# Gambler’s Fallacy – Definition, Meaning And Examples

15.11.23 Time to read: 9min
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Fallacies are reasoning errors within arguments that explain or prove events with wrong or misleading premises. There is a vast variety of fallacies that are prominent in debates or everyday dialogue. One of them is the Gambler’s fallacy. The name of this fallacy originates from the sector it most commonly occurs: Gambling. Everything you need to know about the Gambler’s fallacy will be explained in this article.

## Definition: Gambler’s fallacy

The classic Gambler’s fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, describes an erroneous belief and perception of the event of streaks of randomness. It’s the misconception that a random permutation of events will always even out over time, regardless of the non-existing correlation between the current and the history of events.

In other words, this fallacy is the mistaken belief that random outcomes are more or less likely to occur based on the outcome of previous events. This assumption is false, since all subsequent events and their respective future outcomes in this scenario are independent of each other and not influenced by previous events.

The Gambler’s fallacy is a common event in games of chance, like rolling a die or flipping a fair coin. The name Monte Carlo fallacy originates from a game of roulette in the Monte Carlo Casino in 1913, where with a degree of randomness, the ball landed on black 26 times consecutively. Based on this nature of randomness, the favourable outcome for many gamblers during the game was that there was a higher probability that it would land on red after such a long streak of black. The ideal outcome of their belief turned out to be false, which resulted in a loss of large sums of money.

Example

You roll a die three times, and every time it lands on 6.

Now you have to answer this question:

If you roll the die for a fourth time, is it…

• … more likely than in the previous events that it lands on 6?
• … less likely than in the previous events that it lands on 6?
• … as likely as in the previous events that it lands on 6?

If the ideal outcome for you is that there is a higher or lower degree of randomness, then you just experienced the Gambler’s fallacy.

The correct answer is that the chance of rolling a 6 again has the same likelihood as the previous three rolls. The probability of rolling a 6 is 1 in 6 on every roll, and the previous rolls do not influence the next one.

Rolling a die is a random event, meaning that an event of streaks does not impact the future outcome of the subsequent events and determine what number it will show. The nature of randomness stays the same, meaning the current roll will also not impact the chance process of rolling a certain number in the future.

The erroneous belief that a current outcome was the result of a previous event or will influence the next one is false as long as it regards independent events.

Note: To understand the Gambler’s fallacy, you need to know the meaning of independent and random events in relation to this fallacy:

• Independent event
An event that is not influenced by any former or future events. All outcomes of an independent event are unrelated.
• Random event
A random outcome lacks any predictable pattern or order, meaning every outcome has an equal probability of happening.

## Psychology behind the Gambler’s Fallacy

The false assumption which originates in our cognitive thinking can be subdivided into three main aspects, which make us believe in the Gambler’s Fallacy.

#### Search for Order and Meaning

The psychological reason behind the Gambler’s fallacy lies in the human tendency to perceive patterns and, as a result, expect randomness to align with a pattern over time. Our minds naturally seek order and meaning in random events, leading us to assume that if a certain outcome has not occurred for a while, it becomes more likely to happen in the near future.

#### Law of Small Numbers

The Gambler’s fallacy can be attributed to a cognitive bias called the “law of small numbers.” This bias causes individuals to assume that small samples will reflect the same distribution as a larger population. In gambling, this manifests as a mistaken belief that a short streak of losses or wins will somehow correct itself and align with the expected probabilities.

#### Illusory Correlation

The Gambler’s fallacy can be influenced by the concept of “illusory correlation.” This refers to the tendency to perceive a relationship between random events that are actually unrelated. When experiencing a history of events of loss, gamblers may mistakenly associate certain external factors or personal behaviours with their losses, leading them to believe that changing those factors will increase their chances of winning.

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## Examples

The following examples will outline common events with the Gambler’s fallacy in motion, providing further explanations in the context of real-life situations.

#### Psychology of coin toss

Example 1

A series of coin tosses shows ten consecutive successive heads. For the eleventh toss, you bet on tails because you think this result is overdue.

Wrong. The probability of successive heads or tails is always 50% or 1 in 2.

#### Psychology of Roulette

Example 2

You are part of a roulette game and realize that the last five results were all “black” and place a huge amount of money for the next round on “red”. You think this outcome is likely to occur after the ball landed on “black” for the last five rounds.

Your assumption is wrong. The probability for the result “red” is still 47.4% or 18 in 38.

#### Psychology of Lottery Gambling

Example 3

Person A has already won three times in a weekly lottery, in which one number out of 50 gets drawn and declared as the winner. In this lottery, everybody is only allowed to purchase one ticket with one specific number. Person B has never won this lottery. Person B thinks that she is more likely to win next time than Person A because he has already won three times.

