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Encyclopedia > Gambler's fallacy

The gambler's fallacy, also known as the Monte Carlo fallacy, is an informal fallacy. It is the incorrect belief that the likelihood of a random event can be affected by or predicted from other, independent events. Image File history File links This is a lossless scalable vector image. ... In Philosophical logic, an informal fallacy is a pattern of reasoning which is false due to the falsity of one or more of its premises. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ...


The gambler's fallacy gets its name from the fact that, where the random event is the throw of a die or the spin of a roulette wheel, gamblers will risk money on their belief in "a run of luck" or a mistaken understanding of "the law of averages". It often arises because a similarity between random processes is mistakenly interpreted as a predictive relationship between them. (For instance, two fair dice are similar in that they each have the same chances of yielding each number - but they are independent in that they do not actually influence one another.) The law of averages is a lay term used to express the view that eventually, everything evens out. ...


The gambler's fallacy often takes one of these forms:

  • A particular outcome of a random event is more likely to occur because it has happened recently ("run of good luck");
  • A particular outcome is more likely to occur because it has not happened recently ("law of averages" or "it's my turn now").

Similarly Random redirects here. ...

  • A particular outcome is less likely to occur because it has happened recently ("law of averages" or "exhausted its luck");
  • A particular outcome is less likely to occur because it has not happened recently ("run of bad luck").

A more subtle version of the fallacy is that an "interesting" (non-random looking) outcome is "unlikely" (eg that a sequence of "1,2,3,4,5,6" in a lottery result is less likely than any other individual outcome). Even apart from the debate about what constitutes an "interesting" result, this can be seen as a version of the gambler's fallacy because it is saying that a random event is less likely to occur if the result, taken in conjunction with recent events, will produce an "interesting" pattern.

Contents

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. Coin flipping or coin tossing is the practice of throwing a coin in the air to resolve a dispute between two parties. ...


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 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 (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. It has been suggested that this article or section be merged with Martingale (probability theory). ... Example of eight random walks in one dimension starting at 0. ...


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, 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. This article or section does not cite its references or sources. ... For other uses, see Heuristic (disambiguation). ... The representativeness heuristic is a heuristic wherein we assume commonality between objects of similar appearance. ... The term pareidolia (pronounced or ), referenced in 1994 by Steven Goldstein,[1] describes a psychological phenomenon involving a vague and random stimulus (often an image or sound) being perceived as significant. ...


Other examples

  • What is the probability of flipping 21 heads in a row, with a fair coin? (Answer: 1 in 2,097,152 = approximately 0.000000477.) What is the probability of doing it, given that you have already flipped 20 heads in a row? (Answer: 0.5.) See Bayes' theorem.
  • Will you eventually come out ahead at roulette by betting double what you lost the previous time, and adding an extra amount? (Answer: given infinite time and funds, yes, you will eventually win on that color in a fair game. However, given finite time and even more finite funds, the chance exists that you will exhaust your money before winning. Regardless of the odds of a color losing (or winning) several times in a row, the probability of the ball landing on that color in a given spin is the number of that color that exist, divided by all possibilities. In the case of a Vegas roulette wheel, the chances of hitting red are 18/38, or ~.47, regardless of previous results.)
  • Are you more likely to win the lottery jackpot by choosing the same numbers every time or by choosing different numbers every time? (Answer: Either strategy is equally likely to win.)
  • Are you more or less likely to win the lottery jackpot by picking the numbers which won last week, or picking numbers at random? (Answer: Either strategy is equally likely to win, but if others choose the same numbers your payout is likely to be less.)

A rational gambler might attempt to predict other players' choices and then deliberately avoid these numbers. Coin flipping or coin tossing is the practice of throwing a coin in the air to resolve a dispute between two parties. ... Bayes theorem (also known as Bayes rule or Bayes law) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ...


Non-examples

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.

In probability theory, the Chernoff bound, named after Herman Chernoff, gives a lower bound for the success of majority agreement for n independent, equally likely events. ... Game theory is a branch of applied mathematics that is often used in the context of economics. ... In search of a new car, the player picks door 1. ...

