The Gambler’s Fallacy Explained

The gambler's fallacy is a mathematical error, arising from a misuse of mathematical laws: statistics and probability, more precisely the law of large numbers and the independence of random events.
This kind of error is also called a sophism, that is is a (clever) false argument with fallacious logic. In other words, arguments which have the appearance of a precise and rigorous argument but which is in reality erroneous: we are not being fooled for nothing !
It may therefore appear normal to be a "victim" of this type of gambler's fallacy, to be fooled, because the reasoning clearly seems, on appearance, logical and mathematical. However, a serious and knowledgeable bettor must be able to thwart this type of error.
In this article, we explain the phenomenon of gambler's fallacy in more detail: what are these errors and their explanations ? and above all, finally, how to avoid them?

Gambler's Fallacy or bettor's sophism

As we will see, there are several errors, or biases, which should worry bettors. Among all these, the "gambler's fallacy" or "gambler's sophism" specifically refers to an error of logic, of reasoning, which consists of believing that past draws influence future draws.

For example, I flipped a coin 100 times and got "a lot more" heads than tails. “Much more” of course means here, much more than 50/50. For example I got Tails 60 times. So I expect to get heads more times in the next throws "to compensate"
This reasoning actually shows two errors: a misuse of the law of large numbers and a misunderstanding of the independence of the results. We will discuss these two mathematical elements in the following.

On the probability of a new occurrence of an already too frequent event

So let's imagine that I flipped my coin 10 times and got tails 10 times. What is the probability of getting Tails again now?
Assuming this coin is well balanced, this probability of now getting Tails once again is
0.510×0.5 = 0.511 = 1/2048
that is less than 0.05%. This probability is indeed very low and really does not make you want to bet!
And yet…
This is exactly one of the aspects of this famous gambler's fallacy or gambler's sophism. Indeed, after having obtained Tails 10 times, what is the probability of now obtaining Tails? This probability is also
0.510×0.5 = 0.511 = 1/2048 ≃ 0.05%
which is therefore just as weak as the other, but above all which is equal to it!

In summary, after getting Tails 10 times out of my 10 throws, the probabilities of getting Heads or Tails are very low and equal.

Explanation of bias, false impression

The mistake made here is to confuse the two probabilities: The first probability is a conditional probability and is worth 1/2, while the second is a fairly low overall probability, around 0.05%.
This bettor's error can here be summed up as the mathematical error of confusing these two events and their probability: getting "again" tails after 10 times in a row tails is very rare, but what is unlikely is not the "heads again" but the previous "10 heads in a row".

The last flip of the coin (like all the previous ones) is independent of the others. This notion of independence is very important for the calculation and estimation of probabilities. Any random events are not always independent. We now detail this notion of independence:

Principle of independence of random outcomes

In mathematics, two random events are said to be independent if knowledge of the outcome of one event does not influence the probability of the other event.
The typical example is that of Heads or Tails with a coin tossed twice: having gotten Tails (for example) on the first toss does not change in any way the probability of getting Heads (or Tails) on the second.

This example is always cited first and often remains the only one cited. Independent events don't just exist for coins or roulette. Consider the following example of a 52-card deck and the three events: The probability of event A is simply, 4 aces out of 52 cards, i.e.: P(A) = 4/52 = 1/13.
Now if I draw a card at random and I know before looking at it that event B has occurred, that is to say I know that it is a spade, then the probability is now, 1 ace out of 13 spade cards: P(A) = 1/13.
The probability has not changed and these two events are therefore independent.
On the other hand now, if I draw a card at random and I know before turning over that it is a face, the probability that I have drawn an ace is now equal to 4 aces on 4×4 faces: P(A) = 4/16 = 1/4.
The probability is now different: events A and C are not independent, knowing whether C happens or not changes the probability of A happening.
In summary, even in games of pure chance, events can be independent or not: the context (coin or roulette thrown several times) or a probability calculation allows us to know this rigorously. Poker players, for example, are accustomed to these probability calculations, as the previous simple example showed.

To this error of possibly confusing an absolute probability (obtaining Tails) with a conditional probability (obtaining Tails knowing that I have already obtained Tails 10 times just before), is also often added a poor understanding of the mathematical law of large numbers. This is what we are going to see.

Wrong use of the law of large numbers: explanations

The law of large numbers tells us that, when the greater the number of flippers of the coin, the more the probabilities of getting Heads and that of getting Tails approach probability 1/2. Hence the following reasoning, for example if after 100 throws I have already obtained the result Heads 60 times, therefore 40 times Heads, then I am at the statistics 60/100 = 0.6 of Heads and 40/100 = 0.4 of Tails. As the two "must balance" around 1/2, obviously, to catch up, Tails must "slow down" a little, and Tails the opposite. So, to do this, Tails would have to catch up in the next draws and its probability is thus now greater.

