The Legendary Series in Monte Carlo and the Birth of the “Gambler's Fallacy”
In the world of gambling and elsewhere, we often encounter situations where our intuition conflicts with the laws of probability. One striking example of such a contradiction is the so-called “gambler's fallacy” – a cognitive distortion that makes us believe that random events are somehow connected to each other, although in fact they are not.
The famous case in Monte Carlo
The history of the “gambler's fallacy” is inextricably linked to a famous incident that took place at the Monte Carlo casino on August 18, 1913. That evening, an incredible event occurred at one of the roulette tables – the color black came up 26 times in a row.
Picture this scene: a crowded casino, a tense atmosphere at the roulette table. After black has come up 10 times in a row, a real frenzy begins among the players. Everyone suddenly decides that now red must definitely come up, and they start betting on this color en masse. But the bad streak continues – 11, 12, 13 times in a row… With each new black coming up, the bets on red become bigger and more desperate.
This amazing series eventually reached 26 blacks in a row. The probability of such an event is extremely low – about 1 in 136.8 million. However, the players, having succumbed to the “gambler's fallacy”, continued to believe that red was about to appear. As a result, the casino was able to earn several million francs that night.
What is the essence of the error?
The gambler's fallacy, also known as the “Monte Carlo fallacy” or “mature odds fallacy,” is the false belief that if some random event occurs more or less often than expected, then it is more likely to occur less or more often in the future, respectively.
In fact, for truly random and independent events, such as the number that comes up in roulette, the probability of each individual outcome remains the same, regardless of previous results. In the case of roulette, the odds of black coming up on any given roll are always 18/37 (in European roulette with a single zero).
Mathematical explanation
Let's look at this using the example of flipping a coin. The probability of getting heads or tails on each flip is 1/2. Let's say we flipped the coin 5 times and each time it came up heads. What is the probability that the sixth time it came up tails?
Many people, succumbing to the gambler's fallacy, will say that the probability of getting tails is now higher. But this is not true. The probability of getting tails the sixth time is still 1/2.
Here's why:
Probability of getting 5 heads in a row: (1/2)^5 = 1/32
The probability of getting 5 heads in a row and then tails is: (1/2)^5 * (1/2) = 1/64
Conditional probability of getting tails after 5 heads: P(tails after 5 heads) = P(5 heads and then tails) / P(5 heads) = (1/64) / (1/32) = 1/2
Thus, despite the previous series, the probability of the next throw does not change.
Psychological roots of delusion
Why are we so prone to this error? Researchers believe that the roots of this misconception lie deep in our psychology and even evolution.
According to theories of Amos Tversky and Daniel Kahnemanthe gambler's fallacy is related to the so-called “representativeness heuristic” – a cognitive bias in which we judge the likelihood of an event by how similar it is to our typical pattern or representation.
Hidden text
Tversky and Kahneman conducted a study in which participants were given the following task:
An accident involving a taxi occurred at night. There are two taxi companies in the city: “Green” (85% of the fleet) and “Blue” (15% of the fleet). A witness claims that the taxi was blue. A reliability check of the witness showed that in such conditions he correctly identifies the color in 80% of cases and is wrong in 20%.
Question: What is the probability that the taxi involved in the accident was actually blue, given the witness's testimony?
Most participants estimated the probability to be above 50%, some even above 80%. However, the correct answer obtained using Bayes' theorem was significantly lower:
Probability of correct identification of blue taxi: 12% (0.15 × 0.80)
Probability of misidentifying a green taxi as a blue one: 17% (0.85 × 0.20)
Overall probability of identifying a taxi as blue: 29% (12% + 17%)
The resulting probability that a taxi identified as blue was actually blue: 41% (12% ÷ 29%)
In the case of roulette, we intuitively expect that even in a short series of throws the ratio of red to black should be approximately equal. Therefore, a long series of one color seems “unrepresentative” to us and we expect that the opposite color should come up soon to “even out” the statistics.
Evolutionary explanation
Interestingly, the tendency to make such errors is found not only in humans, but also in other primates. Experiments with monkeys have shown that they, too, exhibit behavior similar to the “gambler's fallacy,” making choices based on previous successful attempts, even when each event is independent and random.
This suggests that this behavior may have evolutionary roots. In nature, many events are indeed related to each other – for example, finding one ripe fruit on a tree makes you more likely to find others nearby. So the tendency to look for patterns and rely on recent experience may have been a useful survival strategy for much of our evolutionary history.
Gambler's fallacy in real life
Although the gambler's fallacy is most evident in gambling, its effects can be found in other areas of life as well. Let's look at a few examples:
Lotteries
Once a number is drawn in a lottery, players often avoid choosing it in subsequent draws, believing that the chances of it being drawn again have decreased. A study conducted by Charles Clotfelter and Philip Cook in 1991 showed an interesting dynamic: immediately after a number is drawn, its popularity among players drops sharply, but then gradually recovers over the course of about three months.
Let's look at a specific example from their research:
Date | Number | Number of bets |
April 11 | 244 | 41 |
April 12 | 244 | 29 |
April 13 | 244 | 28 |
April 14 | 244 | 134 |
April 15 | 244 | 10 |
We see that after the number 244 came up on April 14, the next day the number of bets on this number dropped sharply from 134 to 10.
Asylum Judges
A study of US asylum judges found that after two consecutive approvals, the chances of a third application being approved dropped by 5.5%.
Baseball Umpires
An analysis of more than 12,000 baseball games found that umpires were 1.3% less likely to call a strike if the previous two pitches were also strikes.
Credit inspectors
Research shows that loan officers who are not interested in monetary gain are 8% less likely to approve a loan if they approved it for a previous customer.
Games
Some video games use a system of “loot boxes” – virtual containers with random in-game items of varying value. Since 2018, this practice has raised concerns among authorities and activists, as it resembles gambling, especially in games for young people.
A number of games employ a “compassion mechanism”: if a player goes a long time without receiving a valuable item from a loot box, the chances of it dropping gradually increase. It is believed that this reinforces the player's delusion, creating the illusion that after a series of unsuccessful attempts, something valuable will definitely drop, as in gambling.
Conclusion
The “gambler's fallacy” is a prime example of how our intuition can fail us in a world of probability and chance. Although this cognitive bias is deeply rooted in our psychology and may even have evolutionary roots, being aware of its existence and understanding the basics of probability theory can help us make more rational decisions both in gambling and in everyday life.
Additional materials:
About loot boxes in games.
About court decisions and baseball
A counter-point based on a study of British lotteries.
All this and much more — TG “Mathematics is not for everyone”
