IN the opening act of Tom Stoppard’s play “Rosencrantz and Guildenstern Are Dead,” the two characters are passing the time by betting on the outcome of a coin toss. Guildenstern retrieves a gold piece from his bag and flips it in the air. “Heads,” Rosencrantz announces as he adds the coin to his growing collection.
Guil, as he’s called for short, flips another coin. Heads. And another. Heads again. Seventy-seven heads later, as his satchel becomes emptier and emptier, he wonders: Has there been a breakdown in the laws of probability? Are supernatural forces intervening? Have he and his friend become stuck in time, reliving the same random coin flip again and again?
Eighty-five heads, 89… Surely his losing streak is about to end.
Psychologists who study how the human mind responds to randomness call this the gambler’s fallacy — the belief that on some cosmic plane a run of bad luck creates an imbalance that must ultimately be corrected, a pressure that must be relieved. After several bad rolls, surely the dice are primed to land in a more advantageous way.
The opposite of that is the hot-hand fallacy — the belief that winning streaks, whether in basketball or coin tossing, have a tendency to continue, as if propelled by their own momentum. Both misconceptions are reflections of the brain’s wired-in rejection of the power that randomness holds over our lives. Look deep enough, we instinctively believe, and we may uncover a hidden order.
Recent studies show how anyone, including scientists, can be fooled by these cognitive biases. A working paper published this summer has caused a stir by proposing that a classic body of research disproving the existence of the hot hand in basketball is flawed by a subtle misperception about randomness. If the analysis is correct, the possibility remains that the hot hand is real.
I was thinking about Guil and the psychologists last week as I walked into the Camel Rock Casino, operated by the pueblo of Tesuque, a few miles north of Santa Fe. With five full-scale gambling operations in a stretch of 30 miles, the highway there has become a kind of elongated Las Vegas Strip.
Gamblers, with their systems and superstitions, sat nearly immobile at video slots, trying to outguess the algorithmic heart beating inside. They were immersed in what the anthropologist Natasha Dow Schüll calls “the machine zone.”
In her book “Addiction by Design,” she describes how modern slot machines are engineered to maximize “gaming productivity” — the velocity with which dollars fly from the players’ pockets. Mechanical levers have been replaced by faster, more efficient electronic buttons, while the simulated reels of cherries, bars and other symbols are programmed to give the illusion that you missed a jackpot by just a hair — fuel for the gambler’s fallacy.
I’d first come to Camel Rock more than 20 years ago while I was writing a book about the human drive to find order in the world — and impose it when it is not really there. In those days there was only a makeshift bingo hall, and all eyes were on a large machine in which the lettered and numbered balls jumped around like popcorn — an analog equivalent of the random-number-generating chips driving today’s slots. I thought of how an omniscient intelligence, like the one imagined by the philosopher Pierre-Simon Laplace, could precisely track the trajectories of the balls, the elasticity of their impacts, the buoyancy of the air — a vast amount of data — and predict the outcome of the game.
We mortals can benefit, at least in theory, from islands of predictability — a barely perceptible tilt of a roulette table that makes the ball slightly more likely to land on one side of the wheel than the other. The same is true for the random walk of the stock market. Becoming aware of information before it has propagated worldwide can give a speculator a tiny, temporary edge. Some traders pay a premium to locate their computer servers as close as possible to Lower Manhattan, gaining advantages measured in microseconds.
But often the patterns we see are illusions. Some research has suggested that more excitable people are likelier to embrace the magic of the hot hand (go, go, go!) while those with “higher cognitive skills,” as the studies put it, are prone to the gambler’s fallacy — the belief that a run of heads will probably be followed by tails. Their swaggering brains think they have psyched out the system, discovering an underlying regularity.
Or maybe they are misapplying a real phenomenon called regression toward the mean. In the long run the number of heads and tails will even out, but that says nothing about how the next flip will fall. A paper this summer in a German economics journal found that in clearly random situations, the tables are turned: People with lower cognitive abilities are likelier than more rational types to be led astray by the gambler’s fallacy.
In a study that appeared this summer, Joshua B. Miller and Adam Sanjurjo suggest why the gambler’s fallacy remains so deeply ingrained. Take a fair coin — one as likely to land on heads as tails — and flip it four times. How often was heads followed by another head? In the sequence HHHT, for example, that happened two out of three times — a score of about 67 percent. For HHTH or HHTT, the score is 50 percent.
Altogether there are 16 different ways the coins can fall. I know it sounds crazy but when you average the scores together the answer is not 50-50, as most people would expect, but about 40-60 in favor of tails.
There is not, as Guildenstern might imagine, a tear in the fabric of space-time. It remains as true as ever that each flip is independent, with even odds that the coin will land one way or the other. But by concentrating on only some of the data — the flips that follow heads — a gambler falls prey to a selection bias.
In an interesting twist, Dr. Miller and Dr. Sanjurjo propose that research claiming to debunk the hot hand in basketball is flawed by the same kind of misperception. Studies by the psychologist Thomas Gilovich and others conclude that basketball is no streakier than a coin toss. For a 50 percent shooter, for example, the odds of making a basket are supposed to be no better after a hit — still 50-50. But in a purely random situation, according to the new analysis, a hit would be expected to be followed by another hit less than half the time. Finding 50 percent would actually be evidence in favor of the hot hand. If so, the next step would be to establish the physiological or psychological reasons that make players different from tossed coins.
Dr. Gilovich is withholding judgment. “The larger the sample of data for a given player, the less of an issue this is,” he wrote in an email. “Because our samples were fairly large, I don’t believe this changes the original conclusions about the hot hand. ”
Flaws in perceptions about randomness affect more than gambling and basketball. When multiple cases of cancer occur in a community, especially among children, it is only human to fear a common cause. Most often these cancer clusters turn out to be statistical illusions, the result of what epidemiologists call the Texas sharpshooter fallacy. (Blast the side of a barn with a random spray of buckshot and then draw a circle around one of the clusters: It’s a bull's-eye.)
Taken to extremes, seeing connections that don’t exist can be a symptom of a psychiatric condition called apophenia. In less pathological forms, the brain’s hunger for pattern gives rise to superstitions (astrology, numerology) and is a driving factor in what has been called a replication crisis in science — a growing number of papers that cannot be confirmed by other laboratories.
For all their care to be objective, scientists are as prone as anyone to valuing data that support their hypothesis over those that contradict it. Sometimes this results in experiments that succeed only under very refined conditions, in certain labs with special reagents and performed by a scientist with a hot hand.
We’re all in the same boat. We evolved with this uncanny ability to find patterns. The difficulty lies in separating what really exists from what is only in our minds.