**Gambling** is fun and sometimes rewarding. Still, it has a dark side, which is visible mostly to experts: People fall prey to the so-called gambling misconceptions, fallacies, and irrational beliefs, which are cognitive distortions that are risk factors for developing problematic gambling behaviour. When analyzing the reasons why such distortions occur in gambling and how they can be corrected, experts found that **the math of gambling is involved with several roles**. In this article, we will see **how math is involved in gambling-related cognitive distortions and to what extent mathematics education can contribute to their correction**.

*TOP*

## What are gambling misconceptions and fallacies?

Almost all people participating in the gambling phenomenon have heard about the **Gambler’s Fallacy**. I prefer to call it **the Monte Carlo Fallacy**, as there are also other different gambling fallacies. **It is the fallacious belief that the likelihood of occurrence of a certain outcome in a random trial (as a game of chance is) depends on the previous occurrences or non-occurrences of that outcome.** This fallacy is qualified as such as it is inconsistent with the laws of probability and the concept of randomness.

**Gambling cognitive distortions are qualified in cognitive psychology as heuristics and biases.** These are mental processes leading to beliefs and expectations that do not follow a complete rational flow of evidence and arguments. Such processes are in the form of shortcuts or simply arbitrary or “intuitive” inclinations toward an option by ignoring others * (Kahneman & Tversky, 1974)*. They are explained neurologically through the features of the brain, for instance, the energy-saving feature

*(causing shortcuts)*or the safety and equilibrium feature

*(causing biases)*.

With this description, we can see that cognitive distortions are of a wide palette, we all are predisposed to them due to our inner biological constitution, and they impact our daily lives.

Gambling-specific cognitive distortions form only a small part of them.

The most popular and mostly affecting regular gamblers are:

**The Monte Carlo Fallacy**– Misconception about randomness and statistical independence and false belief in the behaviour of the random outcomes as dependent on previous occurrences.**The conjunction and disjunction fallacies**– Fallacious beliefs about the likelihood of occurrence of a conjunction (‘and’), respectively disjunction (‘or’) of events, inconsistent with the properties of probability.**Subjective estimations of odds/probabilities**– Under- or over-estimations of the probabilities of the events, based on feelings and “intuitive” reasoning rather than the mathematical definition and properties of the probability.**The near-miss effect**– The false belief that a non-winning outcome looking nearly a winning one indicates a high likelihood for a winning one to occur in the near future*(the reality itself and the effect of “I was that close!”)*.**Illusion of control**– The irrational belief and illusion that one can acquire an ability to control the outcomes of random events as basing on their experience or behaviour of any kind.**Misunderstanding of the gambling language**– Misconception stemming from inadequate semantic commitments, namely meanings and associations of meanings of gambling and gambling-related terms in different contexts, which can lead to false beliefs or nonsensical statements.

## The mathematical feature of gambling cognitive distortions

Looking at the above definitions, one can notice that almost all mention mathematical notions and concepts: *randomness, probability/likelihood, statistical independence, operators and mathematical properties*. Even the last one in the list *(related to language)* **refers indirectly to mathematics**, as it is about terms used in both mathematical and non-mathematical contexts.

The mathematical feature of these distortions should not be surprising. Gambling games are conceived on the basis of mathematical models that guarantee their functionality and profitability (for the house). This is true for all popular games of chance, from casino games to sports bets. Given the mathematical nature of such games, mathematics is implicitly involved in the conception of strategies for playing and, in general, in how players think of these games with respect to their expectations, criticism, decisions or choices. Gambling is a phenomenon investigated by experts in various academic disciplines (psychology, economy, sociology, ethics, and mathematics, to name the most important). Given the same mathematical nature of the games, research in gambling studies cannot ignore the mathematics of gambling either. This math-indispensability principle applies everywhere and every time we come to approach gambling rationally.

