Cognitive aspects of gambler’s fallacy

Gambler’s fallacy refers to the mistaken belief that future toss of a coin or drop of the ball in roulette or roll of the dice in craps is dependent of past events. For example, if you toss a coin and head comes up three times in a row, you may bet against a head coming up for the fourth toss. Similarly, if you see seven rolled on the dice five times in a row, then you are likely to bet against another seven coming up again on the next roll.

People expect events to even out in the short run because they know that events even out in the long run. If the coin is tossed ten thousand times, then, about fifty percent of falls would have been with head up. This is due to the result of even out of events in the long run. People, using their commonsense, think that this even out is applicable in the short run of events also.  This is an informal fallacy. It is also known as the law of averages in the common parlance.

If a coin is tossed repeatedly and tails comes up a larger number of times than is expected, a gambler may incorrectly believe that this means that heads is more likely in future tosses. Such an expectation could be mistakenly referred to as being due, and it probably arises from everyday experiences with scheduled and non-random events such are arrivals of trains, flights etc. For example, when a scheduled train is late, it can be expected that it has a greater chance of arriving the later it gets. In the case of random events such as tossing of coins what is true instead is the law of large numbers. In the long term, averages of independent trials will tend to approach the expected value, even though individual trials are independent.

Child birth and gambler’s fallacy

As in the case of all other sexual species, the sex ratio of humans is approximately 1:1. Naturally, childbirth is a random event as far as the gender of the child is concerned.  A very real-world example of gambler’s fallacy is that mothers and couples trying for another child tend to think that if they have had several children of the same sex previously, it is more likely of finally having a child of the opposite sex. This is similar to what people tend to think of with Henry VIII of England trying so desperately for a son. It is almost always a 50% chance of either sex, despite what parents may hope for their next child.

Monte Carlo Casino

The most famous example happened in a Monte Carlo Casino in Monaco in the summer of 1913, when the ball fell in black 26 times in a row, an extremely uncommon occurrence and gamblers lost millions of francs betting against black black after the black streak happened. Gamblers reasoned incorrectly that the streak was causing an "imbalance" in the randomness of the wheel, and that it had to be followed by a long streak of red.

Psychology behind the gambler’s fallacy

Daniel Kahneman

Cognitive psychologist Amos Tversky and Nobel laureate Daniel Kahneman proposed that the gambler's fallacy is a cognitive bias produced by a psychological heuristic. 

Amos Tversky 

In psychology, heuristics are simple, efficient rules, hard-coded by evolutionary processes or learned, which have been proposed to explain how people make decisions, come to judgments, and solve problems, typically when facing complex problems or incomplete information. These rules work well under most circumstances, but in certain cases lead to systematic errors or cognitive biases.
A cognitive bias is a pattern of deviation in judgment that occurs in particular situations, leading to perceptual distortion, inaccurate judgment, illogical interpretation, or what is broadly called irrational thinking. Implicit in the concept of a "pattern of deviation" is a standard of comparison with what is normatively expected; this may be the judgment of people outside those particular situations, or may be a set of independently verifiable facts. A long and ever-growing list of cognitive biases has been identified over the last six decades of research on human judgment and decision-making in cognitive science, social psychology, and behavioral economics.

According to this view, after observing a long run of red on the roulette wheel most people erroneously believe that black will result in a more representative sequence than the occurrence of an additional red, so people expect that a short run of random outcomes should share properties of a longer run, specifically in that deviations from average should balance out. When people are asked to make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to tails stays closer to 0.5 in any short segment than would be predicted by chance. Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be representative of longer ones.

Hot-hand fallacy

Another psychological perspective states that gambler's fallacy can be seen as the counterpart to basketball's Hot-hand fallacy. In the hot-hand fallacy, people tend to predict the same outcome of the last event - that a high scorer will continue to score. This is positive recency. In gambler's fallacy, however, people predict the opposite outcome of the last event - that, for example, since the roulette wheel has landed on black the last six times; it is due to land on red the next. This is negative recency. Peter Ayton, City University, London and Ilan Fischer, University of Haifa, Israel have theorized that people display positive recency for the hot-hand fallacy because the fallacy deals with human performance and that people do not believe that an inanimate object can become "hot" like human hands.