Gambler's fallacy

The gambler's fallacy is one of many common misunderstandings which arise in everyday reasoning about probabilities, many of which have been studied in a great detail. It is considered to be a logical fallacy, and can be summarised with the phrase "the coin doesn't have a memory".

The gambler's fallacy can be illustrated by a game in which a coin is tossed over and over again. Suppose that the coin is in fact fair, so that the chances of it coming up heads are exactly 0.5 (a half). Then the chances of it coming up heads twice in succession are 0.5×0.5=0.25 (a quarter); three times in succession, they are 0.125 (an eighth) and so on.

Nothing fallacious so far; but suppose that we are in one of these states where, say, four heads have just come up in a row, and someone argues as follows: "if the next coin flipped were to come up heads, it would generate a run of five successive heads. This is very unlikely; therefore, the next coin flipped is more likely to come up tails."

This is fallacious. The probability of a run of five heads is in fact one thirty-second, or 0.03125, from before the first coin is tossed. However, the probability of four successive heads followed by one tails is the same: one thirty-second. Therefore, the probability of the result of the next coin toss at this moment is still the same as any toss: it is 0.5. The present toss is independent of what has happened in the past. 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 idea that a run of luck in the past somehow influences the odds of a bet in the future. Related fallacious ideas are inherent in such phrases as "a lucky streak" or "a winning streak" or a "break".

Since the odds of a run of five heads are indeed very low, one might wonder where the fallacy lies. The point is that those odds are low given no prior information. But at the point when the fallacy is formulated, four of those heads are already tossed; there is no uncertainty about them at all. Given that we are already in a state which itself had a probability of only one sixteenth (the odds of getting four heads in a row), we can be sure that the next state will in fact have an uncertainty of one thirty-second no matter what the next toss is. The odds of a heads on the next toss (for a fair coin) are even, no matter what the past history of the gamble has been.

Sometimes, gamblers argue like this: "I just lost four times. Since the coin is fair and therefore in the long run everything has to even out, if I just keep playing, I will eventually win my money back." However, it would be irrational to look at things "in the long run" starting from before he started playing; he ought to consider that in the long run from where he is now, he could expect everything to even out to his current point, which is four losses down.

Mathematically, the probability is equal to one that gains will eventually equal losses and a gambler will return to his starting point; however, the expected number of times he has to play is infinite, and so is the expected amount of capital he will need! A similar argument shows that the popular doubling strategy (start with $1, if you lose, bet $2, then $4 etc., until you win) does not work; see St. Petersburg paradox. Situations like these are investigated in the mathematical theory of random walks. This and similiar strategies either trade many small wins for a few huge losses (as in this case) or visa 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.

Notice that the gambler's fallacy is quite different from the following path of reasoning (which comes to the opposite conclusion): the coin comes up heads more often than tails, so it is not a fair coin, so I will bet that the next toss will be heads also. This is not fallacious, though the first step - the argument from a finite number of observations to a statement of likelihood - is a very delicate matter, and is itself prone to fallacies of its own peculiar kind.

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!"

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

Gambler's fallacy

See also