In particular, the principle applies to the field of problem gambling, belonging to psychology, in which gambling cognitive distortions are investigated from multiple perspectives. Problem gambling researchers have found that gambling misconceptions and fallacies are important risk factors in the development of problematic or pathological gambling behaviour, and hence, their prevention or correction should be given utmost importance * (Fortune & Goodie, 2012)*.

Summing up the description of gambling cognitive distortions, we see that **they have a psychological and a mathematical dimension**. The former dimension reflects our inner biological constitution, while the latter is our mathematical knowledge and cognition related to gambling.

## How gambling cognitive distortions work

People’s beliefs, decisions, and behaviour in daily life are the result of the continuous interaction between the left and the right sides of the brain. **The left one is responsible for emotions and creativity, and the right one for rational thinking and logic.** No belief or decision can be formed or taken entirely rational just because the left side of the brain is always there. It’s in human nature to be emotional and subjective, as it is to be rational and objective.

Psychologists found that people develop cognitive distortions as a way of coping with adverse life situations, which fuels back the involvement of the emotional side of the brain. Moreover, research suggested that humans developed cognitive distortions as a kind of evolutionary method of surviving * (Gilbert, 1998)*.

As such, cognitive distortions are not any kind of disease, but they are natural and triggered by life itself.

Gambling cognitive distortions are a result of the same left-right side interaction, and each gambling misconception or fallacy has its own pattern of formation and manifestation * (Orlowski et al., 2020)*. Obviously, these patterns differ across the various profiles of the individuals, including with their education.

Among the information processed by the rational side of the brain is, of course, the mathematical information about the gambling phenomenon *(in the form of knowledge)* that is subject to distorted cognition, as well as other expert knowledge. The power of this knowledge to take control over the emotional, subjective side of the brain gives a raw measure of the distortion * (Leonard & Vokey, 2015)*. This being said, it would appear that

**the more educated we are about the mathematics of gambling, the less likely we are to develop gambling cognitive distortions**

*. However, things are not that simple and prevention and correction of these distortions are not straightforward, as we will see further.*

__(Pelletier & Ladouceur, 2007)__## Why gambling cognitive distortions are special

In everyday life, we may fall prey to all kinds of superstitions, irrational beliefs and misconceptions. Poor rational knowledge of powerful emotions causes them. If we want to believe in something that makes us happy, we come to do that despite objective information telling us not to believe. **They are cognitive distortions that can usually be corrected through appropriate education.**

Games of chance and gambling, as well as the associated strategies of playing, are based on probability and statistical models, which, in turn, essentially rely on the mathematical concept of probability and the notions and results of probability theory. It happens that probability theory and its concepts may be very tricky for those unaware or unfamiliar with it. That is because no other mathematical concepts are as predisposed to empirical interpretation and prediction as probability. However, interpretation is by nature subjective, and prediction is based on that interpretation. The effect is that abstract concepts such as probability, expected value and other statistical averages, core notions for probability theory applied in gambling, come to be inadequately interpreted in the real life of gambling and lead to false or nonsensical predictions and expectations.

This poses a serious problem to the issue of correcting the gambling cognitive distortions and makes them special because acquiring formal mathematical education *(applicable to gambling)* and even having a good grasp of it is no longer sufficient as a cognitive asset. An adequate cognitive intervention for correcting them should also cover the relationships of the mathematical notions and models with and their adequate interpretation in the real life of gambling. This requires the specialized expertise of the instructors and/or of the resources used beyond mathematics.

## Correcting distortions with mathematical and non-mathematical knowledge

The educational factor of correcting gambling cognitive distortions pertains to the acquisition of gambling-mathematics knowledge, as well as non-mathematical knowledge related to gambling mathematics.