The assumption is wrong. The probability of winning is still 2% or 1 in 50.

## Wrong use of the Gambler’s fallacy

You always need to keep in mind that the Gambler’s fallacy can only be applied to random, independent events. If the result isn’t random or independent, the Gambler’s fallacy can’t occur. There are basically two possibilities that deny the Gambler’s fallacy:

• The event isn’t independent.
This case occurs when the result of the current event is influenced by a previous or future event.
• The event isn’t random.
This case occurs when the result is predictable and the probability of different outcomes varies.

Example

You draw a card from a full card deck. For the draw, only the symbol of the card is relevant, not the value. The deck consists of the following distribution sequence:

 Clubs Spades Hearts Diamonds 13 13 13 13 25 % 25 % 25 % 25 %

You draw one card from the deck, and it’s a heart. Since you don’t put this card back into the deck, the representative sequence of probability for the next draw is as follows.

 Clubs Spades Hearts Diamonds 13 13 12 13 25.5 % 25.5 % 23.5 % 25.5 %

This shows that without putting the drawn card back in the deck, every draw is a dependent event, which is influenced by the previous draw. Therefore, it can be stated that the probabilities of drawing the symbols are not equal in the second draw, which makes it a not random event.

## Types of Gambler’s fallacy

The classic Gambler’s fallacy is the most widely recognized and strongest Gambler’s fallacy form, which can be subcategorized into various nuances and related misconceptions in terms of probability. The following will elaborate on the various types of Gambler’s fallacy.

### Inverse Gambler’s fallacy

The inverse Gambler’s fallacy refers to the law of small numbers in a psychological aspect. It describes the irrational belief that a rare event must have had many prior attempts for it to happen without evident or supportive observation. In other words, if someone attributes a larger sample size to the occurrence of a rare event, the inverse gambler’s fallacy occurs.

Example

Someone has, at first glance, observed the rare event of the ball landing on “32” in a roulette game. Based on this, they assume that the wheel has been spinning over and over again until “32” came up, disregarding the possibility that it was the first spin of the day.

### Retrospective Gambler’s fallacy

This fallacy refers to a misunderstanding of the nature of randomness, where someone reflects on past permutations of events and, based on this, concludes that they have produced a favorable outcome in the present. In other words, the mistaken belief that a current outcome is a product of previous events.

Example

Someone observes several successive heads on a coin flip and concludes that there is a high probability that the previous flips were tails.

### Reverse Gambler’s fallacy

In contrast to the inverse Gambler’s fallacy, the reverse Gambler’s fallacy refers to the law of large numbers. It is the fallacious belief that when a sequence of trials carries repetitive outcomes of past events, it is more probable for a sequence of similar nature to happen in the future.

Example

A gambler has a consistent tendency to flip tails and concludes from this that he has a higher probability of success of flipping tails continuously. With this mistaken belief, he doesn’t think about betting on heads next time, although the events are independent of each other.

## How to avoid the Gambler’s fallacy

There is no real way of avoiding the Gambler’s fallacy, as there are no instructions about randomness. The most effective strategy is to be aware of the actual process. If you have a closer look at the gambling conditions, you can change your role of experience and learn to identify whether an event is random or predictable.

Example

If you are gambling in a roulette game and the result is black three times in a row and your mind tells you that it’s more likely to be red next time, be aware of the Gambler’s fallacy.

If you remember its existence and think about it, you will realize that every round is independent and random.

The key is to analyze every situation on its own to avoid the Gambler’s fallacy. However, overcoming the psychological aspect behind this fallacy can be a difficult task, as you have to actively manipulate and train your instincts and mind.

## FAQs

#### What is the Gambler's fallacy?

The Gambler’s Fallacy is a misunderstanding of dependence and the randomness of a streak of events in gambling. After a row of the same result, people think it’s more likely for another result to appear next, but this is a wrong belief.

#### Why does the Gambler's fallacy exist?

It’s a psychological reason because the human mind seeks for patterns and expects randomness to even out. Together with the belief that a small number will be representative for a whole, it is the psychological foundation for the fallacy.

#### What is an example for the Gambler's fallacy?

An example of the appearance of the Gambler’s Fallacy is roulette. Every round is independent and random and if the result is black for 5 rounds in a row, it’s still as likely as before that the sixth result is black.

#### How do you avoid the Gambler’s fallacy?

The key to avoiding the Gambler’s fallacy is to be fully aware of it. However, there are tools such as seeking independent research, keeping a diary, or asking for external feedback.

#### What is the Hot-hands fallacy?

The Hot-hands fallacy or hot-hand belief refers to a type of experience of success. It is a type of reverse gambler’s fallacy and involves the mistaken belief that a successful streak is likely to continue. For example, if a hockey player has made two consecutive goals, he has a higher probability of making a third goal.

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