Taxonomy of the Gambler's Fallacy Alias:
  • The Monte Carlo Fallacy
  • The Doctrine of the Maturity of Chances

Form:

A fair gambling device has produced a "run".
Therefore, on the next trial of the device, it is less likely than chance to continue the run.

Example:

On August 18, 1913, at the casino in Monte Carlo, black came up a record twenty-six times in succession [in roulette]. … [There] was a near-panicky rush to bet on red, beginning about the time black had come up a phenomenal fifteen times. In application of the maturity [of the chances] doctrine, players doubled and tripled their stakes, this doctrine 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.

Source: Darrell Huff & Irving Geis, How to Take a Chance (1959), pp. 28-29. Taxonomy of the Gambler's Fallacy

Exposition:

The Gambler's Fallacy and its sibling, the Hot Hand Fallacy, have two distinctions that can be claimed of no other fallacies:

  1. They have built a city in the desert: Las Vegas.
  2. They are the economic mainstay of Monaco, an entire, albeit tiny, country, from which we get the alias "Monte Carlo" fallacy.

Both fallacies are based on the same mistake, namely, a failure to understand statistical independence. Two events are statistically independent when the occurrence of one has no statistical effect upon the occurrence of the other. Statistical independence is connected to the notion of randomness in the following way: what makes a sequence random is that its members are statistically independent of each other. For instance, a list of random numbers is such that one cannot predict better than chance any member of the list based upon a knowledge of the other list members.

To understand statistical independence, try the following experiment. Predict the next member of each of the two following sequences:

2, 3, 5, 7, __
1, 8, 6, 7, __

The first is the beginning of the sequence of prime numbers. The second is a random sequence gathered from the last digits of the first four numbers in a phone book. The first sequence is non-random, and predictable if one knows the way that it is generated. The second sequence is random and unpredictable—unless, of course, you look in the phone book, but that is not prediction, that is just looking at the sequence—because there is no underlying pattern to the sequence of last digits of telephone numbers in a phone book. The numbers in the second sequence are statistically independent.

Many gambling games are based upon randomly-generated, statistically independent sequences, such as the series of numbers generated by a roulette wheel, or by throws of unloaded dice. A fair coin produces a random sequence of "heads" or "tails", that is, each flip of the coin is statistically independent of all the other flips. This is what is meant by saying that the coin is "fair", namely, that it is not biased in such a way as to produce a predictable sequence.

Consider the Example: 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, since the wheel is fair, the colors 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. If it were possible to predict one color from others, then the wheel would not be a good randomizer. Remember that neither a roulette wheel nor the ball has a memory.

Every gambling "system" is based on this fallacy, or its Sibling. Any gambler who thinks that he can record the results of a roulette wheel, or the throws at a craps table, or lotto numbers, and use this information to predict future outcomes is probably committing some form of the gambler's fallacy.

Sibling Fallacy: The Hot Hand Fallacy

Source:

A. R. Lacey, Dictionary of Philosophy (Third Revised Edition), (Barnes & Noble, 1996).

Resources:

Q&A:

Q: Isn't there a sort-of fallacy embedded in the form of the gambler's fallacy, which is that it depends on the gambling device being "fair", something which (I submit) is technically unknowable and at best inferred through a series of trials? Put another way, while it is a fallacy to expect something other than a chance outcome for the next trial on a fair gambling device, I don't think it's a fallacy to expect a different-from-chance outcome on an unfair device. What if the actual reasoning that occurs "in the wild" and at Vegas is something like this?

P: A series of events has occurred which would be highly-improbable if the device is fair.
C: The device is (probably) not fair.

―David A. Ventimiglia

A: You're right that we test randomizers for fairness primarily through trials, but that's not the only consideration. A die will be a good randomizer if it's a cube and evenly weighted, which can be tested by direct measurement. However, it isn't necessary that it be perfect. No physical randomizer will ever be perfectly fair; for instance, no die will ever be perfectly symmetrical or absolutely evenly weighted. As with other mathematical laws―as well as scientific laws in general, and even the laws of logic―the laws of probability apply to the physical world only approximately. But if a randomizer approximates fairness closely enough, then the gambler's fallacy is, indeed, fallacious. So, to be perfectly precise, the form of the fallacy should be expressed in the following way:

An approximately fair gambling device has produced a "run".
Therefore, on the next trial of the device, it is significantly less likely than chance to continue the run.

There was a famous "man who broke the bank at Monte Carlo" who studied the roulette wheels in the casino and discovered that one seemed to favor certain numbers. By betting those numbers, the man was able to amass a small fortune. When the casino realized what he was doing, it switched the roulette wheels around at night while the casino was closed, and the man began to lose heavily. Nowadays, you can bet that casinos around the world have learned the lesson, which is why there has been no "man who broke the bank" since the 19th century.

However, consider how the man who broke the bank reasoned: Certain numbers on the wheel had come up more often than expected. Therefore, the man bet on those numbers! According to the Gambler's Fallacy, he should have bet against them because it was the other numbers on the wheel that were overdue. Rather than the Gambler's Fallacy, his reasoning actually took the form of its sibling, the Hot Hand Fallacy; that is, he bet that the run would continue.

In cases such as the Monte Carlo one, it may be reasonable to conclude that a gambling device is not fair based upon a highly improbable series of results. In those rare cases, the Hot Hand form of argument is not fallacious. The Gambler's Fallacy, in contrast, always is.

Source: Barbara Mikkelson, "The Full Monte", Snopes, 8/10/2007 fallacist@fallacyfiles.org

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