### The Gambler’s Fallacy

**The Gambler’s Fallacy may occur in every game of chance and is the most known and complex gambling cognitive distortion.**Its causes pertaining to the educational factor have been identified as follows:First, gamblers affected by this distortion have a wrong perception of or knowledge about the concept of randomness of the outcomes

*. They take randomness to be a kind of order*

__(Matarazzo et al., 2019)__*(induced by the mathematical properties of probability)*rather than disorder. This bias is also influenced by the qualification of the concept of randomness by many people as mathematical. Randomness is not a mathematical concept, even though it is a primary one for probability theory

*. Second, they*

__(Barboianu, 2024)__**employ a sort of causal physical dependence instead of statistical independence**in their reasoning about the behaviour of the outcomes. They believe that the outcomes of a game are somehow related to each other as being produced by the same machine. Although this last premise is correct, statistical independence is what counts for prediction, and it is definable exclusively with mathematical notions. Third,

**they equate probability with relative frequency and translate the results of the Law of Large Numbers on limited intervals**

*(the so-called Law of Small Numbers)*. This error consists of both an incorrect application of a mathematical result in an empirical context and a misunderstanding of the concept of statistical average.

While the third cause can be approached in a didactical intervention, mainly with mathematical knowledge describing the phenomenon, the other two causes can only be approached along with other knowledge outside mathematics * (Barboianu, 2022, pp.67-78)*.

### Fallacies of Estimating Probabilities

There are **gambling cognitive distortions for which mathematical intervention suffices for correcting them**. It is the case with the conjunction fallacy, where all that is needed is a good understanding of a property of probability. This distortion mostly occurs in sports betting. The disjunction fallacy stems as well from a misunderstanding of or lack of knowledge about a mathematical property.

Subjective estimations of probability fall within the same category of distortions that are correctable mainly mathematically. “Intuitive” not-mathematically based estimations of probability as likelihood occur so frequently in our daily life, and things should not be different in gambling. Such fallacies stem from not knowing or applying the definition of probability as a mathematical measure (of possibility) or its properties and rather substituting them with “perceptual” measures based on “evidence” and experience * (Barboianu, 2022, pp. 84-100)*.

### The near-miss effect

A **complex cognitive distortion is the near-miss effect**, which is characterized by a special conjunction fallacy and an inadequate perception of the mathematical model of the so called “close to a win” outcome.

The gambler affected by the near-miss effect estimates a subjective high probability of the conjunction of events *‘I will be in that near-miss situation again’* and *‘At that time I will win’*, as basing (incorrectly) on a high probability for the first event and ignoring the time displacement. They mentally split the outcome of a game into a winning and non-winning part, although this is irrelevant to the mathematical model of the game, where all outcomes are of the same nature and have the same probability as elementary events in the sample space. Such an issue can only be addressed cognitively with the topic of the applicability of mathematics in the given situation * (Barboianu, 2019)*.

The near-miss effect occurs in every game of chance and is the most sticking in slots.

## Conclusion

Most of the gambling cognitive distortions in the form of misconceptions, fallacies, or irrational beliefs have a mathematical dimension and a psychological dimension. The two natures reflect our inner biological constitution, namely the interaction between the two sides of the brain – the emotional and the rational.

**Gambling cognitive distortions are not a kind of disease; they are natural and may affect everyone, including educated people.**

Mathematics plays several roles in gambling cognitive distortions. First, it plays a constitutive role since the games of chance and the playing strategies that are associated with these distortions are based on mathematical models. Second, mathematics has an educational role, manifesting in the prevention or correction of these distortions. Yet the educational role of gambling mathematics in a cognitive intervention is not exclusive because the complexity of these distortions *(also stemming from the complexity of the concepts of probability theory)* also requires additional expertise to be employed in the didactical interventions *(psychological and even philosophical)*.

The cognitive-educational factor of the correction of gambling cognitive distortions is yet counterbalanced by the biological-psychological factor, which cannot be removed, but only influenced. The effectiveness of any intervention for correcting the distortions is relative to the individual’s cognitive, education, and psychological profile.

__References:__

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