Last August, my wife and I welcomed our third daughter into the world. It’s wonderful to be the parent of three girls. There is one significant drawback, however: having to field the question, over and over again, from (mostly) well-meaning people: “So are you going to try for a boy now?” There are several solid reasons we are calling it a day in the reproduction department. But if we were interested in having a fourth child, “trying for a boy” would not be the motivation. The idea is preposterous. Having a string of children of one sex does not presage the arrival of a baby of the opposite sex. Each pregnancy brings the same odds of having a boy or a girl, regardless of how previous pregnancies turned out: about 1 in 2.
The inkling that eventually odds come to favor having a baby of the other sex is an application of the “gambler’s fallacy.” This mistake is often explained with the example of coin tosses. Let’s say you flip a fair coin 5 times and it ends up “heads” each time. Many people watching this unbroken string of unlikely flips would bet good money that the sixth flip will bring “tails.” Heads can’t go on forever! What are the chances that there would be six heads in a row? Answer: on the sixth flip, there are even odds of getting heads or tails, just as there were for the first five flips. You’d be a fool to place a big bet on tails—or on heads, for that matter—for any individual coin toss.
In the long run, with millions or billions of flips, a fair coin will produce increasingly even numbers of heads and tails. The numbers will show something very close to a 50/50 split. That’s the Law of Large Numbers. But when you’re dealing with only a few handfuls of flips, the Law of Small Numbers applies: seemingly unlikely strings of coin flips are not that improbable after all. In our example, there is a probability of 1/64 that six flips of a fair coin will result in heads each time (that’s 1 over 2 to the sixth power). Those odds aren’t great; they come out to about a 1.6% chance. The gambler’s fallacy is to look at those meager odds and conclude there is a 98.4% chance the sixth flip will be tails. But here's the fundamental problem: the probability of the first five flips coming up heads is now 100 percent. They have already happened! The only question is what will happen with the next flip, and those odds are, again, 50/50. Here is another way to look at it: any permutation of six coin flips—all heads or all tails or three heads and three tails or one tails and five heads, e.g.—has a probability of 1/64. So it’s just as likely—and just as unlikely—that six flips of a coin will produce six heads, or three tails and three heads—or any of the other 62 possible permutations.
When we zoom in on a string of one or two dozen flips, then, we are likely to find some series of flips that don’t look so random. Such non-random-seeming strings are to be expected from time to time. And this principle holds outside the realm of coin flips; it applies to purportedly amazing coincidences you might experience in your life. I’ll admit to being very surprised when, ten years after graduation, I ran into a college classmate on my way out of the St. Vitus Cathedral in Prague. “How random is this!” I think we exclaimed. The answer: just as random as any other chance encounter. The chances of our meeting were, no doubt, small. But the chances of meeting any of my other college classmates in any other attraction in a foreign city are equally low—and I have never had any other such encounters. Those didn’t happen; this one did. It might be spooky if my entire Freshman year hallway showed up at the same time at a cafe in Vienna, but stumbling across one fellow in one place at one moment is not, statistically speaking, anything remarkable.
It’s clear how a gambler can suffer from this fallacy: he can lose big money. If you throw all your chips on black in a game of Roulette after the ball has landed on red 10 times in a row because it couldn’t possibly wind up there an eleventh time—well, you have a good chance of walking home empty-pocketed. On August 18, 1913, scores of French gamblers left the Monte Carlo casino bereft after falling victim to this mistake: the Roulette ball landed on black 26 times in a row that day; during the run, everybody was betting that the wheel would even itself out and turn to red. But of course the wheel had no memory of its previous spins. Only the irrational bettors thought that previous spins had anything to do with how the next spin would turn out.
A new piece of research shows there are weighty implications of this cognitive bias well beyond the casino floor. In next Friday's Praxis, I will discuss evidence that judges, loan officers and baseball umpires tend to succumb to the gambler's fallacy in their decision making—dramatically expanding the damage the fallacy can cause to innocent bystanders.
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