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d'Alembert gambler's fallacy

The gambler's fallacy is a logical fallacy involving the mistaken belief that past events will affect future events when dealing with random activities, such as many gambling games. It can encompass any of the following misconceptions:

These are common misunderstandings that arise in everyday reasoning about probabilities, many of which have been studied in great detail. Many people lose money while gambling due to their erroneous belief in this fallacy.

Put simply, the chances of something happening the next time are not necessarily related to what has already happened, especially in many gambling games. This is known in probability theory as the memoryless property.

Coin-tossing

The gambler's fallacy can be illustrated by considering the repeated toss of a coin. With a fair coin the chances of getting heads are exactly 0.5 (one in two). The chances of it coming up heads twice in a row are 0.5*0.5=0.25 (one in four). The probability of three heads in a row is 0.5*0.5*0.5= 0.125 (one in eight) and so on.

Now suppose that we have just tossed four heads in a row. A believer in the gambler's fallacy might say, "If the next coin flipped were to come up heads, it would generate a run of five successive heads. The probability of a run of five successive heads is (1 / 2)5 = 1 / 32; therefore, the next coin flipped only has a 1 in 32 chance of coming up heads."

This is the fallacious step in the argument. If the coin is fair, then by definition the probability of tails must always be 0.5, never more (or less), and the probability of heads must always be 0.5, never less (or more). While a run of five heads is only 1 in 32 (0.03125), it is 1 in 32 before the coin is first tossed. After the first four tosses the results are no longer unknown, so they don't count. The probability of five consecutive heads is the same as four successive heads followed by one tails. Tails is no more likely. In fact, the calculation of the 1 in 32 probability relied on the assumption that heads and tails are equally likely at every step. Each of the two possible outcomes has equal probability no matter how many times the coin has been flipped previously and no matter what the result. Reasoning that it is more likely that the next toss will be a tail than a head due to the past tosses is the fallacy. The fallacy is the idea that a run of luck in the past somehow influences the odds of a bet in the future.

As an example, the popular doubling strategy (start with $1, if you lose, bet $2, then $4 etc., until you win) does not work; see Martingale (betting system). Situations like these are investigated in the mathematical theory of random walks. This and similar strategies either trade many small wins for a few huge losses (as in this case) or vice versa. With an infinite amount of working capital, one would come out ahead using this strategy; as it stands, one is better off betting a constant amount if only because it makes it easier to estimate how much one stands to lose in an hour or day of play.

A joke told among mathematicians demonstrates the nature of the fallacy. When flying on an airplane, a man decides to always bring a bomb with him. "The chances of an airplane having a bomb on it are very small," he reasons, "and certainly the chances of having two are almost none!" - this joke was also written into Blackadder Goes Forth - when Baldrick is carving his name into a bullet, Edmund Blackadder asks why and is told "because if I have the bullet with my name on it, I can't get shot with it!".

Some claim that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic.