Expect the unexpected

Even the most outrageously unlikely things make statistical sense – as long as we have the right person to explain, says Jonathon Keats

The Improbability Principle: Why coincidences, miracles, and rare events happen every day by David J. Hand, Farrar, Straus, Giroux, $27 PEOPLE thought it was a fix when the Bulgarian lottery announced the same six winning numbers on two consecutive draws in 2009. The national sports minister launched an investigation, but found no tampering; the repetition was mere chance. How are such coincidences possible? In The Improbability Principle, statistician David J. Hand delves into extreme unlikelihoods, from duplicate lottery results to global financial meltdowns. “We should expect the unexpected,” he explains, in this lucid overview of the mathematics of chance and the psychological phenomena that can make probability seem counter-intuitive to so many. Understanding the Bulgarian lottery coincidence, it turns out, is relatively straightforward. Although the odds of two specific draws matching is 1 in 13,983,816, the chance of any two draws matching increases with the number of draws, and reaches  a probability of greater than 50 per cent by the 4404th round. Hand concedes that the odds  of consecutive draws being identical are significantly less,  but points out that there are  many lotteries globally, so we shouldn’t be surprised when winning numbers occasionally repeat consecutively somewhere. He dubs this “the law of truly large numbers”, and applies it to lightning strikes, plane crashes and even Bible “codes”, where the sheer number of letters means anyone searching for a “hidden” message will inevitably find one. “With a large enough number of opportunities, any outrageous thing is likely to happen,” writes Hand. Because we tend to focus on specific instances instead of the broader context, we fail to recognise the true probability of an event. Not all outrageous events depend on truly large numbers. Even small numbers will suffice, if probabilities are ascribed after the fact. Consider how dreams  can seem to miraculously foretell future events. US president Abraham Lincoln and the Roman emperor Caligula both dreamed of dying shortly before they “  The law of large numbers applies to lightning strikes, plane crashes  and even the Bible". were assassinated. That seems pretty impressive until you consider the likelihood they dreamed of death on other occasions. Who would bother to record them? Unfulfilled prophesies are not newsworthy. Referring to this as the “law of selection”, Hand notes it doesn’t only afflict doomed heads of state. In fact, selection bias is one of the most pernicious problems in research. Take drug trials, where people whose symptoms don’t improve have a tendency to drop out, potentially making the drug look more effective than it is. Even if the dropout bias is accounted for, publication bias may skew the data. Researchers rarely write up experiments that fail to show an effect, and few journals will publish such papers. We only care about dramatic results. But knowing about only recorded dreams or published experiments, you have no basis for measuring probability. Other improbable phenomena are explained by different facets of Hand’s improbability principle. His “law of near enough”, for example, explains what happens when “events that are sufficiently similar are regarded as identical” – a slip exploited by parapsychologists who support their vague hypotheses with pseudoscientific experiments. In a classic case from the 1930s, a researcher at University College London asked 160 people to guess which card he had chosen from a deck. When they failed to guess, he included the cards before and after his selection. He continued to loosen his standards until the experiment appeared to demonstrate psychic ability. The concepts underlying Hand’s laws will be familiar to  all who have studied probability. His improbability principle is simply a conceit for assembling them in a book. But as an organisational scheme, it serves the purpose exceptionally well. Hand has written a superlative introduction to critical thinking, accessible to everybody, regardless of mathematical ability – and essential if your probability of taking a university-level maths class is less than zero.  ■