The Gambler’s Fallacy Explained: Why streaks have no significance

Published: Feb 27, 2026 14:09

by Catalin Barboianu

10 min read

The gambling activity may be rewarding, either with the goal of entertainment or making profit. Such rewards come with a cost, as gambling is associated with risk factors that can turn it into problem gambling. Among these factors, gambling-related cognitive distortions can lead to false expectations and problematic gambling behavior. The most popular such cognitive distortion is the so-called Gambler’s Fallacy, which can manifest in any game of chance and even beyond gambling.

In this article you will find what is the Gambler’s Fallacy, what its nature is, and how it can be corrected.

Definition of the Gambler’s Fallacy

In general, the Gambler’s Fallacy is defined as one’s erroneous belief that the likelihood of a particular random event happening depends somehow on the previous instances of that event. More precisely, it is the belief that if a particular event has happened during the past more frequently than its probability indicates (considered as the “normal” standard), then it is less likely to happen in the near future, or vice versa (less frequently during the past – more likely in the near future). This belief stands despite that it has otherwise been established that the probability of that event does not depend on its occurrences in the past and those past instances are independent to each other.

Looking at this definition, we see first that the Gambler’s Fallacy falls within the category of beliefs. As such, it has psychological nature. Then, we see that the definition employs mathematical concepts (random event, frequency, probability, independence of events), so there is a mathematical dimension of it.

Overall, the nature of the Gambler’s Fallacy is a mix psychological-mathematical, pertaining to one’s perception of how mathematics actually applies in the gambling context in conditions of randomness.

The Gambler’s Fallacy in action

Imagine one throws a coin repeatedly and the coin lands on heads for 20 times. Would you bet on tails for the next throw just because on the heads streak? If yes, you have fallen prey to the Gambler’s Fallacy. The probability of landing on heads at the 21st throw is still 1/2, evaluated after the 20th throw. The probability of having a streak of 21 heads in a row is 1 in 2,097,152, so you might not bet on this outcome; however this is irrelevant for the moment just before the 21st throw.

The same applies for a red/black decision (or whatever even bet) in roulette (with slightly different probabilities).

In fact, the longest succession of the same color in roulette was recorded at the Monte Carlo Casino in 1913, when the ball stubbornly landed on black 26 times in a row, causing big losses to many players who predicted red for the next spin.

If we think deterministically, there is the possibility of having, say, 1,000 or even 10,000 unfavorable outcomes in a row in whatever game, as there is no physical factor or theoretical argument that may prevent that. This is true for whatever game and type of outcomes, whether we talk about coin or dice faces, numbers in roulette, combinations of stops in slots, or of cards in blackjack.

One may think “1,000 times? That’s crazy!” or “That’s impossible!” Actually, the correct estimation is “It’s almost impossible.” as such an event has a probability close to zero, but still positive and is very likely that will not occur during that person’s lifetime. If one accepts this improbable possibility, it is a first step for correcting their Gambler’s Fallacy, if present.

Examples of streaks and their probabilities

The mathematics behind the Gambler’s Fallacy

The Gambler’s Fallacy stems from misconceptions, fallacies, and errors we can have or make regarding the mathematical facts of gambling, including how mathematics actually applies in the real world of gambling.

Independence of events

First, the Gambler’s Fallacy is about a misconception about the independence of random events (but it’s not the only cause, as we will see further). In probability theory, two such events are called independent if the probability of their conjunction (that is, the event that both will occur) is the product of their probabilities; this is equivalent to a relation in terms of conditional probability: P(A | B) = P(A), reading ‘The probability of event A conditioned by event B equals the probability of event A.’

However, understanding that as applied to gambling is not straightforward. Many people think that in a game the outcomes “have something in common” as being produced by the same device. They actually do have that, however this is a sort of “physical” dependence and not a statistical dependence. The outcomes of a gambling device are statistically independent as the events producing them are random (except of course fraudulent or biased devices), where the statistical independence is expressed by the mathematical relation above. This statistical independence comes from the premise that the outcomes as elementary events are equally probable because all the physical (deterministic) factors of the experiment have been objectively ignored in that probability field. It is that latter kind of independence that keeps the probability of the next outcome the same regardless of the past outcomes.

The Law of Large Numbers

Probability is an abstract ideal notion. It is abstract as it applies to any field of events, provided they belong to a certain mathematical structure. It is ideal as it can be applied in the real world only under ideal conditions, and the main one is randomness. If we throw a dice knowing that number 1 has a 1/6 probability of occurrence, this does not mean that number 1 will occur once in 6 throws or 10 times in 60 throws. Its probability is a kind of average, but not arithmetical average.

