accessible to intelligent laypeople without prior knowledge of probability theory. (Probability theory is the basis of statistical analysis.)

By discovering the laws of chance – the laws by chance– probability theory is one of the greatest achievements of the human mind. A testimony to the beauty of probability theory is the name Mike Lynch gave to his superyacht who sank along the Italian coast last week and claimed his own life and that of many of his guests: ‘the Bayesian’. Thomas Bayes was an 18th-century statistician who developed an important theorem of probability theory.

Tim Harford notes that “in 2010, the British Court of Appeal rejected the use of Bayes’ Theorem as a tool for evaluating how to construct a collage of evidence.” He adds that “a little bit of statistical education for the legal profession would go a long way.”

If you are charged with a crime, your freedom may depend on lawyers and judges’ understanding of the theory of probability. A necessary condition is that it is known to the educated public, the pool from which legal professionals emerge. But this is generally not the case. If that were so, conspiracy theories would have to overcome another hurdle besides the outright obstacle of rational choice. Politicians don’t know anymore.

There is an ongoing debate in Britain about the public’s poor knowledge of science, mathematics and statistics. The recent trial of British nurse Lucy Letby, who was sentenced to multiple life sentences for allegedly killing many patients, after a trial with weak factual evidence and again misleading statistical estimates. Many statisticians have expressed sharp criticism. The Economist writes about former Prime Minister Boris Johnson (“The trial of Lucy Letby has shocked British statisticians“, August 22, 2024):

Mr Johnson is an example of what went wrong. He is – despite what his actions often imply – not a stupid man and certainly not, after Eton and Oxford, a poorly educated man. His education was etiolated; it was not fruitless. He could read Archimedes in the original; he could not understand Archimedes’ mathematics. He is the product of something [the late physicist and novelist C.P.] Snow cited Britain’s “fanatical belief in educational specialization.” And that belief, says David Willetts, former universities minister, is “as acute as ever”.

I would argue that the problem is just as serious when it comes to people who only know science and are illiterate in the humanities and economics. This applies especially and emphatically to people who, be it in their official position or through their voting rights, intend to coercively intervene in the lives of other people.

I don’t think a court should ever convict a person solely or mainly on the balance of probabilities: there should be some weighty factual and testimonial evidence. But the lesson of the case of Sally Clark and others is that when probabilities are appealed to, they must be calculated correctly. The cure for “bad statistics,” Harford argues, “is not ‘no statistics’ – it is using statistical tools appropriately.”

This problem relates to the presumption of innocence and the requirement that guilt must be proven by the prosecution ‘beyond a reasonable doubt’. We owe our freedom, however imperfect, largely to these legal principles in the Western tradition. But whatever the allowable level of reasonable doubt – which in a free society must correspond to a small chance of error – lawyers and judges need to understand statistical theory enough to get a sense of the probabilities involved.

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