Most of us are highly likely to get probability wrong

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Why do we find the mathematics of probability so hard? And why this should make us hesitant to trust lawyers or doctors – and even ourselves

The Monty Hall problem is named after the host of a 1970s quiz show, Let’s Make a Deal. The successful contestant chooses from three closed boxes. One contains the keys to a car and the other two a picture of a goat. The choice made, Monty opens one of the other doors to reveal – a goat. He taunts the guest to change the decision. Should the guest switch to the other closed box?

When the solution was published in an American magazine, thousands of readers – including professors of statistics – alleged an error. Paul Erdös, the great mathematician, reputedly died still musing on the Monty Hall problem. But the answer is, indeed, yes: you should change.

This is not the only case where intuition does not correspond to the mathematics of probability. One person in a 1,000 suffers from a rare disease. A friend has just tested positive for this illness and the test gives a correct diagnosis in 99 per cent of cases. How likely is it that your friend has the disease? Not at all likely. In random groups of 1,000 people an average of 10 would display false positives and only one would be correctly diagnosed with the disease. But most people, including most doctors, think otherwise. “The human mind,” said science writer Stephen Jay Gould, “did not evolve to deal with probabilities.”

Last month, the General Medical Council struck off Professor Sir Roy Meadow, the paediatrician, from the medical register. He had given misleading evidence in the criminal prosecution of Sally Clarke, whose two infants died in their cots. When Mrs Clarke was charged with their murder, Sir Roy told the jury that the chances of two successive cot deaths in the one family was “one in 73m”.

But although the disciplinary committee heard evidence from distinguished statisticians, it does not appear that they understood the application of probability theory to such cases any better than Sir Roy. The committee found that he had underestimated the incidence of cot deaths, and that he had not taken account of genetic and environmental factors that mean a household that experiences one cot death is more likely than average to suffer another. But even if you recognise these effects, his key conclusion remains valid. It is unlikely that such an accident would have happened at all. It is very unlikely indeed that such an accident could have happened twice in the same family.

Of course it is unlikely. The events that give rise to criminal cases are always unlikely, otherwise the courts would be unable to deal with the backlog. If Osama bin Laden is ever brought to justice, the question will not be “is it likely that two aircraft hit the World Trade Center on September 11?” – to which the answer is no – but “given that two aircraft did hit the World Trade Center on September 11, is it likely that bin Laden was responsible?” Confusion of these two separate issues has become known as “the prosecutor’s fallacy”.

A cot death in a family increases the probability that there will be another, but a murder in a family may well increase the probability of another murder by even more: wicked parents may continue to be wicked. Sir Roy might have been right to conclude that two cot deaths were more suspicious than one. But the Court of Appeal, releasing Mrs Clarke, was certainly right to have concluded that this statistical evidence could never, on its own, establish guilt beyond reasonable doubt.

You should not trust doctors, or lawyers, with probabilities; and be very hesitant about trusting yourself. Adversarial legal proceedings are a bad forum for unravelling technical issues. And we cannot expunge collective responsibility for mistakes by excoriating selected individuals.

The business and financial system, more than Bernie Ebbers and Henry Blodget, was to blame for the dotcom boom and bust. Failures in legal processes, rather than over-confident professors, led to the unjust conviction of women such as Sally Clarke. But scapegoating has a long history – at least since Leviticus: “Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel . . . and the goat shall bear upon him all their iniquities unto a land not inhabited”.

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