Gambler’s Fallacy

Also known as the Monte Carlo Fallacy, or the Fallacy of the Maturity of Chances, the Gambler’s Fallacy is the misleading notion that if one event occurs repeatedly then another event becomes more likely to occur for each time that it does not occur. For example, if you throw heads on a coin four times in a row, you might feel that the odds of throwing tails becoming increasingly favourable for each subsequent throw, but this, mathematically, is a false belief, since throws of a coin are independent events and thus the probability of throwing heads or tails remains 1 in 2 regardless of the outcome of any previous throws.

In other words, if one throws four heads in a row one may be tempted to think that the odds of throwing a fifth heads (0.5 x 0.5 x 0.5 x 0.5 x 0.5 = 0.03125 = 1 in 32) are significantly less than the odds of throwing a tails, but that is to assume that there exists a causal/logical relationship between throws. The 1 in 32 chance is only theoretically valid before the first coin is thrown. For each new throw, the odds remain 1 in 2. Similarly, the probability of throwing 21 consecutive heads may indeed be 1 in 2 097 152, but the probability of throwing heads on the 21st throw, even if the previous 20 throws all landed on heads, is still 0.5 i.e. 1 in 2.

Gambler’s Fallacy when playing the lottery

The same kind of thinking described above – where players believe that an entirely random set of short-run results determine the chances of subsequent results – is also very commonly applied when playing the lottery. For example, many lottery players stick to the same set of numbers in the belief that the chances of this combination of numbers coming up increases for each time that the combination does not come up. As lottery draws are independent events, each lottery draw has no relation to any previous or future lottery draws. Therefore, the probability of a certain combination coming up remains exactly the same for each draw. A better strategy for playing the lottery is to avoid choosing combinations that could be commonly chosen by other players (sequences, patterns, etc), so as to prevent your winnings from being significantly divided if such a combination were to be drawn.

Some Gambler’s Fallacy humour

A popular joke among mathematicians that points out the inherent flaw of the Gambler’s Fallacy goes like this:

When flying on an aircraft, a man decides always to bring a bomb with him. “The chances of an aircraft having a bomb on it are very small,” he reasons, “and certainly the chances of having two are almost none!”

Read our Gambling Mathematics glossary for an overview of common concepts relating to the mathematics of gambling and casino games.