The *gambler’s fallacy* is the mistaken belief that if an event occurred more frequently than expected in the past then it’s less likely to occur in the future (and vice versa), in a situation where these occurrences are independent of one another. For example, the gambler’s fallacy can cause someone to mistakenly assume that if a coin that they tossed landed on heads twice in a row, then it’s likely to land on tails next.

It’s important to understand the gambler’s fallacy, since it plays a crucial role in people’s thinking, both when it comes to gambling as well as when it comes to other areas of life. As such, in the following article you will learn more about the gambler’s fallacy, understand the psychology behind it, and see what you can do to avoid it.

Contents

## Explanation of the gambler’s fallacy

The gambler’s fallacy involves manifests in two connected ways:

- Through the belief that if a certain independent event occurred
*more*frequently than expected in the past, then it’s*less*likely to occur again in the future. - Through the belief that if a certain independent event occurred
*less*frequently than expected in the past, then it’s*more*likely to occur again in the future.

These beliefs both represent an underlying expectation of systematic reversal in random sequences of independent events, which is mistaken, since when events are independent of one another, their future occurrences are unaffected by their past occurrences by definition, even if people’s intuition leads them to expect otherwise.

For example, consider a situation where you roll a pair of dice, which both land on 6. The odds of this happening in a fair roll are 1/36, since the odds of each die landing on a 6 are 1/6.

Here, the gambler’s fallacy could cause someone to assume that the odds of both dice landing on 6 again on the next roll are lower than 1/36. However, in reality, on each individual roll, the odds of the dice landing on double 6’s are still 1/36. This continues to be true regardless of how many times we roll the dice, since the dice can’t remember what they landed on last time. Essentially, there is no way for the last dice roll to affect the next one, which is why it’s incorrect to assume that these independent events affect each other.

When considering this, it helps to understand the difference between the odds of getting a certain string of outcomes, and the odds of getting a certain outcome given an independent prior string of outcomes. For example, the odds of having a fair coin land on heads 5 times in a row are 0.5^5; this represents the odds of getting a certain string of outcomes. However, the odds of having a fair coin land on heads any single time are always 0.5, regardless of what number toss it is, since each toss is independent of the prior string of outcomes, meaning that it is unaffected by the previous tosses.

As one book on the topic explains:

“If there are 10 balls in the pool bottle, and we want to draw the 1 ball, it is 9 to 1 that we don’t get it; but after five men have drawn balls ahead of us, and none of them have got the 1 ball, it is only 4 to 1 that we don’t get it, because there are only 5 balls left in the bottle.

But if you have drawn five times, not five balls, without getting the 1 ball any time, it is still 9 to 1 against your getting it on the sixth draw, if there are 10 balls in the bottle. Even if you had drawn twenty times, it would still be 9 to 1 against you, as long as 10 balls remained in that bottle…

Some persons imagine that because the odds are so great against any event happening a certain number of times in succession, that when it has happened so many times it is very unlikely to happen again…

If you will toss a coin and put down all the times that it comes one way five times running, you will find that in just half those cases it will go the same way again. Note all the times that it goes six times one way, and you will find that in half of them it will go seven.”

—From “Hoyle’s Games” (by Edmond Hoyle, 1914)

## Examples of the gambler’s fallacy

One example of the gambler’s fallacy is the mistaken belief that if a coin lands on heads multiple times in consecutive coin tosses, then it’s due to land on “tails” next. A similar example of the gambler’s fallacy is the mistaken belief that if a die landed on the same number (e.g. 6) multiple times in a row, then it’s less likely to land on that same number the next time.

In general, as its name suggests, the gambler’s fallacy is most commonly associated with how people think when they gamble. Beyond the previous examples of this, with coins and dice, another example of this is the incorrect belief that if a certain number was recently drawn in a lottery, then it’s less likely to be drawn again in an upcoming draw.

