Belief in the Gambler's Fallacy is Trivial
[The Grandmother] “Just now I heard the flaxen-haired croupier call out ‘zero!’ And why does he keep raking in all the money that is on the table? To think that he should grab the whole pile for himself! What does zero mean?”
[Alexei Ivanovich, The Gambler and narrator] “Zero is what the bank takes for itself. If the wheel stops at that figure, everything lying on the table becomes the absolute property of the bank…
“Then I should receive nothing if I were staking?”
“No; unless by any chance you had PURPOSELY staked on zero; in which case you would receive thirty-five times the value of your stake.”
“Why thirty-five times, when zero so often turns up? And if so, why do not more of these fools stake upon it?”
“Because the number of chances against its occurrence is thirty-six.”
“Rubbish! Potapitch, Potapitch! Come here, and I will give you some money.” The old lady took out of her pocket a tightly-clasped purse, and extracted from its depths a ten-gulden piece. “Go at once, and stake that upon zero.”
“But, Madame, zero has only this moment turned up,” I remonstrated; “wherefore, it may not do so again for ever so long. Wait a little, and you may then have a better chance.”
“Rubbish! Stake, please.”
The above exchange in Dostoevsky’s The Gambler (1866) between Alexei Ivanovich (the protagonist and titular character) and “the grandmother” (the mother of the general whose children Ivanovich tutors) provides an interesting case. The grandmother has never played roulette, and Ivanovich serves as her guide, explaining how the game works and attempting to steer her away from poor choices. Despite his protests against betting on zero, she continues to do so. By the end of the night, she wins a small fortune in a run of luck that captures the entire casino’s attention. The next day, she returns and loses it all, eventually losing everything she has. Her episode foreshadows what will soon happen—more than once—to Ivanovich himself, ultimately leading to his ruin.
Recall from the previous post that the gambler’s fallacy involves the mistaken belief that chance will self-correct for an improbable series of past events, even when those events are independent.1 Can you spot the gambler’s fallacy in the passage above? Think about it for a moment, then scroll down for the answer.
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Ivanovich expresses the fallacy when he says, “Wait a little, and you may then have a better chance.” The implication is that because zero has come up recently, the likelihood of it occurring again is lower than if one were to wait until zero had not appeared for some time. In fact, assuming the wheel is fair, the probability of the ball landing on zero is constant, regardless of how recently or frequently it has appeared.
Dostoevsky himself struggled with gambling addiction linked to roulette. The story of Ivanovich’s descent into compulsive gambling is partly autobiographical. Notably, it is this more experienced gambler, Ivanovich, who invokes the gambler’s fallacy to dissuade the inexperienced grandmother from what he perceives to be a poor strategy. The grandmother, however, appears wholly unconvinced and instead embrace a different false belief—that the zero is more likely than other numbers because it occurred more frequently than chance would predict.
This passage raises several questions about the gambler’s fallacy. Did Dostoevsky himself believe in it, as implied by Ivanovich’s endorsement of it while attempting to rein in the grandmother? Or do the two characters' subsequent experiences—fabulous winnings followed by devastating losses—suggest that gamblers’ beliefs, and reasoning more broadly, play a minor role in compulsive gambling? During the manic gambling episodes in the novella, the gambler’s fallacy is unmentioned. Instead, these episodes seem driven by compulsive behavior, with a desperate and irrational search for meaningful patterns overshadowing any consistent beliefs about the relationship between past and future outcomes.
The previous Substack essay recounted the famous historical event that gave the gambler’s fallacy its alternate name, the Monte Carlo fallacy. The ball landed on black 26 times in a row, and belief in the gambler’s fallacy reportedly earned the casino a fortune. That event, the enduring name of the fallacy, several studies in decision science (e.g., Ayton & Fischer, 2004; Caruso et al., 2010; Croson & Sundali, 2005; Keren & Lewis, 1994), and the apparent endorsement of the fallacy by a temporarily sensible Ivanovich all suggest that belief in the gambler’s fallacy is widespread in games of chance.
