The game of roulette is the source of so many fascinating stories and educational examples that it worth studying it from a scientific perspective.

The early inventor of roulette was mathematician, physicist, and philosopher Blaise Pascal in the 17^{th} century and such an interest of a mathematician for a game of chance seems not to be accidental, since probability theory was born from a discussion between mathematicians Pascal and Fermat regarding a problem related to a game of chance.

Anyway, I won’t talk in this article about roulette or its mathematics, but about a particular behavior of the players regarding the outcomes of roulette and not only, which can be generalized beyond gambling.

Imagine someone observing the roulette ball landing on black numbers for a few times in a row (say, five or more). This person might think that it is *very likely* for the next coming number to be red, given that succession of blacks, and perhaps place a bet on red. This reasoning is however inconsistent with the mathematical fact that a red number has the same probability of occurrence in the next spin as it had in any of the previous spins and as it would have if that was the first spin (namely, slightly less than 50%), since the roulette spins are independent.

This sort of belief of our hypothetical player is termed *the gambler’s fallacy*. It is briefly defined as one’s fallacious belief that the likelihood of the occurrence of a random event is influenced by previous instances of that type of event. It is the belief that if a particular event has occurred more frequently than “normal” (with respect to its probability) during the past, it is less likely to happen in the future or near future (or vice versa), when it has otherwise been established that the probability of that event does not depend on what has happened in the past.

Ok, you may ask: “So what? Despite having a fallacious belief, the player would have placed a bet anyway. Does it matter if red or black occurs? He loses or wins with the same probability.”

If a player places only one, two, say three bets, loses then he stops playing and goes home, that would be the case – the gambler’s fallacy would be just a theoretical topic for psychology. However, think that our player believes that he is in the possession of some additional information beyond the mathematical facts, which actually is thought to change the evidences expressed by mathematical probability, and such a belief urges him to continue betting. And it is more than that: Some players use progressive betting systems, where they increase their stake with each lost bet. Such systems are based exactly on the idea that any succession of similar outcomes will come to an end and when this happens the player’s overall profit will be positive. However, these players need a solid bankroll to sustain a long succession of failures and the possible cumulated loss may be very big. As such, the gambler’s fallacy may have real concrete negative consequences.

The longest succession of the same color in roulette was registered at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. (This is why the gambler’s fallacy is also called the Monte Carlo fallacy.) You can imagine what bankruptcy this event caused to those who played progressive systems betting on red.

However, no one really knows whether 26 is actually the record for the same color, as games outside a casino or unregistered such “oddities” were possible at any time and any place.

Although the most popular example is that of roulette, we can see the gambler’s fallacy in action in every game of chance and in everyday life as well.

For instance, after having multiple children of the same sex, some parents may believe that it is likely for their next child to be the opposite sex. While the probability of having a child of either sex is regarded as near 50%, the bias for one sex or another is the result of taking into account the previous births and as such is a gambler’s fallacy. Such a parent might be disappointed in his or her expectation (or even go shopping in advance for boy or girl stuff, which might not match the sex of the new born). Again, the gambler’s fallacy has effects.

Before seeing where this fallacious belief comes from and how it can be corrected, let’s say that the *physical* possibility of a long or very long succession of “unexpected” outcomes (like the black in our example) really does exist. There is no rational reason to believe that this is not the case. Black might occur 10, 100, or 1,000 times in a row and perhaps the player knows that. Still, they might doubt the possibility (“100 times?! That’s crazy!”) and form a personal belief that involves the prediction that the unfavorable series will end shortly.

If the player is convinced about the *physical* possibility of the occurrences of, say, 10,000 unfavorable outcomes in a row in a game (even though this is unlikely to happen during his lifetime), it is a first step in correcting their gambler’s fallacy. But the picture is more complex than that.

The popular description of the causes of the gambler’s fallacy is that it results from a misconception or non-understanding of the notions of randomness and independence. We could take them one at a time, but they are tightly related.

The outcomes that trigger the gambler’s fallacy are *random* events. In probabilistic/statistical terms, they are called random because they are the result of trials of the same general random experiment: Spinning the roulette wheel (the random experiment) is done several times (the trials), which means that they are independent of each other. Some may think that they are not independent, since they are the results of *the same* experimental setup, but independence of events must be understood in a statistical sense here and not in that of physical causality or relationship: The trials as actions may not be independent as performed by the same person or machine, but their outcomes, as elementary events, are independent because the possibility (or probability, if you like) of occurrence of one does not depend on another.

This statistical independence comes from the premise that the outcomes as elementary events are equally possible because all the physical factors of the experiment that may determine an outcome or another have been objectively ignored.

Once we have understood the notion of statistical independence, we have to clear up the concept of randomness in the gambler’s fallacy context.

Independence relates to randomness, but mathematics or physics does not define randomness. Randomness is assumed as a perquisite for probability theory, a way of abstracting from experience and the conceptual means by which we are able to postulate the ‘equally-possible’ attribute of the elementary events of a probability field. Like in our roulette example, each number is equally possible to occur at a spin just because we are not able to quantify all the physical factors related to that experiment.

Philosophically, the concept of randomness has been viewed as dependent upon the concepts of possibility, chance, necessity, and indeterminacy. For Aristotle, randomness is a possibility, in the sense of a deviation from the laws of nature: Randomness occurs when the purpose of nature is not attained, when hindering causes corrupt the operations of nature.

Similarly, Pierre de Laplace (an early father of probability theory and philosopher of probability) viewed randomness as opposite to law. For him, the theory of probability pertained to natural science rather than to mathematics, and its goal was not the study of mathematical objects, but the discovery of the laws of nature.

Beyond philosophy, mathematics has not yet succeeded in providing a rigorous definition for randomness. Randomness, as a feature of the real world, is not defined or introduced in the mathematical context or content of probability theory despite the various mathematical attempts to do that.