In addition, another notable example of the gambler’s fallacy in the context of gambling occurred in a 1913 incident, at a roulette game at the Monte Carlo Casino, where the ball fell on the color black 26 times in a row since this was such a rare occurrence, gamblers lost millions of dollars betting that the ball will fall on red throughout this streak, in the mistaken belief that the ball was due to land on it soon.

As one book notes on this phenomenon:

“If you will toss a coin and put down all the times that it comes one way five times running, you will find that in just half those cases it will go the same way again. Note all the times that it goes six times one way, and you will find that in half of them it will go seven. As they roll about 4,000 times a week at Monte Carlo, or 200,000 a year, it ought to come red fifteen times in succession at least once during that time…

Any person who offers to give odds on account of the maturity of the chances, is betting against himself. If a coin has been tossed five times heads, and a man offers to bet 2 to 1 that it will not come heads again, he is just as foolish as if he offered to bet 2 to 1 against the first toss of all.

It is by knowing the folly of such bets, and taking them up at once, that some men get rich, whether the odds are in business or in gaming. It is the acceptance of unfair odds that makes the keeping of a gambling house so profitable. If a person offers you odds that are not fair, it is your own fault if you accept them. The science of betting is to offer odds that look well but that give the bettor a little the best of it in the long run.”

—From “Hoyle’s Games” (by Edmond Hoyle, 1914)

Furthermore, the gambler’s fallacy can also influence people’s thinking and decision making in other areas of life beyond gambling. For example, in the case of childbirth, the gambler’s fallacy means that people often believe that someone is “due” to give birth to a baby of a certain gender, if they have previously given birth to several babies of the opposite gender. A similar phenomenon was described by French scholar Pierre-Simon Laplace, in the first published account of the gambler’s fallacy:

“I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls ought to be the same at the end of each month, they judged that the boys already born would render more probable the births next of girls.

Thus the extraction of a white ball from an urn which contains a limited number of white balls and of black balls increases the probability of extracting a black ball at the following drawing. But this ceases to take place when the number of balls in the urn is unlimited, as one must suppose in order to compare this case with that of births.”

— From “A Philosophical Essay on Probabilities”, by Pierre-Simon Laplace (as translated by Truscott & Emory from the sixth edition of “Essai philosophique sur les probabilités”, which was originally published in 1814)

Finally, the gambler’s fallacy has been shown to affect the judgment, decision-making, and behavior of various professionals, such as loan officers, sports referees, judges, and even psychologists, despite the fact that many of them are well aware of its influence.

*Note:* the gambler’s fallacy is sometimes referred to as the *Monte Carlo Fallacy*, as a result of the aforementioned incident at the Monte Carlo casino, or as the *doctrine of the maturity of chances*.

## The psychology behind the gambler’s fallacy

The gambler’s fallacy is a cognitive bias, meaning that it’s a systematic pattern of deviation from rationality, which occurs due to the way people’s cognitive system works. It is primarily attributed to the expectation that even short sequences of outcomes will be highly representative of the process that generated them, and to the view of chance as a fair and self-correcting process.

Essentially, people often assume that streaks of outcomes will even out in the short-term in order to be representative of what an ideal and fair random streak should look like. In the case of a fair coin toss, for example, the gambler’s fallacy can cause people to assume if a coin just landed on heads twice in a row, then it will now land on tails in order to even out the streak and maintain an equal ratio of heads to tails.

As one key study notes:

“People expect that a sequence of events generated by a random process will represent the essential characteristics of that process even when the sequence is short.

In considering tosses of a coin for heads or tails, for example, people regard the sequence H-T-H-T-T-H to be more likely than the sequence H-H-H-T-T-T, which does not appear random, and also more likely than the sequence H-H-H-H-T-H, which does not represent the fairness of the coin… Thus, people expect that the essential characteristics of the process will be represented, not only globally in the entire sequence, but also locally in each of its parts. A locally representative sequence, however, deviates systematically from chance expectation: it contains too many alternations and too few runs.

