The Law of Large Numbers and the Misconception That Chance Corrects Previous Unlikely Outcomes
A common misconception about the law of large numbers among experienced gamblers, and its relationship to "the Gambler's Fallacy" and "Belief in the Hot Hand"
Table of Contents
Event Independence and the (False) Assumption that Devices Are Fair
The Law of Large Numbers
Last week’s newsletter considered the concept of expected value (EV), the long-term average return from a risky investment or gamble, assuming the same choice could be repeated over and over again an infinite number of times. This is another way of describing the “law of large numbers,” a theorem proven by the mathematician Jacob Bernoulli and published posthumously in his seminal work, Ars Conjectandi, in 1713 (noted here in part because two of his nephews will be mentioned in an upcoming essay). Bernoulli proved that as the size of a sample of independent events increases, the sample mean will tend to approach and converge to the population mean. The law of large numbers says, for example, that as you increase the number of flips of a fair coin, the proportion of heads will converge to 50% over the long run. This principle helps explain why larger datasets provide more reliable estimates in statistics and probability.
Event Independence and the (False) Assumption that Devices Are Fair
Bernoulli’s proof starts with the assumption that events are mathematically independent. “Event independence,” by definition, means previous outcomes do not impact future outcome likelihoods. That assumption works for hypothetical scenarios, including laboratory experiments where participants can be told outcome likelihoods and told to assume that events are independent. But when making decisions in the wild—including among skeptical students participating in lab experiments that might involve deception—assumptions about outcome likelihoods and event independence are more problematic.
Card games like blackjack or poker are good examples where events are transparently dependent rather than independent. Card outcomes are negatively dependent: previous outcomes of one type of card make future outcomes of that type less likely. If I get a blackjack (an ace and a 10-value card) in the game of blackjack, an outcome that pays a bonus of 3:2 (meaning, for example, that for every 2 units bet the player will win 3 units instead of the usual even money win of just 2 units), then that immediately lessens the likelihood of subsequent blackjacks, reducing my expected value in the game. If I instead get cards other than aces and 10s, that increases the relative frequency of remaining aces and 10s, and a future blackjack becomes more probable.
On first consideration, it might seem equally obvious that events are independent for many other decisions, and in particular games of chance common in casinos. The flip of a fair coin is the standard example. Coins are used in many professional sports to randomly choose between two sides in the competition, since there is essentially a 50:50 chance that the coin will land on one side or the other (but note the slight bias to land on the same side that was facing up at the point when the flip was initiated). In the casino, the repeated rolls of a fair pair of dice in the popular American casino game craps, the repeated outcome of the spin of a fair roulette wheel, or the repeated outcome of the spin of a fair slot machine might all seem like non-controversial real-world examples of repeated independent events.
But note the qualifier “fair” in all of the above statements. If, by definition, the coin, the roulette wheel, and the dice, are fair, and if “fair” means that each outcome has the implied odds—50:50 for the coin; 18:20 for American roulette and 18:19 for European roulette1; and, for example, 1:35 for “snake eyes” (a pair of ones) or “box cars” (a pair of sixes) in craps2—then, yes, we can assume events are independent and match the promised outcome likelihoods. But that points back to the hypothetical but questionable assumption that the devices are fair, as defined above. Outside of human-designed environments or devices—casinos or coins—event independence is rare. If you find a fruit on a tree or catch a fish in a certain spot in the sea, it suggests that more fruits and more fish may be available. But once the availability of fruits or fish starts to noticeably diminish, the likelihood of finding more of them decreases for every additional fruit you pick or fish you catch. Indeed, in the wild, even with games of chance designed to appear fair—such that outcome likelihoods are transparent and outcomes are independent—they rarely are. Three reasons why this tends to be the case follow.
1. The Imperfection of Nature Means That Mechanical Gaming Devices are Never Fair
First, even if events are independent (from one roll of the dice or spin of the roulette wheel to the next), the outcome likelihoods will almost never perfectly match the likelihoods suggested by the game’s design. Coins have approximately a 50:50 chance of landing on heads or tails, and the empirical difference from 50:50 tends to be so small that it is irrelevant and imperceptible to the decision makers themselves. But no coin is perfectly balanced. Casinos go through a great deal of trouble to ensure that their dice are well balanced so that they have an equal likelihood of landing on any number. Roulette wheels are carefully designed so that each number has an equal likelihood, and casinos follow strict protocols to ensure the wheels remain balanced and that the croupiers do not develop a detectable “signature” that might allow gamblers to predict where the ball will land based on when it was released. This includes using different sized balls and requiring croupiers to switch the direction and speed of the ball. In practice, these methods are enough to ensure that the devices are “fair enough,” such that the vagaries of chance have a far larger impact on differences in outcomes across trials than any inherent bias in the devices themselves. But the biases still exist, no matter how small or imperceptible.
