Ignoring Probability Theory Is Dangerous

In 1999, Sally Clark, a young British lawyer, was convicted of murdering her two newborn children within a two-year period and sentenced to life in prison. A pediatrician had testified to the prosecution that the odds of the two boys dying from Sudden Infant Death Syndrome (SIDS) or “death by skin” were about 1 in 73,000,000. This was the only real evidence of the crime.

But the probability assessment, which influenced the judge, was wrong. It was assumed that the two deaths were statistically independent events, indicating the multiplicative probability of both events occurring: 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 odds that a second one will occur. Furthermore, a professor of statistics at the University of Salford, Ray Hill, later calculated that the odds of two SIDS in the same family are between 4.5 and 9 times the odds of the siblings being killed. (See Ray Hill, “Mass Unexpected Child Death—Coincidence or Coincidence?” [Paediatric and Perinatal Epidemiology 2004, 18, 320–326].)

The conviction of Ms. Clark was acquitted on appeal after spending three years in prison. Mathematician David Hand notes that he “never recovered from his ordeal” and, in 2007, was found dead “from an overdose.” As Tim Harford, u Financial Times“an underground economist,” he put it, “drank himself to death at the age of 42.” It is a very sad and disturbing story.

There have been other recorded cases of decisions to kill due to possible ignorance.

For the possibility of mutual understanding, I recommend Professor Hand’s book The Impossible Principle: Why Coincidences, Miracles, and Extraordinary Events Happen Every Day (Science America and Farrar, Straus and Giroux, 2014). I think this book is accessible to intelligent readers with no prior knowledge of probability theory. (Probability theory is the basis of statistical analysis.)

By discovering the laws of danger—i rules of opportunity-probability theory is one of the greatest achievements of the human mind. A testament to the beauty of probability theory is the name Mike Lynch gave to his large boat that sank off the coast of Italy last week, taking his life and the lives of many of his guests: “Bayesian.” Thomas Bayes was an 18th century mathematician, who developed an important theory 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 the combination of evidence.” He adds that “a little math education for the legal profession will go a long way.”

If you are charged with a crime, your freedom may depend on understanding the probability theory of lawyers and judges. A necessary condition is that it is familiar to the educated community, the field from which legal experts emerge. But this is not usually the case. If this were the case, conspiracy theories would have another obstacle to overcome, besides the direct obstacle of rational choice. Politicians know more.

A debate continues in the United Kingdom about the public’s low knowledge of science, mathematics, and statistics. The recent case of British nurse Lucy Letby, who was sentenced to life in prison for allegedly killing many patients, after a trial with weak evidence and misleading statistics. Many mathematicians have voiced strong criticism. The Economist writes of former prime minister Boris Johnson (“Lucy Letby’s case shocks British figures,” August 22, 2024):

Mr Johnson is stunned at what went wrong. He is not—despite what his actions often suggest—stupid and certainly not, after Eton and Oxford, uneducated. His education was supported; it was not in vain. He could read Archimedes from the first; he could not understand Archimedes’ calculations. You are a product of what [the late physicist and novelist C.P.] Snow called Britain “a fanatical believer in the field of education”. And that belief, says David Willetts, the former universities minister, is “as strong as ever”.

I can say that the problem is very serious regarding people who know only science and are not educated in humanities and economics. This is especially true and with emphasis on people who, either in their official duties or through their votes, intend to forcefully intervene in other people’s lives.

I don’t think that the court should convict someone only or mainly because of chance: there should be strong factual evidence and evidence. But the lesson of the case of Sally Clark and others is that if opportunities are to be used, they must be calculated accordingly. The cure for “bad math,” says Harford, “isn’t ‘math’—it’s using the tools of math properly.”

The problem is related to the presumption of innocence and the requirement that the prosecution prove the charge “beyond a reasonable doubt.” We owe our freedom, however imperfect, to these legal principles of Western culture. But regardless of the permissible level of skepticism—which, in a free society, must correspond to a small probability of error—lawyers and judges need to understand statistical theory well enough to get a sense of the probabilities involved.

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