Expecting Too Much from Randomness: The Gambler’s Fallacy

Author: Dr. Catalin Barboianu

The gambler’s fallacy is 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” in the past, it is less likely to happen in the future or near future (or vice versa).

The gambler’s fallacy is the most known gambling cognitive distortion, and it also manifests in daily life beyond gambling. It is also known as the Monte Carlo fallacy. I prefer this term, as other cognitive distortions also occur in gambling. Gambling is a complex phenomenon in which specific cognitive distortions easily develop, in the form of misconceptions, fallacious or irrational beliefs and expectations, and inadequate estimations. They occur because people have difficulty perceiving, understanding, and applying the abstract concepts of probability theory in the realm of actual gambling.

The Gambler’s Fallacy in Roulette

Perhaps the most popular example of the gambler’s fallacy in action is in roulette. Assume a player is betting on red just because a succession of black numbers occurred in the previous spins. If a black number occurs again, the player continues to bet on red and perhaps does that again and again if a red is stubborn to appear, due to the belief that the long series of black is likely to come to an end, so the “normal” frequency given by the probability of occurrence of red and black (slightly less than 50%) is restored or fairly balanced.

Of course, the long succession of black may result in a big loss of money for the roulette player, in case of either static bets or progressive, martingale system-based bets, if the player keeps betting on red. In the latter case, it may result in even bigger losses and ruin. Therefore, the player’s belief that a red has to appear soon leads to poor decision-making and problematic gambling behavior.

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 where the name of this cognitive distortion comes from. However, no one really knows if 26 is actually the record, as games outside of casinos and unregistered “oddities” of this type were possible at any time and any place. Of course, many players got ruined on that day in that casino. For example, a succession of only 8 unfavorable outcomes for a player playing a martingale with an initial wager of $10 and doubling at each loss would result in a total loss of $2,550.

That historical event in roulette had a probability of (18 / 37)²⁶, which is about 0.000000007308, or 137 million to 1 -- that is almost zero. Still, the event occurred at some moment among the billions of roulette spins over its history. Read our article on mathematics behind roulette to understand the principles of the wheel better.

Of course, the gambler’s fallacy refers to and applies to any kind of gambling outcome: coin faces, die faces, cards, lottery or bingo numbers, slot lines, or scratch-ticket lines. It may manifest in a single play, in a succession of plays, or even in the succession of stages of the same game (like blackjack, other card games, or bingo).

What Causes the Fallacy?

First, let’s say that the physical possibility of a long or very long succession of “unexpected” outcomes (like the black in our example) does really exist. There is no rational reason or law of nature to make us 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, the player doubts the possibility (“100 times?! That’s crazy!”) and forms a personal belief that the unfavorable succession will come to an end shortly.

If the player was 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 or even in his or her children’s lifetimes), it would be a first step in correcting player’s fallacy. But the picture is more complex than that.

Two types of factors cause this fallacy: educational and psychological-cognitive.

Educational Factors

This category of factors is related to a poor or inadequate understanding of the phenomenon associated with the fallacy. The popular description of the causes of the gambler’s fallacy is that it is caused by a misconception or lack of understanding of the notions of randomness and independence. Nothing is truer than that; however, understanding these two concepts only by their dictionary definitions and even the probabilistic/statistic definition of independence of events does not suffice in either explaining the formation of the fallacy or correcting it.

The educational causes of the gambler’s fallacy consist of three interdependent fallacies, misconceptions, and errors, detailed below.

Physical Versus Statistical (In)dependence

Employing physical causal (in)dependence instead of statistical (in)dependence in reasoning about gambling outcomes is a fallacy. Obviously, the outcomes that trigger the fallacy are random events. In probabilistic/statistical terms, they are random because they are the results 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 the independence of events must be understood in the statistical sense here and not in that of physical causality. The trials as actions are 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 have been objectively ignored.

Randomness - Order or Disorder?

