The gambler's fallacy bias, in its simplest form, is a misconception that an occurrence is 'overdue' simply because it hasn't happened for a long time.
Monte Carlo fallacy
The bias, which is sometimes referred to as the Monte Carlo fallacy, originated in Monte Carlo's Le Grande Casino over one hundred years ago, in August 1913. At the roulette wheel, records state that the ball landed on black 29 times in a row. The incident has since become immortalised as the gambler's fallacy, simply because of the vast amounts of money lost on red. Once black had come up a 10th time, patrons at the casino began waging more and more on red, making significantly higher bets based on the false assumption that the ball couldn't possibly land on black again.
That assumption is false, because the wheel doesn't have a memory. Every time there's a new spin, the odds of landing on black are exactly the same - assuming the roulette wheel is of the single zero variety, the chances of black coming up is 18 in 37 every time. By the time the casino closed its doors for the night, the owners of the casino were said to be more than 10 million francs better off.
Probability and chance
A lot of gamblers and traders alike fall into the trap of mistaking probability and chance. The reality is that they are very different. If you take the toss of a coin as an example, the chances of the coin landing on heads or tails is 50%. The chance of either outcome never changes, so it makes no difference whether it's the first toss or the 10,000th. This is down to the fact that the odds are entirely defined by the chance ratio of one outcome against the other. In this case, heads has one chance and tails has one chance.
Probability is defined differently, and it makes it seem to some as if the probability of repeated outcomes diminishes over time. The chance of tails on the first toss is 1/2, as is the probability. When it comes to the following tosses, the chances are multiplied progressively in order to shape the probability. That means the probability of getting tails twice in two tosses is 1/4, three heads in a row is 1/8, and five heads in a row has a probability of 1/32. The result of this is that a lot of people erroneously calculate that, after four successive outcomes, the probability of a fifth is one in 32.
The fallacy exposed
By confusing the theoretical probability with chance, the gambler mistakenly believes that a fifth heads outcome is highly unlikely to happen. The fact is that when it comes to chance, probability is only in operation before the first outcome. After the fourth flip is completed, all the previous outcomes (four heads), merge into one single outcome, this is known as a 'unitary quantity', which we can think of as one. That means we can apply the rule of one, so all we have to do is subtract the odds of 50% on the next toss from one and we have the true probability. Put simply, it resets to 1/2 one more time, and will continue to do so for every toss.
So, any reasoning that a certain outcome is more likely based on past outcomes, and that what has gone before can influence the odds on what will happen next is entirely false, and is therefore, a fallacy.