2. Gamblers Are Often Mistaken About Seemingly Transparent Outcome Likelihoods: the Example of Slot Machines
Second, players might not correctly assess outcome likelihoods, even when those likelihoods seem transparent. Previous outcomes can be a good guide for updating likelihood estimates given that potential for error. You might tell me you are flipping a fair coin, but if the coin lands on heads 100 times in a row, I would perhaps be foolish to take your promise at face value and to assume I had encountered a particularly improbable streak of heads. In that sense, outcomes can be considered dependent to the extent that previous outcomes can rightly inform and change our assessment of subsequent outcome likelihoods.3 Since many casino games are misleading about outcome likelihoods, this process of updating likelihood estimates based on observed outcomes becomes important. Slot machines provide a great example of just how nontransparent outcome likelihoods can be (for a more detailed discussion, see the sections on slot machines in this chapter).
Left: The “Liberty Bell,” the father of the contemporary slot machine (image courtesy of Marshall Fey), released to the public in 1899 (Legato, 2004). Right: A contemporary 25¢ banking slot machine with a siren light on top (image courtesy of Paul and Sarah Gorman).
From, Bennis et al., 2012, “Designed to fit minds: Institutions and ecological rationality.”
The surface structure of a slot machine is partly a historical artifact based on the original mechanical design. The first slots had three reels and each reel had a set number of stops (say 20) with a distinct symbol on each stop. Some symbols could be more common on the reel than others (so there might be seven cherries but only one red 7 among the 20 stops on each reel). Three red sevens would have a likelihood of just 1 in 8,000 (1/20 * 1/20 * 1/20 = 1/8,000) along with a jackpot payout, whereas three cherries—given the hypothetical example—would be 343 times more likely (7/20 * 7/20 * 7/20 = 343/8,000 ), but it would also have a much smaller payout. Before the computer age, players could calculate the EV of a slot machine. They could count the number and type of outcomes on each reel to determine the likelihoods of each winning combination. Then they could combine that information with the payouts for each type of win to calculate the precise expected loss for playing a particular slot machine (assuming, of course, that the machine was “fair”).
Today’s slot machines are designed to preserve that perception: a series of spinning reels, each with a set number of symbols equally spaced (often with “blank” between each symbol that the reel could stop on as well), with each reel seemingly stopping randomly on any of the equiprobable symbols or blanks. Today, however, that is just window dressing. The stops are still random, but they are programmed onto a computer chip, and the virtual reel can have thousands or millions of stops mapped onto each physical reel. There may be just one of a million virtual stops mapped onto the red seven jackpot symbol, but half of all stops mapped onto the blanks just above and below the red sevens. As long as the outcomes on the invisible virtual reel are random and possible—allowing the jackpot and various other pay lines to occur, however rarely—such machines are legal by Nevada standards. There are other requirements. Nevada slots must pay out a certain minimum percentage, and they usually pay out far more than the minimum because they are competing for business, the overall casino slot payout percentages are published, and higher payouts encourage customer loyalty. The only way to assess outcome likelihoods of specific machines is through inside information or experience playing. The surface structure of the outcomes (the proximity to a red 7, for example) provides little indication of how close players actually were to a jackpot. Importantly, frequent slot machine players know this very well. It is non-gamblers and beginner slot machine players who need to learn about the disconnect between the surface structure and the actual design of the machines.
3. The Games Could Always Be Rigged
Events may be dependent even on games that present outcomes as independent if the games themselves are rigged. The above discussion has assumed that casinos offer fair games. That is probably a good assumption in American casinos today: First, they are usually owned by publicly traded companies rather than private individuals and so both the motivation and opportunity to fix the games, and to do so undetected, would seem… more complex. Second, they already have a good system in place for making a profit by offering fair games that favor the casino without the need to risk cheating. Third, they have a large enough turnover to almost guarantee they reliably get into the long-run each month so that their actual returns correspond, more or less, to their expected returns.4 Finally, they have a lot at stake if they get caught cheating. Historically, however, casinos and other gambling halls did cheat. Loaded dice, magnetic roulette wheels, and stacked decks could all be used when bets are highest and casino losses are imminent. In smaller casinos markets, where minimizing the risk of big losses is paramount, fair games may be the exception rather than the rule, especially historically, when external measures to ensure fair games were less prevalent. This may also be true of today’s less regulated (and less regulatable) online casinos.
One might expect Czech casinos to be more prone to cheating for these reasons. They are less regulated, are more overtly associated with organized crime, and have fewer customers. As a result, getting into “the long run”—or simply earning enough profit each month to cover expenses, given expected returns—may be far less assured. Similarly, the competition for loyal customers the drives much of casino behavior in large gambling markets like Las Vegas is essentially non-existent. My own experience in Czech casinos supports such a conjecture. Dealers sometimes intentionally cheated by peaking at upcoming cards and helping us win to improve their tips (tips are uncommon in Czech, unlike in US casinos). There are other examples I will not go into here, but I have little doubt that cheating was widespread in some Prague casinos, at least in the early 2000s when I conducted my research.