Emile Borel (mathematician, important contributor to measure theory) stated that, unlike other objects from the surrounding reality for which the creation of theoretical models assumes an idealization that preserves their properties, this idealization is not possible in the case of randomness. The fact that “reason cannot reproduce the randomness,” as Borel said, still remains a principle that not even philosophers have contradicted.

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Randomness is conceptualized first as a *disorder* (of the occurrences of the events for which causes are not known in their entirety). This disorder reflects our lack of knowledge (or ignorance), and as such, it is first a feature of our reasoning and second of the phenomenological world, if this world is non-deterministic.

But randomness exists as a *special type* of disorder and is a sort of *total* disorder, characterizing all factual reality as seen through our reason. The ‘total’ attribute may be expressed through ‘equally possible’ or ‘equally unknown’ or just ‘independent.’ For science and mathematics randomness is just a convenient conceptual perquisite for probability theory and for making the probabilistic/stochastic method operational and effective in scientific reasoning.

However, all this latter characterization makes randomness in turn to be an *order*. The uniformity of randomness is in fact order rather than disorder. Moreover, the *infinite* feature of randomness also strengthens the qualification as order. Indeed, infinity is present in the concept of randomness. We cannot talk of something random without imagining it in an infinite context, in an infinite number of instances or infinite possibilities. Accepting randomness as both an order and a disorder should not twist our mind in any way, as this is not an inconsistency at all, but just the mere nature of randomness.

When a person affected by the gambler’s fallacy expects a succession of non-favorable events to end as having a reason for that, that person actually is strongly inclined to take randomness exclusively as *order*, because the reason behind that expectation is the belief that what seems to be disorder must be restored as order – or at least come to the tendency of restoring. The order in this case is expressed through either the mathematical probability of that event or any registered, memorized, or known average frequency of that event in the past experience of the person or of other persons.

In the case that the person knows the probability of the event, their expectation is for the relative frequency of that event to approach this probability. In the opposite case, the expectation is for the current relative frequency to match an average relative frequency recorded statistically in that person’s own experience or the history of that event.

Equating probability with relative frequency over the short-to-medium run is obviously a mathematical error. The mathematical result that links probability and relative frequency is the Law of Large Numbers, in the sense that probability is the limit of the sequence of the relative frequencies calculated when the number of trials increases.

Equating the current relative frequency with an average based on a large enough number of past recordings is a mathematical error as well, as the Law of Large Numbers does not provide such kind of relationship – it is only about the limit, and the terms of a sequence may differ from each other in whatever range over a finite interval without affecting the limit of the sequence.

Flipping a coin gives a probability of 1/2 of either head or tails, but this does not mean that we should expect the last 100 throws to show near 50 heads, nor to show about as many heads as in the previous 100 throws. They could be in any number.

In the above imagine is a simulation of coin tosses, showing that the proportion of red versus blue approaches 50-50 (the law of large numbers), but the difference between red and blue does not systematically decrease to zero.

The Law of Large Numbers grants randomness a “minimal” order in the sense of convergence, and our hypothetical roulette player inclines to grant it even “more ordered” order, by translating the result of this law over a finite interval of trials or his or her short experience. However, all concepts of probability theory are based on infinity, while all our experiences are finite.

Summing up, the causes of the gambler’s fallacy consist of three interdependent fallacies, misconceptions, and errors:

1) employing physical causal independence instead of statistical independence in reasoning (a fallacy);

2) taking randomness to be order and not disorder (a misconception);

3) equating probability with relative frequency (a mathematical error).

So, the question arises, would a cognitive intervention based on in-depth learning about randomness and essentials of probability theory correct the gambler’s fallacy for a person who is subject to it? Not necessarily. The correction of this fallacy in practice is not straightforward and is not only dependent of the level of education of the individual. This fallacy has a strong cognitive-neurological dimension, besides the mathematical and philosophical ones.

Psychologists Amos Tversky and Daniel Kahneman proposed in 1971 that the gambler’s fallacy is a cognitive bias produced by a psychological heuristic called the *representativeness heuristic*, which assumes that people evaluate the likelihood of occurrence of an event by assessing a degree to which it is similar in essential characteristics to its parent population, and it reflects the salient features of the process by which it is generated. This is actually a belief in a sort of “law of small numbers,” leading to the erroneous belief that small samples must be representative of the larger population.

Moreover, even within a cognitive intervention based on learning, there is a cognitive-psychological component that usually prevents attaining the goal of the assumed intervention – the *perception* of the math-related concepts involved. On the one hand, the notions of probability theory may be tricky for those unfamiliar with it, especially when applied to real-life contexts. Besides, the concepts of this theory are subject to various interpretations, including scientific, and we have to stick with their right meaning in the given context. On the other hand, the adequate understanding of the phenomenon requires a non-mathematical perception of potential infinity. Potential infinity is hard to perceive, even by mathematicians, and it has posed serious problems in the foundation of various concepts and theories during the history of mathematics. For all people, both actual and potential infinity is hard to perceive or understand, just because all our experiences are finite in number.

So even a mathematician could be subject to the gambler’s fallacy in some contexts, since the specific problems of perception are embedded in our inner biological constitution and this fallacy has also non-mathematical dimensions.

A roulette ball landing 10,000 times in a row on the same color remains possible and it will happen within the next 100,000 or 1 million or 1 billion years or more, but it might happen from tomorrow either (it might have been already happened in the quantum multiverse, to make fun of it a bit). We cannot exclude this possibility only because the probability of that happening is close to zero, like we cannot exclude it only because the ball already landed on that color in the previous spins.

Catalin Barboianu is mathematician and philosopher of science. He is the founder of PhilScience.