Suppose you toss a coin and it comes up heads five times in a row. Which side would you want to bet on for the sixth toss? Many people feel that ‘tails is due now.’ Since heads has come up so often, it seems as if tails must come up to even things out.
But a coin does not remember what happened before. On the sixth toss, too, the probability of heads or tails is the same, one half. This mistaken belief that past results change the next result in events that do not affect one another is called the ‘gambler's fallacy.’
In this article, we will look in turn at what the gambler's fallacy is, why we fall into this illusion, what forms it takes in everyday life, and how to avoid it.
Understanding the Gambler's Fallacy
The Illusion That Past Results Change What Comes Next
To understand the gambler's fallacy, you first need to know the concept of ‘independent events.’ Independent events are cases in which the outcome of one event has no effect at all on the probability of another. Tossing a coin, rolling a die, and spinning a roulette wheel are typical examples. Even if you roll a die ten times without getting a single 6, the probability of rolling a 6 next is still one in six.
Of course, the probability of tossing a coin ten times and getting heads all ten times is very low, 1 in 1,024. But that is a calculation made before anything has been tossed. If heads has already come up nine times in a row, the probability of heads on the remaining toss is simply one in two. Moreover, ten heads has exactly the same probability, 1 in 1,024, as any other specific sequence that mixes heads and tails. It only looks special to us; in terms of probability, it is not special at all.
This fallacy is also called the ‘Monte Carlo fallacy.’ The name comes from the story that on August 18, 1913, at the Monte Carlo Casino in Monaco, the roulette ball landed on black 26 times in a row, and people bet large sums that red would come up this time and lost. However, since no primary records from the time have been confirmed, it is unclear whether it really happened, and it is widely passed on as a famous example for explaining probability.
Why Do We Fall Into This Illusion?
Several psychological causes overlap in the gambler's fallacy.
(1) The law of small numbers
In 1971, the psychologists Amos Tversky and Daniel Kahneman showed that people tend to believe the overall proportion should appear even in the results of a small number of trials. The law of large numbers means that the average approaches the expected value only after a great many trials, but people try to apply it to just a few.
(2) A mistaken picture of randomness
People think that if something is random, heads and tails should come out evenly mixed. So when the same result keeps repeating, they feel something has gone wrong. In real random results, however, streaks appear more often than we think.
(3) The wish for balance
The wish to believe that you can win back all your losses on the next try also feeds this illusion. Such expectations become even stronger when money is at stake.
The Gambler's Fallacy in Everyday Life
The gambler's fallacy often appears outside the casino as well.
- Lottery: Some people choose numbers that have not been drawn for a long time, believing they will come up soon. But in every draw, all numbers have the same probability.
- Sex of a child: Parents with three daughters may expect that this time a son is more likely, but the sex of earlier children does not affect the next child.
- Investment: Judging that a stock is due to rise because it has fallen for several days in a row can also stem from a similar illusion.
There is also an illusion in the opposite direction: the ‘hot hand’ phenomenon, in which people believe that a basketball player who has made several shots in a row will make the next one too. Whereas the gambler's fallacy holds that ‘it's time for a change,’ the hot hand holds that ‘the streak will continue,’ so the two are opposites.
How to Avoid the Illusion
The first step in avoiding the gambler's fallacy is to consider whether the events you are dealing with are truly independent of each other. If they are independent, like coins or dice, you should forget past results and think of each try as having the same probability.
On the other hand, not everything is independent. In card games, the cards already dealt change the makeup of the remaining cards, so the next probability changes too. There are also cases, such as the weather or the momentum of a game, in which earlier conditions really do affect later ones. So what matters is not to ignore results unconditionally, but to check whether there really is a connection between them.
It also helps to make a habit of not drawing conclusions from just a few results and instead gathering enough data before judging. Be especially careful about raising your stakes to make up for losses. The surest defense is to decide in advance how much you can spend and when to stop, and to stick to those limits whatever the results.
The gambler's fallacy arises from the natural human urge to find order even in randomness. But probability does not move to match our expectations. Knowing this lets us judge much more calmly, whether in lotteries, investments, or small everyday choices.
Remembering that the coin does not remember: that is the first step toward making wise judgments in the face of chance.
