Ignoring Probability Theory Is Dangerous

In 1999, Sally Clark, a young British lawyer, was convicted of killing her two newborn babies over a period of two years and she received a life sentence. A pediatrician had testified for the prosecution that the probability that the two boys had died from the Sudden Infant Death Syndrome (SIDS) or “crib death” was about 1 over 73,000,000. This was the only real evidence of the crime.

But the probability estimate, which persuaded the jury, was defective. It assumed that the two deaths were statistically independent events, justifying the multiplication of their respective probabilities for both events to happen: 1/8543 × 1/8543 is approximately equal to 1/73,000,000. In reality, however, two SIDS deaths in the same family are not independent events: one such death increases by 10 the medical probability that a second one will happen. Moreover, a professor of mathematics at the University of Salford, Ray Hill, later calculated that the probability of two SIDS in the same family is between 4.5 and 9 times the probability of two infant siblings being murdered. (See Ray Hill, “Multiple Sudden Infant Deaths—Coincidence or Beyond Coincidence?” [Paediatric and Perinatal Epidemiology 2004, 18, 320–326].)

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

There have been other documented cases of murder verdicts resulting from 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 lay readers with no prior knowledge of probability theory. (Probability theory is the basis of statistical analysis.)

By discovering the laws of chance—the laws of chance—probability theory is one of the great achievements of the human mind. A testimony to the beauty of probability theory is the name that Mike Lynch had given to his superyacht which sank along the Italian coast last week, claiming his own life and the lives 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 UK Court of Appeal ruled against the use of Bayes’ Theorem as a tool for evaluating how to put together 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 liberty may depend on the understanding of probability theory by lawyers and judges. A necessary condition is that it be familiar to the educated public, the pool from which legal professionals come. But this is not generally the case. If it were, conspiracy theories would have another hurdle to overcome, besides the straight obstacle of rational choice. Politicians don’t know more.

A debate is going on in the United Kingdom on the public’s poor knowledge of science, mathematics, and statistics. The recent trial of British nurse Lucy Letby, who was condemned to several life sentences for allegedly murdering many patients, after a trial with weak factual evidence and again misleading statistical estimates. Many statisticians have voiced strong criticisms. The Economist writes about former prime minister Boris Johnson (“The Trial of Lucy Letby Has Shocked British Statisticians,” August 22, 2024):

Mr Johnson is paradigmatic of what has gone wrong. He is not—despite what his actions often imply—a stupid man and certainly not, after Eton and Oxford, an ill-educated one. His education was etiolated; it was not ineffectual. He could read Archimedes in the original; he could not begin to understand Archimedes’s maths. He is the product of what [the late physicist and novelist C.P.] Snow called Britain’s “fanatical belief in educational specialisation”. And that belief, says David Willetts, a former universities minister, is “as acute as ever”.

I would argue that the problem is as serious regarding people who know only science and are illiterate in the humanities and in economics. This is especially and emphatically true for people who, be it in their official functions or with their votes, intend to coercively intervene in the lives of other people.

I don’t think that a court should ever condemn a person only or mainly on the basis of probabilities: there should be some weighty factual and testimonial evidence. But the lesson of the Sally Clark case and others is that if probabilities are invoked, they should be calculated correctly. The cure for “bad statistics,” Harford argues, “isn’t ‘no statistics’—it’s using statistical tools properly.”

This problem is related to the presumption of innocence and the requirement that guilt be proven by the prosecution “beyond a reasonable doubt.” We largely owe our liberty, however imperfect it is, to these legal principles in the Western tradition. But whatever is the allowed level of reasonable doubt—which, in a free society, must correspond to a tiny probability of error—lawyers and judges need to understand statistical theory enough to get a sense of the probabilities involved.

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