The Gambler’s Fallacy

This post first appeared on Lernabit.com.

The gambler’s fallacy is a fallacy in which someone incorrectly concludes that if a random event has occurred more frequently than expected in the near past, then it will occur less frequently than expected in the near future. Or, if a random event occurred less frequently in the near past, then it will occur more frequently in the near future.

An example of this fallacy can be seen by flipping a coin. Imagine you flip a coin many times, and it lands on heads 10 times in a row. Because there is a 50/50 chance of landing on heads or tails, you might conclude that the next flip is more likely to be tails. In other words, because it landed on heads 10 times in a row, you might believe it is “due” to land on tails. But this is actually wrong. Regardless of how many times the coin landed on heads, the odds of the next flip being tails is still only 50%, assuming a fair coin. Over the long term, it is safe to assume that the law of large numbers will hold up, and overall, the results will be around 50/50. But this cannot be used to predict the result of the next individual coin flip.

The gambler’s fallacy can be better understood by looking at the origins of its name. The gambler’s fallacy is also known as the Monte Carlo fallacy, after an unusual event in a casino involving the fallacy.

On August 18, 1913 at the Monte Carlo Casino, a roulette ball landed on black 26 times in a row. The odds of this happening are about 1 in 67 million. A lot of the players who witnessed this unusual occurrence saw the ball repeatedly landing on black, and concluded that the next ball must land on red. But the streak continued, with the players losing millions of Francs betting against black. This is a classic example of the gambler’s fallacy, because even if the roulette wheel landed on black 25 times in a row, the odds of the 26th toss landing on red are still only 50%.

In the case of flipping a coin or rolling a roulette ball, the events are said to be independent of one another. In other words, the outcome of one toss does not influence the outcome of the next. It is important to keep that in mind, because the gambler’s fallacy does not apply if the events are not independent. If the outcome of each event does in fact influence the next event, the gambler’s fallacy doesn’t hold.

For example, in a card game in which cards are removed from the deck without being replaced, the result of each card draw influences the odds of the next draw. The gambler’s fallacy does not apply here, because the outcome of each draw actually does influence the probability of the next draw. The gambler’s fallacy only applies when each round is independent of those before and after it, such as with a coin toss.

It is worth emphasizing again that all of these examples assume fair odds. If someone is cheating, the laws of statistics are less applicable, and the gambler’s fallacy doesn’t really apply.