Another consequence of the belief in local representativeness is the well-known gambler’s fallacy. After observing a long run of red on the roulette wheel, for example, most people erroneously believe that black is now due, presumably because the occurrence of black will result in a more representative sequence than the occurrence of an additional red.

Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not ‘corrected’ as a chance process unfolds, they are merely diluted.”

— From “Judgment under uncertainty: Heuristics and biases” (Tversky & Kahneman, 1974). Note that, in addition to viewing

chanceas a self-correcting process, people sometimes also implicitly treat certain devices, such as coins and dice, as intentional systems, with volition, memory, and ability to affect outcomes.

As this shows, the issue underlying the gambler’s fallacy is the incorrect belief in *local representativeness*, which is people’s expectation that small samples (or small parts of large samples) will be representative of their parent population, since they expect the essential characteristics of the population to be represented not only globally in the entire population (or in large samples), but also locally in all its parts. This is also referred to as the *law of small numbers*, which is the incorrect belief that small samples are likely to be highly representative of the populations from which they are drawn, similarly to large samples.

Furthermore, additional explanations have been proposed for the gambler’s fallacy. This includes, for example, a gestalt approach to assessing strings of events, which involves the belief that upcoming independent random events will be connected to prior ones, as a result of the tendency to perceive patterns and connections where there are none.

These explanations, together with the representativeness explanation, generally revolve around the concept of *heuristics*, which are mental shortcuts that can be beneficial in some cases, but that can also lead to erroneous judgments in others. Because of this, and because the gambler’s fallacy occurs as a result of belief in local representativeness, this bias is closely associated with the *representativeness heuristic*, which is the tendency to evaluate probabilities by the degree to which one thing is representative of another (i.e., the degree to which a sample is representative of its parent population or an event is representative of the process that generated it).

## How to avoid the gambler’s fallacy

To avoid the gambler’s fallacy, you must first be aware that it’s about to be used, in your reasoning or in someone else’s, or that it has been used already. However, research shows that simply being aware of the gambler’s fallacy is often not enough, by itself, in order to avoid it, which suggests that additional debiasing techniques are needed.

One such technique is to emphasize the independence of the different events in question, by highlighting their inability to affect each other. For example, when it comes to the odds of a pair of dice landing on double 6’s in an upcoming roll, after they have landed on double 6’s in the previous roll, you should internalize the fact that the second roll is independent of the previous one, by considering that:

- The dice have no way of remembering previous rolls.
- The dice have no way of influencing future rolls.

When doing this, you can either explain this issue when it comes to the specific scenario under consideration, or you can illustrate the concept of event independence using a simple and intuitive generic example, such as that of a dice roll or a coin toss.

In addition, you can further internalize this concept by asking yourself or whomever you’re trying to help avoid this fallacy to explain *how* the dice might be able to influence the roll. This can be beneficial, since asking people to think through the process, instead of simply explaining it to them, can increase the likelihood that they will understand why this belief is false.

Finally, you can also benefit from other, more generalized debiasing techniques. This can involve, for example, slowing down the reasoning process, or optimizing the decision-making environment by removing distractions that make it harder for people to think clearly.

Overall, to avoid the gambler’s fallacy, you should become aware that it’s playing a role in someone’s thinking, and then demonstrate the independence of the events in questions, by showing that they cannot possibly affect each other. You can also explain why this type of reasoning is flawed, illustrate its issues using relevant examples, and implement general debiasing techniques, such as slowing down the reasoning process.

### Remember that events aren’t always independent

It’s important to keep in mind that, in some cases, an unlikely outcome suggests that events aren’t truly random and independent from one another.

For example, consider the example of a coin flip. The odds of a fair coin landing on heads 5 times in a row are roughly 3 in 100. This isn’t too unlikely, and so, if we toss a coin and it ends up landing on heads 5 times in a row, it shouldn’t necessarily cause us to be suspicious. Furthermore, even if you keep tossing a coin, and it lands on heads 10 times in a row, this doesn’t mean that you should necessarily assume that it’s not a fair coin, since the odds of a fair coin doing that are approximately 1 in 1,000.