Dostoevsky’s rich gambling description, however, suggest that the issue is more complex. While some individuals, sometimes believe in the gambler’s fallacy, it is equally undeniably that others—or even the same individuals at different moments—form different false beliefs about the relationship between past and future outcomes in games of chance. Later in The Gambler (Chapter 14), Ivanovich (reflecting Dostoevsky’s own experience) presents a different perspective on the gambler’s fallacy, suggesting that naive, inexperienced gamblers are the ones most likely to believe in it, whereas seasoned gamblers understand its flaws. The passage, written almost 50 years before the Monte Carlo streak, serves as a fictional prelude to the fallacy’s famous historical namesake event2).
As though of set purpose, there came to my aid a circumstance which not infrequently repeats itself in gaming. The circumstance is that not infrequently luck attaches itself to, say, the red, and does not leave it for a space of say, ten, or even fifteen, rounds in succession. Three days ago I had heard that, during the previous week there had been a run of twenty-two coups on the red — an occurrence never before known at roulette — so that men spoke of it with astonishment. Naturally enough, many deserted the red after a dozen rounds, and practically no one could now be found to stake upon it. Yet upon the black also — the antithesis of the red — no experienced gambler would stake anything, for the reason that every practised player knows the meaning of “capricious fortune.” That is to say, after the sixteenth (or so) success of the red, one would think that the seventeenth coup would inevitably fall upon the black; wherefore, novices would be apt to back the latter in the seventeenth round, and even to double or treble their stakes upon it — only, in the end, to lose.
Here Ivanovich explicitly states that only a novice would bet as though they believed in the gambler’s fallacy. He does not necessarily claim the belief is false. Instead, he suggests an opposing force whereby luck attaches itself to things, such as colors in roulette.
B.F. Skinner’s classic 1948 paper, “‘Superstition’ in the pigeon” describes how even pigeons learn random associations between their actions and rewards. If a pigeon happens to be standing on one leg when it receives food, it will repeat that action in the expectation that it may increase the chances of getting more food. This aligns with the grandmother’s shifting superstitions in The Gambler, where her beliefs about winning evolve based on random past experiences.
The previous Substack essay described three interconnected false beliefs contributing to the gambler’s fallacy. Those beliefs do not, however, explain why people hold the fallacy, just as they do not explain why some people do not believe in it. If you are reading this Substack, you likely do not believe in the gambler’s fallacy, just as I do not.
So, who actually beliefs in the gambler’s fallacy? Is the belief widespread among gamblers, as suggested by the Monte Carlo event and academic studies? Do only experienced gamblers believe in it (as implied by Ivanovich in the first passage), while novices like the grandmother hold different misconceptions? Or is it the opposite—that experienced gamblers recognize the fallacy’s flaws, while only novices fall for it (as suggested by Ivanovich in the the second passage)? Alternatively, is the gambler’s fallacy just one of many superstitious false beliefs that arise when gambling but that shift depending on what—by chance—worked in the past (as seen in the grandmother’s fickle convictions and Skinner’s superstitious pigeons)?
One partial answer emerges from the discussion so far. The term, “the gambler’s fallacy” is misleading and oversimplifies the issue. It overgeneralizes the belief by suggesting that a tendency exhibited by some gamblers as certain times applies to all gamblers. Conversely, it underspecifies the belief, as many non-gamblers also exhibit the belief under conditions conducive to the fallacy. Dostoevsky’s depiction underscores a reality of gambling: there is a broad spectrum of beliefs about luck and probability, shaped by individual and cultural differences, education, and gambling experience. While belief in the gambler’s fallacy certainly exists among gamblers, this fact is somewhat trivial—little different from observing that many gamblers and many non-gamblers also believe in conspiracy theories or pseudoscience.
This diversity in gambling beliefs is worth emphasizing. People often attribute specific capacities—such as believing in the gambler’s fallacy—to entire groups, when in reality, these beliefs fluctuate among individuals and over time. While some gamblers fall for the gambler’s fallacy, others do not. Some even hold contradictory beliefs simultaneously.
