Gambler's fallacy

From Wikipedia, the free encyclopedia

The gambler's fallacy, also known as the Monte Carlo fallacy, is the false belief that the probability of an event in a random sequence is dependent on preceding events, its probability increasing with each successive occasion on which it fails to occur. If a fair coin is tossed repeatedly and tails comes up many times in a row, a gambler may believe, incorrectly, that heads is more likely on the following toss.^[1] This is an informal fallacy.

The inverse gambler's fallacy deals with the belief that a particular outcome is less likely to occur because it has happened recently ("law of averages" or "exhausted its luck"), or because it has not happened recently ("run of bad luck").

1 An example: coin-tossing
2 Psychology behind the fallacy
3 Other examples
4 Non-examples of the fallacy
5 References
6 See also
7 External links

[edit] 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 (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 <math>(1/2)^5=1/32</math>; 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 do not 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. This kind of logic would only work if we had to guess all the tosses' results 'before' they are carried out. Let's say we are gambling on a HHHHH result, that is likely to constitute the significantly lesser chance to succeed.

As an example, the popular doubling strategy of the Martingale betting system (where a gambler starts with a bet of $1, and doubles their stake after each loss, until they win) is flawed. 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.

[edit] Psychology behind the fallacy

Some claim that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic, and a related phenomenon called pareidolia. There is an argument that we are programmed to look for patterns in chaos ("Is that a tiger half-hidden in the trees?" "Is that a bunch of ripe fruit half-hidden in the leaves?") and are actually biased towards spotting patterns when none exist. An animal that is prone to over-imagining patterns (e.g., never misses real tigers, but sometimes sees imaginary ones) is far more likely to pass on its genes than a cousin which ignores just one real tiger.

There are two major factors that some consider to be the cause of the gambler's fallacy within people, these factors are the clustering illusion and the illusion of control. The clustering illusion basically states that human beings are more apt to notice patterns in evens. Such as four heads being flipped in a row and then believing that tails has a better probability of turning up. When in fact the odds of getting a heads or tails is still the same 1/2 or 50% on a fair coin. Illusion of control shows that the typical gambler when playing craps for example will throw the dice harder to get a high number and throw the dice softer to get a lower number. However if it is a fair set of dice then the probability for every number on the die is the same, which is 1/6.

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!".

The gambler assumes that historical outcomes will affect future outcomes. This holds true when the occurrence of an event changes the probability of a future event. For instance, if a bag holds 50 black marbles and 50 white marbles, the chances of choosing a black marble the first time are 50%. The second time, now that one marble is already removed (49 black marbles and 50 white marbles), the chances of choosing a black marble are 49.5%. This is not the case when flipping a coin, where the previous occurrence does not affect future occurrences.

[edit] Other examples

The probability of flipping 21 heads in a row, with a fair coin is 1 in 2,097,152, but the probability of doing it after having already flipped 20 heads in a row is only 0.5. This is an example of Bayes' theorem.

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.

[edit] Non-examples of the fallacy

There are many scenarios where the gambler's fallacy might superficially seem to apply but does not, including:

When the probability of different events is not independent, the probability of future events can change based on the outcome of past events. 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%).
When the probability of each event is not even, such as with a loaded die or an unbalanced coin. The Chernoff bound is a method of determining how many times a coin must be flipped to determine (with high probability) which side is loaded. As a run of heads (or, e.g., reds on a roulette wheel) gets longer and longer, the chance that the coin or wheel is loaded increases.
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. See Game Theory.
Many riddles trick the reader into believing that they are an example of Gambler's Fallacy, such as the Monty Hall problem.

[edit] References

^ Colman, Andrew (2001). Gambler's Fallacy - Encyclopedia.com. A Dictionary of Psychology. Oxford University Press. Retrieved on 2007-11-26.

[edit] See also

v • d • e Informal fallacies
Special pleading • Red herring • Gambler's fallacy and its inverse • Fallacy of distribution (Composition • Division) • Begging the question • Many questions
Correlative-based fallacies:	False dilemma (Perfect solution) • Denying the correlative • Suppressed correlative
Deductive fallacies:	Accident • Converse accident
Inductive fallacies:	Hasty generalization • Overwhelming exception • Biased sample • False analogy • Misleading vividness • Conjunction fallacy
Vagueness and Ambiguity:	False precision • Slippery slope • Continuum fallacy
Equivocation:	Equivocation • False attribution • Fallacy of quoting out of context • Loki's Wager • No true Scotsman • Reification
Questionable cause:	Correlation does not imply causation (Cum hoc) • Post hoc • Regression fallacy • Texas sharpshooter • Circular cause and consequence • Wrong direction • Single cause
Other types of fallacy