Lottery tickets and long shots

We humans have a curious relationship with chance. We’re prepared to place bets that events with incredibly small probabilities will occur — such as the 1 in 259 million chance that a ticket will win the Mega Millions lottery. Yet, we go about our everyday lives happily ignoring far larger probabilities — that we might get killed by a lightning strike, for example, which has a chance of about 1 in 3 million in the United States each year.

The great mathematician Emile Borel said that sufficiently improbable events never occur. If he were right, then while we shouldn’t worry about being killed by lightning, we should certainly never play the Mega Millions lottery.

Of course, Borel didn’t mean that such things were actually impossible. What he really meant was that we should regard as impossible any event that had so small a probability that we were unlikely to see it in our lifetimes. Like winning the lottery.

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And yet … And yet … People do win lotteries. People do get struck by lightning. And even less probable events do occur, such as the same lottery numbers coming up in consecutive weeks and people winning the lottery twice. (So unfair!) If these events are so improbable that we won’t see them in our lifetime, so improbable that we should regard them as impossible, how is it that we see such things happening again and again?

The explanation lies in what I call “the improbability principle.” In brief, this says that extremely improbable events are commonplace. That, in complete contrast to what Borel said, highly unlikely events keep happening, and indeed that they do so with an almost tedious regularity.

To resolve the apparent contradiction between Borel’s assertion and the improbability principle, we need to look at the bigger picture.

The improbability principle is composed of five basic laws, which combine with each other and then interact with human behavior, perceptions and understanding. The five laws are the law of inevitability, the law of truly large numbers, the law of selection, the law of the probability lever and the law of near enough.

You may already be familiar with some of these laws.

For example, the law of inevitability says that if you predict that the outcome will be one of all the possible outcomes, you are certain to be right: You can predict that a thrown die will certainly show one of the numbers 1 to 6. If this sounds simple, it is. But this basic idea has been taken and made profitable use of: in guaranteeing lottery wins (though not the Mega Millions), in stock picking and in other areas.

For the lottery wins, for example, all you have to do is contrive to buy all possible ticket combinations. In 1992 a group of people did exactly this for the Virginia state lottery. Though it is indeed simple in theory, the practice may not be so straightforward. After all, they had to buy some 7 million tickets in just a few days to guarantee holding the ticket with the jackpot numbers, an exercise that in fact proved too complicated — they managed to buy only 5 million. Fortunately for them, after much nail-biting searching, these 5 million were found to include the winning ticket. But even then there are further risks. Just buying the winning ticket in a lottery doesn’t guarantee you’ll win it all; you may have to split a jackpot with another winner.

A second law, the law of truly large numbers, says that if you have enough opportunities then even the most improbable of events is likely to happen. Given the number of poker hands dealt around the world each year, it would be nothing short of amazing if a royal flush didn’t come up occasionally. After all, the probability that a randomly dealt five cards is a royal flush is about 1 in 650,000 and there are far more poker games than that played around the world each year.

Those two laws are straightforward enough, but the other three are deeper, and can lead to some very surprising and often counterintuitive conclusions: That the best-scoring student is probably not the most able; that the long-term statistical average (according to a U.S. National Academies of Sciences report) works out to 91 people killed per year from meteorite strikes — and yet hardly any such deaths have been recorded throughout the whole of human history; and that positive test results of psychic powers never translate into anything that is practically useful.

The five laws of the improbability principle are powerful enough in themselves to produce extraordinarily unlikely events. But we must also factor in the human brain.

The brain has evolved to recognize patterns: that a particular shape and color of fruit is good to eat, that a particular hissing sound means danger. These patterns are abstracted from the constant turmoil of events boiling around us. And often that abstraction has to be done quickly — or, for example, the snake will strike. So we have evolved heuristics, simple, quick rough approximations to condense the confusion into signals that can be acted on — Nobelist Daniel Kahneman’s “fast thinking.”

But such simple approximations come at a cost. We overestimate small probabilities, we selectively remember things that seemed interesting, we confuse the probability that a suspect is guilty given the evidence with the probability of this evidence given that they are guilty, we think that just because A happened after B then A must have been caused by B, and we look back and curse ourselves for not seeing what is obvious only in hindsight.

When such heuristic failures in the way we think are coupled with the five laws of the improbability principle, then no wonder we fail to recognize that unlikely events will keep happening. No wonder we fail to see that highly improbable events are commonplace.

Given all this, it shouldn’t have surprised me when, on reading about my book, an author I didn’t know contacted me. John Ironmonger told me that his novel, “Coincidence,” was about to be published in the U.S. It is about a London-based professor who studied and wrote about coincidences, who was born on June 30 (my birthday); the female protagonist worked at the University of London, where my wife works.

It shouldn’t have surprised me, but it did.

David J. Hand, an emeritus professor of mathematics at Imperial College, London, is the author of “The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day.”