There is another sense in which belief in the gambler’s fallacy—and similar ideas about the relationship between choices and outcomes in games of pure chance—is trivial. In such games, a gambler's beliefs ultimately make no difference. After an improbable streak of red outcomes, it does not matter whether I mistakenly believe black or red is more likely, or correctly understand that each outcome remains equally probable. The grandmother’s expected value was not impacted by whether or not she staked on zero (the strategy that she thought followed from reason) or on some other number that had not occurred for quite some time (the strategy that Ivanovich seemed to find more reasonable). In the end, it all boiled down to luck.3
When I play roulette, I like to bet on the number 29. Why? Because when I was 29 years old and in a casino, I placed bets on and around that number and walked away with one of my few big wins at the roulette table. Do I believe that betting on 29 improves my chances of winning? In a certain sense, yes. It makes the game more enjoyable if I can attach meaning to my bets, if I can pretend that my choice has significance. In moments of high emotion or when large sums are at stake, I might even feel as though I truly believe it.
Yet, in a deeper and more important sense—one in which I am not indulging in fantasy to make the game more meaningful—I do not believe it at all. Since my expected return remains unchanged regardless of my bets, embracing this illusion, even to the point of temporarily convincing myself it is real, enhances the experience. For someone like me, who understands that my beliefs do not affect the outcome, this kind of playful self-deception adds to the utility of gambling.4
The above discussion points appropriately to the triviality of the fact that gamblers believe in the gambler’s fallacy. Yes, sometimes they do, but sometimes they don’t. Sometimes non-gamblers do and do not as well, such that it is not in fact a gambler’s fallacy, except in the sense that the opportunity for the fallacy is rare outside the gambling context. And it mostly does not matter what gamblers believe in that context, because the success or failure of one’s choice is just luck anyway.
That, however, is an unsatisfying conclusion, begging for clarification. How widespread is belief in the gambler’s fallacy? Does it vary by game type? Are gamblers more prone to it than non-gamblers? Does it influence who chooses to gamble in the first place? Or who becomes addicted once that choice to gamble has been made? How does belief in the fallacy depend on the gamblers’ level of experience? Does belief in the fallacy impact the size and likelihood of wins and losses, as suggested by Footnote 2?5 The upcoming Substack essays will explore these issues in some depth, ultimately confirming Ivanovich’s observation in the second passage above: belief in the gambler’s fallacy is, for the most part, a mistake of the uninitiated, uncommon common among seasoned gamblers.
If the description of the gambler’s fallacy does not sound like a fallacy, then you are not alone. Many people do not believe it’s a fallacy. That’s part of the reason it is a well-known fallacy: it is widespread. In that case, please start with this earlier essay about the law of large numbers. It serves as background on the series of essays on the gambler’s fallacy and explains why the law of large numbers does not imply that chance must self-correct after an unlikely streak. Then continue from that essay on to the first essay in the-gambler’s-fallacy series. That essay defines and describes just what is required for belief in the gambler’s fallacy.
Fun fact, a streak of 26 blacks or reds in a row (mirroring the 26 blacks in 1913) will occur 1 in 68.4 million tries on a fair European roulette wheel (American wheels have an extra “00” that makes such a streak half as likely).
This is not entirely true. There are some variants of roulette where even-money bets have higher expected value than all other bets (such betting on a single number). Furthermore, while how one bets may not impact expected value, it does impact outcome variability (the size and frequency of winning and losing sessions), despite the fact that the mean loss over the long run will be the same). Betting on every number, including the zero, will result in zero variability: the gambler will lose 2.7% on every round in European roulette, removing any role for luck. Betting on just a single number each round, however, has very large variability: 36 of 37 times, on average, the gambler will lose their bet, but on one of 37 times, the gambler will win 35 times their bet (in addition to getting their original bet back). The long-term expectation is still 2.7%, but that particular return during a single roulette session will be uncommon.
The exception to this observation is that such beliefs can, and do, impact how much money one is willing to risk on each round, which can have disastrous consequences. If I truly believe that red is authentically more likely on the next spin than black, then it behooves me to continue gambling and, indeed, to bet more than I would if I thought the odds were even.
Spoiler alert: it does! All other things being equal, it makes losing sessions less common but more extreme than they would be without the belief. It also makes winning sessions more common but less extreme they they would be without the belief.