Gambler’s fallacy

The Gambler’s fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a Monte Carlo casino in 1913) or the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future. For example, if a fair 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. This is an informal fallacy. It is also known colloquially as the law of averages.

The gambler’s fallacy implicitly involves an assertion of negative correlation between trials of the random process and therefore involves a denial of the exchangeability of outcomes of the random process.

The reversal is also a fallacy, the inverse gambler’s fallacy, in which a gambler may instead decide that tails are more likely out of some mystical preconception that fate has thus far allowed for consistent results of tails; the false conclusion being: Why change if odds favor tails? Again, the fallacy is the belief that the “universe” somehow carries a memory of past results which tend to favor or disfavor future outcomes.

An example: coin-tossing

The gambler’s fallacy can be illustrated by considering the repeated toss of a fair coin. With a fair coin, the outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is exactly ¹⁄₂ (one in two). It follows that the probability of getting two heads in two tosses is ¹⁄₄ (one in four) and the probability of getting three heads in three tosses is ¹⁄₈ (one in eight). In general, if we let A_i be the event that toss i of a fair coin comes up heads, then we have,

.

Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is only ¹⁄₃₂ (one in thirty-two), a believer in the gambler’s fallacy might believe that this next flip is less likely to be heads than to be tails. However, this is not correct, and is a manifestation of the gambler’s fallacy; the event of 5 heads in a row and the event of “first 4 heads, then a tails” are equally likely, each having probability ¹⁄₃₂. Given the first four rolls turn up heads, the probability that the next toss is a head is in fact,

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While a run of five heads is only ¹⁄₃₂ = 0.03125, it is only that before the coin is first tossed. After the first four tosses the results are no longer unknown, so their probabilities are 1. Reasoning that it is more likely that the next toss will be a tail than a head due to the past tosses, that a run of luck in the past somehow influences the odds in the future, is the fallacy.

Explaining why the probability is 1/2 for a fair coin

We can see from the above that, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2,097,152. However, the probability of flipping a head after having already flipped 20 heads in a row is simply ¹⁄₂. This is an application of <i>Bayes’ theorem</i>.

This can also be seen without knowing that 20 heads have occurred for certain (without applying of Bayes’ theorem). Consider the following two probabilities, assuming a fair coin:

• probability of 20 heads, then 1 tail = 0.5²⁰ × 0.5 = 0.5²¹
• probability of 20 heads, then 1 head = 0.5²⁰ × 0.5 = 0.5²¹

The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. Therefore, it is equally likely to flip 21 heads as it is to flip 20 heads and then 1 tail when flipping a fair coin 21 times. Furthermore, these two probabilities are equally as likely as any other 21-flip combinations that can be obtained (there are 2,097,152 total); all 21-flip combinations will have probabilities equal to 0.5²¹, or 1 in 2,097,152. From these observations, there is no reason to assume at any point that a change of luck is warranted based on prior trials (flips), because every outcome observed will always have been equally as likely as the other outcomes that were not observed for that particular trial, given a fair coin. Therefore, just as Bayes’ theorem shows, the result of each trial comes down to the base probability of the fair coin: ¹⁄₂.

Other examples

There is another way to emphasize the fallacy. As already mentioned, the fallacy is built on the notion that previous failures indicate an increased probability of success on subsequent attempts. This is, in fact, the inverse of what actually happens, even on a fair chance of a successful event, given a set number of iterations. Assume you have a fair 16-sided die, and a win is defined as rolling a 1. Assume a player is given 16 rolls to obtain at least one win (1−p(rolling no ones)). The low winning odds are just to make the change in probability more noticeable. The probability of having at least one win in the 16 rolls is:

However, assume now that the first roll was a loss (93.75% chance of that, ¹⁵⁄₁₆). The player now only has 15 rolls left and, according to the fallacy, should have a higher chance of winning since one loss has occurred. His chances of having at least one win are now:

Simply by losing one toss the player’s probability of winning dropped by 2%. By the time this reaches 5 losses (11 rolls left), his probability of winning on one of the remaining rolls will have dropped to ~50%. The player’s odds for at least one win in those 16 rolls has not increased given a series of losses; his odds have decreased because he has fewer iterations left to win. In other words, the previous losses in no way contribute to the odds of the remaining attempts, but there are fewer remaining attempts to gain a win, which results in a lower probability of obtaining it.

The player becomes more likely to lose in a set number of iterations as he fails to win, and eventually his probability of winning will again equal the probability of winning a single toss, when only one toss is left: 6.25% in this instance.

Some lottery players will choose the same numbers every time, or intentionally change their numbers, but both are equally likely to win any individual lottery draw. Copying the numbers that won the previous lottery draw gives an equal probability, although a rational gambler might attempt to predict other players’ choices and then deliberately avoid these numbers. Low numbers (below 31 and especially below 12) are popular because people play birthdays as their so-called lucky numbers; hence a win in which these numbers are over-represented is more likely to result in a shared payout.

A joke told among mathematicians demonstrates the nature of the fallacy. When flying on an aircraft, a man decides to always bring a bomb with him. “The chances of an aircraft having a bomb on it are very small,” he reasons, “and certainly the chances of having two are almost none!”

A similar example is in the book The World According to Garp when the hero Garp decides to buy a house a moment after a small plane crashes into it, reasoning that the chances of another plane hitting the house have just dropped to zero.

The most famous example happened in a Monte Carlo casino 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 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 a long streak of red.

Non-examples of the fallacy

There are many scenarios where the gambler’s fallacy might superficially seem to apply but does not. When the probability of different events is not independent, the probability of future events can change based on the outcome of past events (statistical permutation). Formally, the system is said to have memory. An example of this is cards drawn without replacement. For example, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be of another rank. Thus, the odds for drawing a jack, assuming that it was the first card drawn and that there are no jokers, have decreased from ⁴⁄₅₂ (7.69%) to ³⁄₅₁ (5.88%), while the odds for each other rank have increased from ⁴⁄₅₂ (7.69%) to ⁴⁄₅₁ (7.84%). This is how counting cards really works, when playing the game of blackjack.

The outcome of future events can be affected if external factors are allowed to change the probability of the events (e.g., changes in the rules of a game affecting a sports team’s performance levels). Additionally, an inexperienced player’s success may decrease after opposing teams discover his or her weaknesses and exploit them. The player must then attempt to compensate and randomize his strategy.

Many riddles trick the reader into believing that they are an example of the gambler’s fallacy, such as the Monty Hall problem.

Non-example: unknown probability of event

When the probability of repeated events are not known, outcomes may not be equally probable. In the case of coin tossing, as a run of heads gets longer and longer, the likelihood that the coin is biased towards heads increases. If one flips a coin 21 times in a row and obtains 21 heads, one might rationally conclude a high probability of bias towards heads, and hence conclude that future flips of this coin are also highly likely to be heads. In fact, Bayesian inference can be used to show that when the long-run proportion of different outcomes are unknown but exchangeable (meaning that the random process from which they are generated may be biased but is equally likely to be biased in any direction) previous observations demonstrate the likely direction of the bias, such that the outcome which has occurred the most in the observed data is the most likely to occur again.

Psychology behind the fallacy

Amos Tversky and Daniel Kahneman proposed that the gambler’s fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic. According to this view, “after observing a long run of red on the roulette wheel, for example, 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 close to 0.5 in any short segment more so 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.

The representativeness heuristic is also cited behind the related phenomenon of the clustering illusion, according to which people see streaks of random events as being non-random when such streaks are actually much more likely to occur in small samples than people expect.