However, let’s say you keep tossing the coin, until you get 50 tosses in a row. Since the odds that the coin will keep landing on heads 50 times in a row are about 1 in 1,126,000,000,000,000 (1 in 1.126 quadrillion), if this outcome occurs then it’s reasonable to assume that the coin isn’t a fair one, since the likelihood of receiving this outcome is so low otherwise. Accordingly, you can be relatively certain that the next time you toss that coin, it’s still going to land on heads, as it will the next time after that, based on the previous tosses.

As such, while it’s important to be aware of the gambler’s fallacy, and to avoid assuming that independent events can affect one another, it’s also important to remember that in some cases, certain unlikely outcomes suggest that events aren’t truly independent of one another, which you should take into account when making decisions.

*Note*: there are various statistical approaches, such as *Bayesian inference*, that can be used to assess the likelihood that supposedly independent events are not actually independent of one another.

## Related concepts

### The inverse gambler’s fallacy

The *inverse gambler’s fallacy* (sometimes referred to as the *retrospective gambler’s fallacy*) is the mistaken belief that a random process is likely to have occurred many times in the past, after an outcome of it that is perceived as rare is observed. For example, the inverse gambler’s fallacy can cause someone who sees a pair of dice landing on double 6’s to assume that the person rolling them has rolled them several times beforehand, just because this outcome is perceived as rare, and as unlikely to occur on the first roll.

### The gambler’s fallacy fallacy

One study on the topic of the gambler’s fallacy, proposes several variants of this fallacy, which are defined as follows:

“The

gambler’s fallacyis the belief that the probability for an outcome after a series of outcomes is not the same as the probability for a single outcome. The gambler’s fallacy is real and true in cases where the events in question are independent and identically distributed.The

gambler’s fallacy fallacyis our argument that, contrary to the standard account of the gambler’s fallacy, probabilities of sequences of outcomes can be epistemically rational in situations where the gambler’s fallacy might arise. This is the case when (and only when) the odds of the probabilities of the relevant sequences of outcomes are compared to each other. Those odds are the same as the odds of the singular outcomes at the end of those sequences.The

gambler’s fallacy fallacy (fallacy)is the irrational belief that the probability for a series of outcomes is the same as the probability for the last outcome in that series of outcomes. The gambler’s fallacy fallacy (fallacy) is a variant of the gambler’s fallacy that arises from an irrational implementation of the gambler’s fallacy fallacy argument.”— From “The gambler’s fallacy fallacy (fallacy)” by Kovic & Kristiansen (2019)

### Type I and Type II gambler’s fallacies

A distinction is sometimes drawn between two different types of the gambler’s fallacy:

“Visitors to casinos may notice that it is not uncommon to observe players who laboriously record the outcomes of the roulette wheel and suddenly decide to bet on a certain color or number. This is often the point at which they decide that the random process has deviated too much (e.g., “red” did not appear for several rounds) and thus expect nature to “correct” itself. In other words, they believe that the chance of red on the following trial is larger than on previous trials (i.e., larger than .5). This is a manifestation of what we will refer to in this paper as the Type I gambler’s fallacy…

Note that underlying the Type I gambler’s fallacy is the implicit assumption that the roulette wheel is indeed perfectly calibrated, implying that each number is equally likely to occur (i.e., in a roulette wheel with 37 numbers, the probability for each number is the same and equals 1/37)…

Many gamblers believe that detecting a favorable number is a relatively simple task… Evidently, many gamblers (and nongamblers alike) share an erroneous belief regarding the number of observations needed to reliably detect a favorable number. We label this belief as the Type II gamblers’ fallacy…

The two types of the fallacy are, in many respects, similar, thus representing two sides of the same coin. Yet, they are also different: Type I assumes a perfect wheel (in a way, accepting the null hypothesis) and reflects mainly the failure to comprehend statistical independence. Type II fallacy occurs while searching for a bias and as such incorporates a process of hypothesis testing. The main deficiency observed in this case is the insensitivity to statistical power. This is a weakness that is apparently shared by many researchers in the social sciences as pointed out by Cohen ( 1988). Both fallacies stem from misconceptions of randomness. People apparently fail to appreciate the amount of “noise” (or variance) that can be introduced into a system by random effects and that in turn gives rise to the representativeness heuristic.”

