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Gambler’s Fallacy – in a nutshell.

April 16, 2019

The Gambler’s Fallacy, also known as the Monte Carlo Fallacy, is the proposition that people, instead of accepting an actual independence of successive outcomes, are influenced in their perceptions of the next possible outcome by the results of the preceding sequence of outcomes – e.g. throws of a die, spins of a wheel. Put another way, the fallacy is the mistaken belief that the probability of an event is decreased when the event has occurred recently, even though the probability of the event is objectively known to be independent across trials.

This can be illustrated by considering the repeated toss of a fair coin. The outcomes of each coin toss are in fact independent of each other, and the probability of getting heads on a single toss is 1/2. The probability of getting two heads in two tosses is 1/4, of three heads in three tosses is 1/8, and of four heads in a row is 1/16. Since the probability of a run of five successive heads is 1/32, the fallacy is to believe that the next toss would be more likely to come up tails rather than heads again. In fact, “5 heads in a row” and “4 heads, then tails” both have a probability of 1/32. Since the first four tosses turn u heads, the probability that the next toss is a head is 1/2, and similarly for tails.

While a run of five heads in a row has a probability of 1/32, this applies only before the first coin is tossed. After the first four tosses, the next coin toss has a probability of 1/2 Heads and 1/2 Tails.

The so-called Inverse Gambler’s Fallacy is where someone entering a room sees an individual rolling a double six with a pair of fair dice and concludes (with flawed logic) that the person must have been rolling the dice for some time, as it is unlikely that they would roll a double six on a first or early attempt.

The existence of a ‘gambler’s fallacy’ can be traced to laboratory studies and lottery-type games (Clotfelter and Cook, 1993; Terrell, 1994). Clotfelter and Cook found (in a study of a Maryland numbers game) a significant fall in the amount of money wagered on winning numbers in the days following the win, an effect which did not disappear entirely until after about sixty days. This particular game was, however, characterized by a fixed-odds payout to a unit bet, and so the gambler’s fallacy had no effect on expected returns. In pari-mutuel games, on the other hand, the return to a winning number is linked to the amount of money bet on that number, and so the operation of a systematic bias against certain numbers will tend to increase the expected return on those numbers.

Terrell (1994) investigated one such pari-mutuel system, the New Jersey State Lottery. In a sample of 1,785 drawings from 1988 to 1993, he constructed a subsample of 97 winners which repeated as a winner within the 60 day cut-off point suggested by Clotfelter and Cook. He found that these numbers had a higher payout than when they previously won on 80 of the 97 occasions. To determine the relationship, he regressed the payout to winning numbers on the number of days since the last win by that number. The expected payout increased by 28% one day after winning, and decreased from this level by c. 0.5% each day after the number won, returning to its original level 60 days later. The size of the gambler’s fallacy, while significant, was less than that found by Clotfelter and Cook in their fixed-odds numbers game.

It is as if irrational behaviour exists, but reduces as the cost of the anomalous behaviour increases.

An opposite effect is where people tend to predict the same outcome as the previous event, resulting in a belief that there are streaks in performance. This is known as the ‘hot hand effect’, and normally applies in the context of human performance, as in basketball shots, whereas the Gambler’s Fallacy is applied to inanimate games such as coin tosses or spins of a roulette wheel. This is because human performance may not be perceived as random in the same way as, say, a coin flip. 

Exercise

Distinguish between the Gambler’s Fallacy, the Inverse Gambler’s Fallacy and the Hot Hand Effect. Can these three phenomena be logically reconciled?

References and Links

Gambler’s Fallacy. Wikipedia. https://en.wikipedia.org/wiki/Gambler%27s_fallacy

Gambler’s Fallacy. Logically Fallacious. https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/98/Gambler-s-Fallacy

Gambler’s Fallacy. RationalWiki. https://rationalwiki.org/wiki/Gambler%27s_fallacy

Inverse Gambler’s Fallacy. Wikipedia. https://en.wikipedia.org/wiki/Inverse_gambler%27s_fallacy

Inverse Gambler’s Fallacy. RationalWiki. https://rationalwiki.org/wiki/Gambler%27s_fallacy

Hot Hand. Wikipedia. https://en.wikipedia.org/wiki/Hot_hand

Clotfelter, C.T. and Cook, P.J. (1993). Notes: The “Gambler’s Fallacy” in Lottery Play, Management Science, 39.12,i-1553. https://pubsonline.informs.org/doi/abs/10.1287/mnsc.39.12.1521

https://www.nber.org/papers/w3769.pdf

Terrell, D. (1994). A Test of the Gambler’s Fallacy: Evidence from Pari-Mutuel Games. Journal of Risk and Uncertainty. 8,3, 309-317. https://link.springer.com/article/10.1007/BF01064047

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