The Ambiguity of Event Independence in the Wild Complicates the Law of Large Numbers
These three examples illustrate how (a) events might not be independent, or (b) assessing outcome likelihoods might otherwise depend on information from previous outcomes. The key point is that the theoretical assumption of event independence, required by Bernoulli’s proof of the law of large numbers, is rarely true in non-hypothetical situations. Consider a gambling-related example. If there were such a thing as luck (where luck refers to a condition that can be used to predict future outcomes), then even seemingly independent events—such as whether a roulette ball lands on red or black—could actually be dependent on recent previous outcomes. They would depend on the state of luck. In such a scenario, the law of large numbers would not predict average expected returns, at least in the short term.
For example, if luck were negatively dependent—meaning that a lucky previous outcome (or series of outcomes) predicted a shift in luck to the opposite direction—then even the results of fair coin flips would not be independent. That would not disprove Bernoulli and the law of large numbers, however. His proof is mathematical and assumes event independence. Bernoulli’s proof is not an empirical claim about whether or not particular repeated events in nature are in fact independent. The discovery of a negative dependence associated with uncanny luck would just demonstrate that event independence does not exist in nature (at least when luck gets involved). On the other hand, if luck were cyclical, like the seasons, with extended periods of “hot” and “cold,” then unlikely streaks associated with wins and losses would suggest that such streaks are apt to continue, at least in the short term, until the “season” changes. Luck presumably does not work in either of those two contradictory ways. But the point is that this is an empirical question rather than a purely mathematical or theoretical question. Whether luck is interdependent or independent is an empirical question, not something that can be deduced a priori. Of course, it is difficult to imagine how in practice that might work (whose luck is being tracked and for how long?) or why scientists would not have been able to measure its impact up to now.
Gamblers’ (Mis)Conception about the Law of Large Numbers, and Its Relationship to Two Fallacies Widely Associated With Gambling
The two empirical possibilities just discussed—concerning the nature of luck and its potential to negate event independence—are presumably false. However, they correspond to a common misconception about the law of large numbers among casino gamblers. That misconception has implications for two well-known fallacies associated with casino gamblers: The gambler’s fallacy and belief in the hot hand. The gambler’s fallacy is the belief that chance must correct for an improbable sequences of events. Belief in the hot hand is the opposite, the false belief that unrepresentative recent outcomes will continue into the future.
From my interviews and conversations with casino gamblers, it was evident that many long-time gamblers are familiar with the term “the law of large numbers.” They often referred to it—incorrectly—to explain their beliefs about how past unrepresentative patterns (such as an unlikely streak of wins or losses, or of red or black outcomes in roulette) influence future outcome likelihoods. Their misconception is that the law of large numbers indicates that these unrepresentative patterns must self-correct over time so that the long-term average will match the outcome likelihoods that are designed into the structure of the game. For example, if a roulette ball lands on red ten times in a row, many gamblers (and even non-gamblers) will invoke the law of large numbers. They use it to suggest that the streak must eventually be corrected, so that black outcomes catch up with reds, “since the law of large numbers says that with a large enough sample, red and black should each occur 50% of the time.”
To be clear, that understanding of the law of large numbers is mistaken. The difference between red and black outcomes could remain 10 forever, or even increase indefinitely. Even so—assuming the coin or roulette wheel are fair—subsequent outcome likelihoods remain unchanged: 1/2 in the case of a fair coin and 18/38 for red or black outcomes in American roulette. Furthermore, the relative frequency of reds to blacks will nonetheless approach the average expected outcome as the sample size increases, even after unlikely previous patterns in outcomes, despite common intuitions to the contrary.
To illustrate this, consider a small sample of just ten roulette wheel outcomes (ignoring the green zeros in roulette for simplicity): nine red and one black (a difference of eight). That’s 90% reds and only 10% black, not the 50:50 that would be expected over the long run. Now consider a sample with 1 billion outcomes, but now with 1000 more reds than blacks: 500,000,500 red outcomes and 499,999,500 black outcomes. That’s 50.00005% red and 49.99995% black. You’d be right to say I’m being ridiculous by not rounding to 50% each, even though red now has a 1000-outcome advantage over black, compared to the eight-outcome advantage after the first ten spins. In other words, the fact that large samples approach average expectation for independent events does not imply that previous outcomes impact future outcome likelihoods. On the contrary: by definition independent events do not impact future event likelihoods. With large samples, the relative frequency of each outcome approaches 50:50 even if the difference in the frequency of each outcome continues to increase over time. Due to the slight imperfections in nature, even with the most carefully designed roulette wheels, we can expect the difference in red and black outcomes to increase in the direction of any natural bias in the wheel. This can happen even as the relative frequency of red and black approaches something closer to 50:50.