— From “The two fallacies of gamblers: Type I and Type II” by Keren & Lewis (1994)

### The hot-hand fallacy

The *hot-hand fallacy* is the mistaken belief that a string of similar outcomes signals that additional similar outcomes are likely to follow, in a situation where these outcomes are independent of one another. For example, the hot-hand fallacy can cause someone to believe that if they rolled double 6’s twice in a row, then they are likely to get double 6’s again if they roll the dice a third time.

Though this phenomenon appears to represent an opposite effect than the gambler’s fallacy, the two are not always viewed as contradictory or as a simple inverse of one other, and various distinctions have been drawn between them. For example, one study states the following:

“The gambler’s fallacy is a belief in negative autocorrelation of a non-autocorrelated random sequence. For example, imagine Jim repeatedly flipping a (fair) coin and guessing the outcome before it lands. If he believes in the gambler’s fallacy, then after observing three heads his subjective probability of seeing another head is less than 50%. Thus he believes a tail is ‘due,’ that is, more likely to appear on the next flip than a head.

In contrast, the hot hand is a belief in positive autocorrelation of a non-autocorrelated random sequence. For example, imagine Rachel repeatedly flipping a (fair) coin and guessing the outcome before it lands. If she believes in the hot hand, then after observing three correct guesses in a row her subjective probability of guessing correctly on the next flip is higher than 50%. Thus she believes that she is ‘hot’ and more likely than chance to guess correctly.

Notice that these two biases are not simply inverses of each other. In particular, the gambler’s fallacy is based on beliefs about outcomes like heads or tails, the hot hand on beliefs of outcomes like wins and losses. Thus someone can believe both in the gambler’s fallacy (that after three coin flips of heads tails is due) and the hot hand (that after three correct guesses they will be more likely to correctly guess the next outcome of the coin toss).”

— From “The gambler’s fallacy and the hot hand: Empirical data from casinos” (Croson & Sundali, 2005)

### Independent and identically distributed variables

The gambler’s fallacy is often mentioned in the context of variables that are *independent and identically distributed* (*i.i.d.)*, such as coin tosses and dice rolls. In this context, *independent* means that the variables (e.g. coin tosses) do not influence one another, and *identically distributed* means that the variables have the same probability distribution (i.e. the same likelihood of resulting in the same outcomes).

However, note that the outcomes of i.i.d. variables do not have to be *equiprobable*, which means that they do not need to have an equal probability of occurring. For example, even if a coin is unfair because it’s more likely to land on heads than tails, a series of tosses of this coin is still identically distributed, because the coin has the same probability of landing on heads each time.

## Summary and conclusions

- The
*gambler’s fallacy*is the mistaken belief that if an event occurred more frequently than expected in the past then it’s less likely to occur in the future (and vice versa), in a situation where these occurrences are independent of one another. - For example, the gambler’s fallacy can cause someone to mistakenly assume that if a coin that they tossed landed on heads twice in a row, then it’s likely to land on tails next.
- The gambler’s fallacy primarily attributed to the expectation that even short sequences of outcomes will be highly representative of the process that generated them, and to the view of chance as a fair and self-correcting process.
- To avoid the gambler’s fallacy, you should become aware that it’s playing a role in someone’s thinking, and then demonstrate the independence of the events in questions; you can also explain why this type of reasoning is flawed, illustrate its issues using relevant examples, and implement general debiasing techniques, such as slowing down the reasoning process.
- When resolving the gambler’s fallacy, remember that in some cases, a sequence of highly unlikely outcomes can indicate that the events in question are not truly random or independent of one another.