The Gambler’s Fallacy Explained
The easiest and most direct explanation of the Gambler’s Fallacy is to call this notion and phenomenon as misconception.
In which way, you might ask?
The very core of this intriguing principle holds a logically incorrect belief that a sequence of past outcomes will affect the probability of future results and that a coin – to put it as simple as possible – is bound to come up heads after a series of tails.
The Gambler’s Fallacy is the mistaken belief appealing to the human mind which occurs in many practical situations that go far beyond a simple game of coin toss, but it is most strongly connected to gambling and betting where such misconceptions most usually happen.
The Monte Carlo Fallacy
The Gambler’s Fallacy often goes under a different name which is related to an episode which occurred in August 1913, in Le Grande Casino, Monaco. A spectacular roulette round yielded the colour black the incredible 29 times in a row, a probability which was later calculated to be of 1 in 136,823,124.
What marked this number so iconic with the subject of the gambler’s fallacy is the amount of money that was eventually lost, due to the players’ misconstrued belief that the laws of probability will result in a wheel coming up in red after a sequence of blacks.
Failing to realise the random nature of sequenced outcomes on a roulette wheel, the players started placing significantly higher bets on red after the tenth black, falsely assuming that the balance of nature needed to be restored.
What Gambler’s Fallacy teaches us, in fact, is that roulette wheel has no memory and that chances of it coming up with red and/or black remained the same entire time 50% - 50%. Under the law of large numbers, the proportion of black vs red will approach the 1/2 odds.
In order to mathematically support the claim of odds growing even with an increasing number of tries, let us illustrate it through a simple game of coin toss.
Coin Toss and Probability vs Chance
The main reason behind our brain’s inability to fight the phenomenon of the Gambler’s Fallacy is a wrong association of probability with chance, which are in fact two different things.
Simply put, the odds represent the ratio of chances a coin toss has on falling on heads against chances of tails. The both results are a chance each, whereas the probability is further on determined as the ratio of ‘chances for’ and ‘total chances’ which can be explained through a definition:
P(x) = (Chances for) / (Total chances)
With the rise of total chances, the probability to get repeated results will diminish and decrease. Rising number of tosses in this case and tries in any other situation will bring the distribution of results closer together, ultimately levelling the odds for each respective result.
In contrast to the Gambler’s Fallacy a concept called Reverse Fallacy also exists.
It is basically a contrast to standard fallacy, which leads player into thinking that, after a consistent tails, the same outcome must be the best result to back out of a mystical perception that fate has done its share to allow consistent runs on the same side of a coin.