Gambler's fallacy

The gambler's fallacy is a logical fallacy which encompasses 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. Although the gambler's fallacy can apply to any form of gambling, it is easiest to illustrate by considering coin-tossing; its rebuttal can be summarised with the phrase "the coin doesn't have a memory".
 * 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.

An example: 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 (a half). The chances of it coming up heads twice in a row are 0.5&times;0.5=0.25 (a quarter). The probability of three heads in a row is 0.5&times;0.5&times0.5= 0.125 (an eighth) 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 $$0.5^5=0.03125$$; 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 .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 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. 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.

Sometimes, gamblers argue, "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 is 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.

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.

Related links
The gambler's fallacy exposed

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)
 * A couple already has two daughters. What is the probability that the next child is a son? (Answer: 0.5) [if we assume the gender of a child is completely random, and that a male or female child is equally likely, either or both of which may be incorrect.]
 * 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. In reality, you may be better off choosing numbers in such a way as to reduce the risk of splitting the jackpot.)

Non-examples
There are many scenarios where the gambler's fallacy might superficially seem to apply, where it in fact 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. An example of this is cards drawn without replacement.  It's true that 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.


 * When the probability of each event is not even, such as with a loaded die, a number which has come up more often in the past may very well continue to do so, if that number is favored by the weighting of the dice. This has been dubbed Nerd's Gullibility Fallacy -- assuming the coin indeed is fair and the gamblers are honest when it isn't the case.  This is an example of Hume's principle: twenty tails in a row indicates that it is far more likely that the coin is loaded than that the coin is fair and the next toss will be fifty-fifty heads or tails.


 * Sporting events and races are also not even, in that some entrants have better odds of winning than others. Presumably, the winner of one such event is more likely to win the next event than the loser.


 * 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).


 * Many riddles trick the reader into believing that they are an example of Gambler's Fallacy, such as the Monty Hall problem. Similarly, if I flip a coin twice and tell you that at least one of the two flips was heads, and ask what the probability is that they both came up heads, you might answer, that it is 50/50 (or 50%).  This is incorrect: if I tell you that one of the two flips was heads then I am removing the tails-tails outcome only, leaving the following possible outcomes: heads-heads, heads-tails, and tails-heads.  These are equally likely, so heads-heads happens 1 time in 3 or 33% of the time.  If I had specified that the first flip was heads, then the chances the second flip was heads too is 50%.