The Law of Large Numbers is the only mathematical result in probability theory connecting the ideal and the real world of applications. It says that the relative frequency of the occurrence of an event in a sequence of trials performed in identical conditions converges toward the probability of that event. This means that in our dice experiment, if we count the occurrences on number 1 and divide that number to the number of throws at each throw, the obtained sequence (of fractions) will approach 1/6 when the number of throws increases.

A player affected by the Gambler’s Fallacy may have the expectation for the current relative frequency to match the probability of the predicted outcome or an average relative frequency recorded statistically in player’s own games or the history of that game. It is this kind of expectation that triggers the feeling that a certain outcome “is due” after a streak missing it.

Equating probability with the relative frequency over the short- to medium-run is a mathematical error, as the Law of Large Number provides an average over an infinite number of trials.

Lacking this mathematical knowledge about the independence or events and/or the Law of Large Numbers or inadequate or incorrect interpretation of these mathematical facts in gambling are the key premises for triggering the Gambler’s Fallacy. They are what we call the cognitive-educational factors determining this fallacy. It seems then that if one have a good grasp of these notions and interpret them correctly they will not fall prey to the Gambler’s Fallacy. Unfortunately, this is not always true because the Gambler’s Fallacy has also other causes, which are deeply rooted in our inner psychobiological constitution, as you will see further. This means that not only gamblers may be affected, but also other people, regardless of their level of education – even mathematicians as well.

Cognitive-educational vs psychobiological causes

The psychology of the Gambler’s Fallacy

Psychologists gathered the causes of the Gambler’s Fallacy under the general label ‘People have a wrong perception of the complex concept of randomness’. Randomness is a concept that grounds probability theory, however it is not a mathematical concept. It is rather a philosophical concept and everyone perceives it in their own way. The way we perceive and understand randomness is related to the complexity of the concept itself and the physiology of our brain.

Randomness as both order and disorder

The nature of randomness in expressed in plain language as a kind of disorder of the occurrences of the events for which causes are not known in their entirety. We think of randomness as the opposite of law, rule, or purpose, of indeterminacy, irregularity, and implicitly unpredictability.

Such attributes of randomness makes it a sort of total disorder. Concepts like ‘equally possible’, ‘equally unknown’ or just ‘independent’ fall within the attribute ‘total’ and suggest a kind of uniformity as characterizing randomness. For science and mathematics randomness is just a convenient conceptual prerequisite for the probability theory to work. However, all this latter characterization makes randomness in turn to be an order and the Law of Large Numbers just reflects an important aspect of this order.

Accepting randomness as both an order and a disorder should not twist our mind in any way, as this is actually the nature of randomness and we have to perceive it as such.

Those manifesting the Gambler’s Fallacy incline to take it as order and believe in its uniformity. When they observe what looks like disorder (the streaks) they have the expectation and belief that order must be restored.

How our brain behaves in conditions of uncertainty

Humans are so equipped by evolution to look for safety and equilibrium and our brain submits to this principle. People do not like incertitude and tend to evaluate things and phenomena by basing on real, determined, and confirmed facts. Our brain is so built and trained to look for patterns and match them with the experiences stored in its memory. The brain is also a huge consumer of energy and has developed several physiological ways to save energy. This is why we always look for causes for facts jumping out of the usual patterns of our experience and of our universe of beliefs, just for reaching a mental state of equilibrium. In psychology, this is called the clustering illusion. This principle is so strong that we may even come no longer believe in the independence of trials of a random experiment, just to “explain” something that we cannot explain otherwise.

Most of the psychologists’ theses regarding the causes of the Gambler’s Fallacy characterize this condition as a cognitive bias produced by a psychological heuristics called the representativeness heuristic.

Can the Gambler’s Fallacy be corrected?

Correcting this cognitive distortion is a complex process, but possible in an expert environment. First, an educational intervention focused on the mathematical notions associated with this fallacy, which to correct misconceptions and errors of interpretation is mandatory. Then, the subject of the intervention must carry its own fight in which they should train their brains such that its left hemisphere (dominant in tasks of logic, language, and analytical thinking) to take control over the right hemisphere (dominant in tasks of creativity and in managing emotions and feelings) for this particular goal, with the help of the knowledge acquired.

As for punctual concrete recommendations one may easily adopt, the following are known as effective:

Practical recommendations for players

Correcting the Gambler’s Fallacy is important for two reasons: First, the misconceptions and fallacies on which it grounds can fuel other gambling-related cognitive distortions, known as risk factors for problem gambling. Second, players with subjective (over)confidence in predicting outcomes are supposed to increase their stakes, which may result in severe loss.