The Gambler’s Fallacy

by Tim Harding

The nature of the fallacy

The Gambler’s Fallacy, also known as the Monte Carlo Fallacy or the Fallacy of the Maturity of Chances, is the mistaken belief that if something happens more frequently than normal during some period, then it will happen less frequently in the future (presumably as a means of balancing nature).  This belief is also sometimes referred to as the so-called ‘Law of Averages’.

Although appealing to the human mind, this belief is false.  The fallacy can arise in many practical situations although it is most strongly associated with gambling where such errors of reasoning are common amongst players, and even more common amongst ‘problem gamblers’ (see below).

The use of the term Monte Carlo Fallacy originates from the most notorious example of this phenomenon, which occurred in a Monte Carlo Casino in on August 18, 1913.[1]  On this occasion, black came up a record twenty-six times in succession on a roulette wheel.  There was a frenzied rush to bet on red, beginning about the time black had come up a phenomenal fifteen times.  In an application of the Gambler’s Fallacy, players doubled and tripled their stakes, the fallacy leading them to believe after black came up the twentieth time that there was not a chance in a million of another repeat.  In the end, the unusual run enriched the Casino by some millions of francs.[2]


The reality is that if the roulette wheel at the Casino was fair, then the probability of the ball landing on black was a little less than one-half on any given turn of the wheel.  Also, the colours that come up are statistically independent of one another, thus no matter how many times the ball has fallen on black, the probability is still the same at every turn of the wheel.  (Remember that neither a roulette wheel nor the ball has a memory).

Almost every so-called gambling ‘system’ is based on this fallacy, or a similar error of reasoning. Any gambler who thinks that he can record the results of a roulette wheel, or lotto numbers or a gaming machine, and use this information to predict future outcomes is probably committing some form of the gambler’s fallacy. An exception is counting the cards in successive deals from an unshuffled card deck. Many casinos now counteract card counting by reshuffling the cards before each deal. Part of the reason for the prevalence of this fallacy could be derived from this former system of card counting.

Problem gambling

Many psychologists treating problem gamblers have identified false perceptions and beliefs as major contributors to problem gambling.  Erroneous beliefs also lead to uninformed decision-making by a significant number of other players.[3]  A survey of gambling attitudes among 1017 Australian young people in 1998 found that such erroneous beliefs include:

  •  the chances of winning are significantly higher than they actually are;
  •  a player’s skill or adopting ‘the right system’ can influence the outcome of a game that is purely a game of chance;
  •  a player will eventually ‘strike it lucky’;
  •  a player is more likely to win with ‘lucky numbers’, by ‘thinking positively’ or by ‘concentrating hard enough’.[4]

According to the Productivity Commission Inquiry Report, one of the most widespread misconceptions is that gaming machine payouts are dependent upon previous outcomes from the same machine (as evidenced by the frequent ‘chasing of losses’).  To counter this common misconception, the Commission quotes the following facts about gaming machines:

  •  The payout tables on gaming machines indicate the winnings that are associated with certain combinations.  They do not tell the player the probability of the combination occurring.
  •  In most jurisdictions, operators must return at least 85 per cent of turnover to players as winnings. It will usually take hundreds of thousands of games for a machine to come close to this average ‘set’ return.
  •  Each game played on a machine is independent of results from past games —machines which have not paid out for some time have no higher chance of paying out now or in the near future (and vice versa).
  •  Actual outcomes on machines are extremely volatile, with player returns and the amount of time that it takes to lose a set amount of money varying between sessions.
  •  If a gambler ‘reinvests’ the winnings, he or she will eventually lose the lot.[5]

In this way, problem gambling is a severe manifestation of preference failure, with potentially dire consequences.  Harm to problem gamblers and their families can range from excessive time-wasting and financial losses, to loss of employment and family breakdown, to bankruptcy or criminal behaviour, clinical depression and even suicide.[6]


[1] Lehrer, Jonah (2009).How We Decide. New York: Houghton Mifflin Harcourt. p.66.

[2] Darrell Huff & Irving Geis (1959) How to Take a Chance, pp. 28-29.

[3] Productivity Commission (1999) Australia’s Gambling Industries Report No. 10, AusInfo, Canberra, p.41.

[4] Moore, S.M. and Ohtsuka, K. (1999) ‘Beliefs About Control Over Gambling Among Young People, and Their Relation to Problem Gambling’. Psychology of Addictive Behaviors, December 1999, Volume 113, Number 4 APA Journals, Washington. D.C.

[5] Productivity Commission (1999) Australia’s Gambling Industries Report No. 10, AusInfo, Canberra, p.42.

[6] Tim Harding & Associates (2005)  Gambling Regulation Regulations 2005 – Regulatory Impact Statement. Department of Justice, Melbourne.

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Filed under Logical fallacies

6 responses to “The Gambler’s Fallacy

  1. Pingback: ROULETTE SPINS 6 | Society Of Mathematics

  2. Pingback: Guide to Game Theory: expected value and independent events | The Final Wager

  3. Steve

    Such a sad problem in society. I’m enjoying reading your posts and hope you don’t mind me sharing them with my friends. I feel like I am getting a bit of free education. (not much of that around these days!).


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