As the earlier discussion suggests, however, this points to the problem of generalizing the law of large numbers to the real world. If events are independent, then previous outcomes do not impact future outcome likelihoods, and casino gamblers would be wrong to suggest otherwise. That said, casino gamblers could theoretically be correct that something in the nature of chance itself is self correcting.
1. The Fallacy of “the Gambler’s Fallacy”
On first consideration, this misconception about the law of large numbers—the idea that chance corrects itself to adjust for unrepresentative past outcomes—might seem like a clear endorsement of the gambler’s fallacy. However, experienced gamblers rarely make choices consistent with the gambler’s fallacy, nor do they invoke the law of large numbers to justify such a belief. This claim that experienced gamblers do not believe in the gambler’s fallacy (or at least do not make choices that reflect such a belief) contradicts several other sources of evidence: popular myth about the gambler’s fallacy5, historical examples suggesting gamblers believe in the hot hand6, and both laboratory and field research on how roulette players react to streaks7. See this later Substack newsletter for a partial discussion as to why non-gamblers and novice gamblers often do believe in the Gambler’s fallacy. An more complete explanation, that also considers why experienced gamblers tend not to believe in it, is still upcoming.
2. “Belief in the Hot Hand”; or, More Accurately, Belief in Cyclical Luck
Rather than using the law of large numbers to justify a belief in the gambler’s fallacy, experienced gamblers often refer to it as part of a nuanced explanation of why they do not believe in the gambler’s fallacy—or at least do not act on that belief. Instead their behavior corresponds to belief in “the hot hand.” The term “belief in the hot hand” originates from an influential and controversial paper by Gilovich, Vallone, and Tversky. That research provided compelling evidence that the widespread belief that basketball players get hot and cold in a way that can be used to predict their near-term future performance is false.
While I suspect the cause of this false perception in basketball is similar to the cause of the false perception in gambling, I believe that applying the term “hot hand” to gamblers is misleading. “Belief in cyclical luck” would be more accurate. The belief in basketball is about cyclical changes in human performance that can be used to predict near-term future performance. Many plausible psycho-physiological models could explain such performance changes. In fact, denying their existence seems to contradict direct subjective experience of our own changing ability to perform at our best depending on the circumstances. That is no doubt part of what makes their findings—that basketball player performance does not appear to be streaky beyond what would be predicted by chance alone—so surprising. The belief in gambling, however, is not about cyclical changes in human psycho-physiological performance at all, or even necessarily about human performance. Instead, in the casino domain, the belief is about cyclical patterns in luck itself. This luck can be associated with a “hot hand”—for example, a particular person rolling the dice in craps, or a particular roulette or blackjack dealer or player. But it is often associated with other, non-human things: black or red on a roulette wheel, the cards running hot or cold, a blackjack table, a lucky charm.
How does the aforementioned misconception about the law of large numbers—the mistaken belief that chance corrects short-term unrepresentative patterns—lead to a belief in cyclical luck? This too will wait for a soon to come future Substack essay. There is plenty to chew on in the current essay already.
Note that these are odds rather than likelihoods. Odds of 50:50 is equivalent to a likelihood of 50/100 or 50%. American roulette wheels have 18 black numbers, 18 red numbers, and two green zeros. Therefore, the odds of getting black or red are 18:20 (18 successful outcomes to 20 unsuccessful outcomes). European roulette has only one zero, improving the player’s odds to 18:19. The corresponding likelihoods are 18/38 and 18/37, respectively.
What counts as a fair slot machine is less transparent, because different slot machines have different expected returns and those returns are not provided by the casino. In Nevada casinos, the programmed likelihood for each outcome is legally required to remain constant from spin to spin.
Some readers will have noticed the connection to Bayes’ Theorem and the notion of subjective probability, a topic that will be explored more in future posts.
That said, even Las Vegas casinos occasionally suffer net losses on their gambling returns due to luck. A single whale (very high stakes gambler) can make a casino millions in a single night. But a couple of particularly lucky whales can sometimes make the difference between a profitable or unprofitable month.
I’ve been skeptically challenged more than once with the question, “Why would it be called the gambler’s fallacy if gamblers didn’t tend to believe it?”
The gambler’s fallacy is also known as the “Monte Carlo fallacy.” The name refers to a historical event where gamblers reportedly lost fortunes betting against an improbable streak streak of black outcomes.
For example, Croson & Sundali (2005). “The Gambler’s Fallacy and the Hot Hand: Empirical Data from Casinos.”