Gambler's fallacy
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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". Although the gambler's fallacy can apply to any form of gambling, it is easiest to illustrate by considering coin-tossing.
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An example: coin-tossing
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 the fallacious step in the argument. If the coin is fair, then by definition the probability of tails must always be .5, never more (or less), and the probability of heads must always be .5, never less (or more). While a run of five heads is indeed unlikely - the probability is one thirty-second, or 0.03125, from before the first coin is tossed - the probability of four successive heads followed by one tails is the same: one thirty-second. It is no more likely. After the fourth head, there are now two possible outcomes with equal probability. In such a situation, given the first 4 tosses, the probability of each equally likely outcome is now .5. Therefore, the probability of tails on the next coin toss at this moment is still the same as any toss: it is 0.5. This is simply another way of showing that 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".
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 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.
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.
The gambler's fallacy then includes the misconceptions that:
- A random event is more likely to occur because it has not happened for a period of time;
- A random event is less likely to occur because it has not happened for a period of time;
- A random event is more likely to occur because it recently happened; and
- A random event is less likely to occur because it recently happened.
Obviously if you fall for the fallacy, you can conclude pretty much anything!
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.
Other examples
- You flip a fair coin 20 times and it comes up heads every time. What is the probability it will come up tails next time? (Answer: 0.5)
- Are you more likely to win the lottery by choosing the same numbers every time, or by choosing different numbers every time? (Answer: you are equally likely with either strategy.)
Counterexamples
- When the probability of different events is not independent, the probability of future events can change based on the outcome of past events.
- 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).