See also

It has been suggested that this article or section be merged into Availability heuristic. ... The clustering illusion is the natural human tendency to see patterns where actually none exist. ... The illusion of control is the tendency for human beings to believe they can control or at least influence outcomes which they clearly cannot. ... The inverse gamblers fallacy is a tempting mistake in judgments of probability, comparable to the gamblers fallacy whence its name derives. ... This article or section does not cite its references or sources. ... The basic meaning of gamblers ruin is a gamblers loss of the last of his bank of gambling money and consequent inability to continue gambling. ... Statistical regularity is a notion in statistics that if we throw a thumbtack onto a table once, we would have a hard time predicting whether the point would touch the surface of the table or not. ... In Philosophical logic, an informal fallacy is a pattern of reasoning which is false due to the falsity of one or more of its premises. ... Special pleading is a form of spurious argumentation where a position in a dispute introduces favorable details or excludes unfavorable details by alleging a need to apply additional considerations without proper criticism of these considerations themselves. ... Look up red herring in Wiktionary, the free dictionary. ... The inverse gamblers fallacy is a tempting mistake in judgments of probability, comparable to the gamblers fallacy whence its name derives. ... A fallacy of distribution is a logical fallacy occurring when an argument assumes there is no difference between a term in the distributive (referring to every member of a class) and collective (referring to the class itself as a whole) sense. ... A fallacy of composition arises when one infers that something is true of the whole from the fact that it is true of some (or even every) part of the whole. ... A fallacy of division occurs when someone reasons logically that something that is true of a thing must also be true of its constituents. ... In logic, begging the question describes a type of logical fallacy, petitio principii, in which the conclusion of an argument is implicitly or explicitly assumed in one of the premises. ... This article or section does not adequately cite its references or sources. ... In logic, correlative-based fallacies, also known as fallacies of distraction, are logical fallacies based on correlative conjunctions. ... The form of the fallacy of false dichotomy as an argument map with the conclusion at the top of the tree. ... The perfect solution fallacy is a logical fallacy that occurs when an argument assumes that a perfect solution exists and/or that a solution should be rejected because some part of the problem would still exist after it was implemented. ... The logical fallacy of denying the correlative is the opposite of the false dilemma, where an attempt is made at introducing alternatives where there are none. ... The logical fallacy of suppressed correlative is a type of argument which tries to redefine a correlative (two mutually exclusive options) so that one alternative encompasses the other, i. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The logical fallacy of accident, also called destroying the exception or a dicto simpliciter ad dictum secundum quid, is a deductive fallacy occurring in statistical syllogisms (an argument based on a generalization) when an exception to the generalization is ignored. ... The logical fallacy of converse accident (also called reverse accident, destroying the exception or a dicto secundum quid ad dictum simpliciter) is a deductive fallacy that can occur in a statistical syllogism when an exception to a generalization is wrongly called for. ... A faulty generalization, also known as an inductive fallacy, is any of several errors of inductive inference: Hasty generalization is the fallacy of examining just one or very few examples or studying a single case, and generalizing that to be representative of the whole class of objects or phenomena. ... Hasty generalization, is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence. ... A biased sample is one that is falsely taken to be typical of a population from which it is drawn. ... This article does not cite any references or sources. ... The logical fallacy of misleading vividness involves describing some occurrence in vivid detail, even if it is an exceptional occurrence, to convince someone that it is a problem. ... The conjunction fallacy is a logical fallacy that occurs when it is assumed that specific conditions are more probable than general ones. ... Ambiguity is one way in which the meanings of words and phrases can be unclear, but there is another way, which is different from ambiguity: vagueness. ... Look up ambiguity in Wiktionary, the free dictionary. ... This article or section does not cite its references or sources. ... In debate or rhetoric, the slippery slope is an argument for the likelihood of one event or trend given another. ... Continuum fallacy, also called fallacy of the beard is a logical fallacy which abuses the paradox of the heap. ... Equivocation, also known as amphibology, is classified as both a formal and informal fallacy. ... The fallacy of a false attribution occurs when an advocate appeals to an irrelevant, unqualified, unidentified, biased or fabricated source in support of an argument. ... It has been suggested that Contextomy be merged into this article or section. ... Lokis Wager is a form of logical fallacy. ... No true Scotsman is a term coined by Antony Flew in his 1975 book Thinking About Thinking – or do I sincerely want to be right?[1]: Imagine Hamish McDonald, a Scotsman, sitting down with his Press and Journal and seeing an article about how the Brighton Sex Maniac Strikes Again. ... Fallacies of questionable cause, also known as causal fallacies, non causa pro causa (non-cause for cause in Latin) or false cause, are informal fallacies where a cause is incorrectly identified. ... Correlation does not imply causation is a phrase used in the sciences and statistics to emphasize that correlation between two variables does not imply there is a cause-and-effect relationship between the two. ... For the episode of the television program The West Wing, see Post Hoc, Ergo Propter Hoc (The West Wing). ... The regression (or regressive) fallacy is a logical fallacy. ... The Texas sharpshooter fallacy is a logical fallacy where a cluster of statistically non-significant data is taken from its context, and therefore thought to have a common cause. ... Circular cause and consequence is a logical fallacy where the consequence of the phenomenon is claimed to be its root cause. ... Wrong direction is a logical fallacy of causation where cause and effect are reversed. ... The fallacy of the single cause, also known as joint effect or causal oversimplification, is a logical fallacy of causation that occurs when it is assumed that there is one, simple cause of an outcome when in reality it may have been caused by a number of only jointly sufficient...

External links

  • The gambler's fallacy exposed
  • The Gambler's Fallacy

  Results from FactBites:
 
Gambler's Fallacy (717 words)
Both versions of the fallacy are based on the same mistake, namely, a failure to understand statistical independence.
Any gambler who thinks that he can record the results of a roulette wheel, or the throws at a craps table, or lotto numbers, and use this information to predict future outcomes is probably committing some form of the gambler's fallacy.
The fundamental mistake in the Reverse form is the same as in the Gambler's Fallacy, that is, the failure to appreciate statistical independence.
Biases in casino betting: The hot hand and the gambler's fallacy (7900 words)
The gambler's fallacy is defined as an (incorrect) belief in negative autocorrelation of a non-autocorrelated random sequence.
For example, individuals who believe in the gambler's fallacy believe that after three red numbers appearing on the roulette wheel, a fl number is "due," that is, is more likely to appear than a red number.
If players bet according to the gambler's fallacy, the probability of their betting on a given number should be negatively related to its hotness measure; numbers which have come up more frequently while they were at the table are less likely to be bet on.
  More results at FactBites »

 
 

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