This reasoning is erroneous and comes from a misuse of the law of large numbers which “only” asserts that the statistics of Heads and Tails will tend towards their probability, here 1/2.
We can for example imagine that from the previous 100 draws, we then draw exactly as many heads and tails (i.e. 50 each), during the following 100 draws, leading to the statistics 110/200 = 0.55 of Heads et 90/200 = 0.45 de Tails.
We thus got very close to 0.5 for both results, without Tails having “caught up”.
We can still imagine that over the next 1000 draws, after the previous 200, Heads goes on "leading" with 510 draws in his favor. We then arrive at the stats 620/1200 ≃ 0.51 for Heads and 580/1200 ≃ 0.49 for Tails, and we are still closer to 0.5 for both.

Be careful, in games of "pure chance" the statistics of the past have no say: mathematically speaking, this is a property of some laws of probability called "law of probability without memory".

This type of mathematical error is now well known, and referred to as the gambler's fallacy or also bettor's sophism. This is now a fairly well-known general cognitive bias which is added to other mathematical cognitive biases and which must of course be known and avoided.

Law of small numbers

If the law of large numbers is indeed a mathematical law, proven mathematically, the so-called "small numbers" law on the other hand designates a belief, another fallacious bias of reasoning.
This law of small numbers is the false argument that a small sample must necessarily be representative of the larger population.
For example, with a fair coin a (small) sample of 10 tosses should be representative.
The psychologists Tversky and Kahneman highlighted this reasoning error as a cognitive bias, that is to say a certain natural distortion of our ability to reason correctly. They gave it the name "representativeness heuristic": people evaluate the probability of an event based on how similar it is to other events they have seen and remember, so following the representation they have in. As our capacity for memory is limited, and in any case we have only experienced a limited number, a small number, of these events our representation is just as limited: our decision-making based on our knowledge alone is necessarily erroneous, because very subjective.

Note also that this “law of small numbers” also induce a cognitive bias of the same nature. The mathematician R.K. Guy humorously quoted that

"There are not enough small numbers for them to meet all the demands placed on them."

In other words, we may sometimes see coincidences between events simply because small numbers (those that we, humans, use easily and commonly) are few and therefore necessarily used a lot.

The same psychologists as previously, Tversky and Kahneman, thus explain and summarize both the gambler's error, the error on the use of the law of large numbers and that of small numbers, Les mêmes psychologues que précédemment, Tversky et Kahneman, expliquent et résument ainsi à la fois l'erreur du parieur, l'erreur sur l'utilisation de la loi des grands nombres et celle des petits nombres,

"People have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative, i.e., similar to the population in all essential characteristics."

Gambler's fallacy: description and explanation by cognitive biases

Cognitive biases

A cognitive bias is a mechanism that reflects a logical processing error, a thought mechanism causing impaired judgment. This results in distorted decision-making. This notion of cognitive bias, studied in the field of cognitive psychology, can be studied and exploited specifically in different fields, that of games of chance and sports betting in particular.
These biases result from our humanly limited capacities: Among the many biases that have been discovered and studied, some are of specific interest to bettors:

Outcome bias

Outcome bias is a cognitive bias that reflects a logical processing error, a mathematical error. It is essential for a serious bettor to be well aware and understand this distortion of logical thinking that we all face, but which can have more unfortunate consequences for a bettor.
This outcome bias is a cognitive bias that leads us to judge the quality of our decision-making based only on the outcome. In other words, thinking that our decision (and the reasoning, the calculations, etc. that led to it) is good and well-founded just because the result is what we hoped for.

This bias is frequently found among gamblers who try to predict events purely by chance: as this is not really possible, a gambler may tend to judge actually independent elements as explanations, because it worked … once …

This cognitive bias actually fuels another bias, also a real gambler's fallacy: the illusion of control bias.

Illusion of control

Its name says it all: the illusion of control describes situations in which the bettor believes he has the power to control, or at least influence, over random outcomes.
In an experiment conducted in 1975 by a professor of psychology at Harvard, people who had chosen the lottery ticket number themselves were more confident of their chances of winning than people who had been given a lottery ticket with a number drawn at random.

This illusion of control bias is also well exploited by gambling companies which ask players to intervene (as in the choice of the ticket number in the previous experiment), for example with a lever to pull in casinos, a scratch box, the choice of numbers, the choice of the scratch ticket, etc. Buying a randomly generated lotto ticket is less attractive, gives less confidence, even if the probability of winning is the same!
So many elements on which the player can bring into play his result bias and illusion of control: I lowered the slot machine lever gently and then suddenly and I won, so the gain is the result of my way of doing things, and I can control this machine like this.

How to avoid gambler's fallacies ?

A serious bettor, who intends to make profits from his bets, regularly and in the long term, must be aware of all these biases and avoid them. How to do ?
The origin of these cognitive biases lies in our cognitive limits: we cannot remember everything, make a link between all events, etc.
Mathematics, statistics and probability then clearly come into play, to surpass our abilities.

Mathematical betting strategies do exist, and are effective, but they require application and rigor so as not to fall back into the realm of our own biases.

We come back (as always) to the fundamental conclusion:

Winning at sports betting? it's a job !

Being a regular and long-term winner requires an investment (including reading and thinking throughout this article!):
Finally, progressing in this knowledge makes it possible to “naturally” avoid the biases, gambler's fallacy, and all kind of errors of bettors, which ultimately arise when making unthought-out, unplanned and not calculated decisions.

This long-term investment is important and will result in increasing gains: time invested for gains? This is called a job !.

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