Many players tend to view randomness as order and not disorder, which is a misconception. Randomness is not a mathematical concept, as it does not have a mathematical description or definition. It is conceptualized as a disorder (of the occurrences of the events for which causes are not known in their entirety) and attempts to define it aim at defining that special kind of disorder.

This disorder reflects our lack of knowledge (or ignorance); as such, it is first a feature of our reasoning and second a feature of the phenomenological world, if non-deterministic. Whether we think of randomness as the opposite of law, rule, or purpose or as chaos, indeterminacy, irregularity, or a form of independence, concepts like prediction, causality, and dependence fall within the concept of randomness. 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 as “equally possible,” “equally unknown,” or just “independent.” This latter characterization makes randomness, in turn, to be an order. Accepting randomness as both an order and a disorder should not twist our minds in any way, as this is not an inconsistency at all but just the mere nature of randomness. To adequately understand randomness, we must not ignore either of these two attributes.

When a player affected by the gambler’s fallacy expects for a succession of non-favorable outcomes to come to an end, s/he is strongly inclined to perceive randomness as an order because the reason behind this expectation is the belief that what seems to be disorder must be restored as order. 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 player or of other players.

Probability Versus Relative Frequency

Many players equate probability with relative frequency, which is a mathematical error. In the case that the player knows the probability of an event, his or her expectation is for the relative frequency of that event over his or her plays or the observed activity of the game in question 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 the player’s own past games or the history of the game. The mathematical theorem that links probability and relative frequency is the Law of Large Numbers, where probability is the limit of the sequence of the relative frequencies calculated when the number of trials increases. The Law of Large Numbers grants randomness a “minimal” order in the sense of convergence, and our hypothetical player inclines to grant it an even “more ordered” order by translating the result of that law over a finite interval of trials or his or her short experience.

Psychological-Cognitive Factors

This category of factors is related to both the inner biological constitution of the individual and his or her empirical experience. These psychological-cognitive factors are deeply rooted in each person’s psychobiological constitution.

Humans are so equipped by evolution to look for equilibrium and safety. We do not like uncertainty, and we tend to evaluate things and situations using sure facts. This may explain why so many people rely on recorded statistics and frequencies rather than on mathematical measures like probability – what has happened in the past is certain, which is what probability seems not to be.

The human brain always looks for causes for events that deviate from the usual patterns of one’s experience and universe of beliefs in order to maintain a state of mental equilibrium. This habit is so strong that sometimes we associate causes with a fact or event when we should not, leaving the true explanation undiscovered. We may even come to not believe anymore in the independence of the trials of a random experiment, just for the sake of “explaining” something that we cannot find a reason for otherwise.

This phenomenon is not specific to gambling. We all manifest cognitive bias, superstitions, supernatural/paranormal beliefs, beliefs in fate/destiny, and so on, in various daily life circumstances. In gambling, such distortions are related to and potentiated by inadequate understanding of the concepts of randomness, independence, and probability.

How to Correct the Gambler’s Fallacy

Given the complexity of the causes that may lead to the gambler’s fallacy, its correction is not straightforward. Gaining the knowledge to eliminate the educational causes of this fallacy requires serious self-study effort, as there are no courses on such topics. Correcting the psychological factors may require counseling correlated with the educational assets.

If I were to draw a few brief pieces of basic practical advice encompassing all of that expert educational-cognitive intervention, I would offer these three:

• Approach each new game as if it is the first and consequently ignore previous outcomes. Avoid counting the occurrences of unfavorable outcomes in succession in your play or the frequency of a certain outcome in any game.
• Do not look for any patterns, balancing, or similarities in outcomes in recorded statistics or in past experience (yours or others’). If you see some, they are circumstantial; do not use them as a premise for any expectation or reasoning involving the prediction of future outcomes, wins, or losses.
• Do not expect any law of averages to manifest in the behavior of the outcomes of the game over the short or medium run.

Whether these exercises come to be effective for a given individual cannot be predicted, just as the outcomes of games of chance cannot. However, knowing the rationale behind them is a step ahead for both the expert and the gambler.