The Gambler’s Fallacy: odds are not in your favor

Play interests me very much,” said Hermann: “but I am not in the position to sacrifice the necessary in the hope of winning the superfluous.

Aleksandr Sergeevich Pushkin

Photo by Pixabay from Pexels

Oh my gambler friends, how many times have you expected a certain outcome in a game of chance based on the previous results? Playing heads or tails, if the coin toss result in 10 straight heads, one usually expects that the next toss MUST be tails. But no, the reality is way crueler than that.

Also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, the Gambler’s fallacy refers to the situation in which an individual believes that an outcome which has occured many times in the previous attempts will have less chances of occurring again and vice versa. Plenty of examples can be given, most of which are independent and identically distributed Bernoulli trials:

When the roulette results in one color for several times in succession, it is expected for the next attempts to result in the other color
In a card game, for instance poker, a player wins several consequent hands, it is expected that she will not again get a good hand in the next round
The price of a stock is going down for several days, an investor may believe that this cannot continue and it will go up in the next trading day
After a lottery number wins, people will be less likely to bet on it again

The term was coined on August 18, 1913 at Le Grande Casino in Monte Carlo, Monaco. In a game of roulette, the color black turned up for a surprising amount of 26 times. After 10 occurrences of black, the gamblers in the casino started betting large amounts of money on the color red, as a result of their almost-sure expectancy that there should be an immediate end to the sequence of the color black. According to “The Universal Book of Mathematics: From Abracadabra to Zeno’s paradoxes”, the probability for this to happen was 1 in 136,823,184. Imagine the profit of Le Grande Casino that night.

Another interesting instance of the Gambler’s Fallacy can be found in A Philosophical Essay on Probabilities by Laplace:

“I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls ought to be the same at the end of each month, they judged that the boys already born would render more probable the births next of girls.”

Therefore, the Gambler’s fallacy leads people to make decisions erroneously. Even though the probability for 10 consequent heads in a heads or tails game is really low, but even when it happens, it has no effect on the 11-th coin toss. This is simply because these are independent events, and past occurrences have no impact on present or future outcomes. In other words, the probability of any of the outcomes to happen is independent from one trial to the next.