# The Gambler's Fallacy

**Taxonomy:** Logical Fallacy > Formal Fallacy > Probabilistic Fallacy > The Gambler's Fallacy

**Sibling Fallacy:** The Hot Hand Fallacy

**Alias:**

- The Maturity of the Chances
^{1} - The Monte Carlo Fallacy
^{2}

### Form:

A fair gambling device has produced a "run"―that is, a series of similar results, such as a series of heads produced by flipping a coin.

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.^{3}

### Exposition:

The Gambler's Fallacy is based on a failure to understand statistical independence, that is, two events are statistically independent when the occurrence of one has no statistical effect upon the occurrence of the other^{4}. 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, 11, __

4, 6, 6, 1, 9, __

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 five 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 of each other.

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, the Hot Hand Fallacy. 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.

### Resources:

- Colin Bruce, "The Case of the Gambling Nobleman", in Conned Again, Watson! Cautionary Tales of Logic, Math, and Probability (Perseus, 2002). This is a Sherlock Holmes short story which explains clearly and entertainingly why the Gambler's Fallacy is fallacious.
- Robert Todd Carroll, "The Gambler's Fallacy", Skeptic's Dictionary.

### Exposure:

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

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

### Where's the harm?:

From a tax court decision:

"In 1991, [the gambler] won approximately $26,660,000 from the California lottery…. [He] elected to receive payment of the lottery proceeds in 20 annual payments of approximately $1,333,000 each…. Since winning the lottery proceeds, [he] has not been employed."[He] gambled infrequently before winning the lottery. After [1996], [he] started playing the slot machines at the casinos frequently…. [He] spent most of his waking hours at the casinos. He had no outside interests, and generally if he was not at the casinos he was at home. A typical day for [him] generally consisted of waking up, showering, going to a 7-Eleven, getting coffee, going to the casinos, gambling, returning home, sleeping, waking up, and returning to the casino immediately thereafter. Occasionally, [he] spent up to 48 hours continuously in the casinos before returning home. [He] spent an average of 20 days per month at the casinos….

"On those days when he was at the casinos, [he] spent 8 to 48 hours continuously in the casinos, averaging approximately 10 hours per day. While at the casinos, [he] exclusively wagered on slot machines…. On the rare occasions when he left the casino with any money, [he] would bring the money back to the casino the following day, and he would then gamble with, and eventually lose (either the next day or shortly thereafter), that money.

"[He] did not get emotionally excited when he won at the slot machines. [He] did not get excited when he won jackpots of $1,200 or greater because the slot machine would freeze or lock up until he was issued his slot machine winnings…by the casino. Furthermore, [he] knew that eventually he would lose any winnings playing the slot machines.

"[He] lived with his girlfriend…. [She] went with [him] to the casinos and watched him gamble away his money. While watching [him] gamble, [she] saw that he did not get excited and did not enjoy playing the slot machines. … In or around 2003, [she] ended her relationship with [him] as he was never home because of his pathological gambling disorder. After she moved out of [his] home, he did not notice that she was gone until 2 or 3 days later."

^{5}

Why did he behave in this mind-boggling and tragic way? According to a psychologist called as an expert witness:

"…[The expert witness] testified that, unlike recreational and problem gamblers, pathological gamblers take the 'gambler’s fallacy' to a delusional level―they believe if they gamble long enough, they will win back all their losses and even more. [The expert witness] also opined that, unless treated for his illness, [the gambler] will gamble until he dies or loses all his money."^{6}

Here's a note of warning for future litigants:

"Respondent attempted to discredit [the expert witness] by claiming her definition of 'gambler’s fallacy' was incorrect. Respondent relies on a definition of 'gambler’s fallacy' he obtained from Wikipedia. Respondent did not call any witness, or expert witness, to counter [the expert witness]’s conclusions. Respondent’s reliance on a definition of 'gambler’s fallacy' found in Wikipedia is not persuasive. [The expert witness]…credibly explained that there is a difference in the definition of 'gambler’s fallacy' depending on the field of study―e.g., psychology versus mathematics. We find [the expert witness] to be credible and rely on her expert opinion."^{7}

The Respondent should've cited The Fallacy Files, instead! If there really is a difference in the definition of "the gambler's fallacy" in psychology as opposed to math, I haven't heard of it, but I'm no psychologist. The Wikipedia entry for the fallacy at the time of the decision is actually reasonably accurate, but the judge couldn't be expected to know that, which is why he was right not to rely on it.^{8}

### 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"^{9} 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.

**Notes:**

- Darrell Huff & Irving Geis, How to Take a Chance (1959), p. 28.
- A. R. Lacey, Dictionary of Philosophy (Third Revised Edition), (Barnes & Noble, 1996).
- Huff & Geis, pp. 28-29. Huff calls the fallacy "the maturity of the chances"―see the Alias, above―and the mistake on which it is based "the doctrine of the maturity of the chances". The doctrine is the notion that a fair gambling device that has had a run must make up the run in order that things will equal out in the long run. For example, in the Monte Carlo story, that the wheel must start coming up red more often than black in order to make up for the lengthy run of black.
- Statistical independence can be defined precisely in probability theory: A and B are statistically independent if and only if p(A | B) = p(A) and p(B | A) = p(B).
- "Gagliardi v. Commissioner of Internal Revenue", United States Tax Court, 1/24/2008, Pp. 4-11 (PDF).
*Ibid.*, p. 26.*Ibid.*- I found this case in: Eugene Volokh, "More Wikipedia Law", The Volokh Conspiracy, 2/11/2009
- Barbara Mikkelson, "The Full Monte", Snopes, 8/10/2007.

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