Gambler's fallacy
The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event has occurred less frequently than expected, it is more likely to happen again in the future. The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more likely to be six than is usually the case because there have recently been fewer than the expected number of sixes.
The term "Monte Carlo fallacy" originates from an example of the phenomenon, in which the roulette wheel spun black 26 times in succession at the Monte Carlo Casino in 1913.
Examples
Coin toss
The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. The outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is . The probability of getting two heads in two tosses is and the probability of getting three heads in three tosses is . In general, if Ai is the event where toss i of a fair coin comes up heads, then:If after tossing four heads in a row, the next coin toss also came up heads, it would complete a run of five successive heads. Since the probability of a run of five successive heads is , a person might believe that the next flip would be more likely to come up tails rather than heads again. This is incorrect and is an example of the gambler's fallacy. The event "5 heads in a row" and the event "first 4 heads, then a tails" are equally likely, each having probability. Since the first four tosses turn up heads, the probability that the next toss is a head is:
While a run of five heads has a probability of = 0.03125, the misunderstanding lies in not realizing that this is the case only before the first coin is tossed. After the first four tosses in this example, the results are no longer unknown, so their probabilities are at that point equal to 1. The probability of a run of coin tosses of any length continuing for one more toss is always 0.5. The reasoning that a fifth toss is more likely to be tails because the previous four tosses were heads, with a run of luck in the past influencing the odds in the future, forms the basis of the fallacy.
Why the probability is for a fair coin
If a fair coin is flipped 21 times, the probability of 21 heads is 1 in 2,097,152. The probability of flipping a head after having already flipped 20 heads in a row is. Assuming a fair coin:- The probability of 20 heads, then 1 tail is 0.520 × 0.5 = 0.521
- The probability of 20 heads, then 1 head is 0.520 × 0.5 = 0.521
Other examples
The fallacy leads to the incorrect notion that previous failures will create an increased probability of success on subsequent attempts. For a fair 16-sided die, the probability of each outcome occurring is . If a win is defined as rolling a 1, the probability of a 1 occurring at least once in 16 rolls is:The probability of a loss on the first roll is . According to the fallacy, the player should have a higher chance of winning after one loss has occurred. The probability of at least one win is now:
By losing one toss, the player's probability of winning drops by two percentage points. With 5 losses and 11 rolls remaining, the probability of winning drops to around 0.5. The probability of at least one win does not increase after a series of losses; indeed, the probability of success actually decreases, because there are fewer trials left in which to win. The probability of winning will eventually be equal to the probability of winning a single toss, which is and occurs when only one toss is left.
Reverse position
After a consistent tendency towards tails, a gambler may also decide that tails has become a more likely outcome. This is a rational and Bayesian conclusion, bearing in mind the possibility that the coin may not be fair; it is not a fallacy. Believing the odds to favor tails, the gambler sees no reason to change to heads. However, it is a fallacy that a sequence of trials carries a memory of past results which tend to favor or disfavor future outcomes.The inverse gambler's fallacy described by Ian Hacking is a situation where a gambler entering a room and seeing a person rolling a double six on a pair of dice may erroneously conclude that the person must have been rolling the dice for quite a while, as they would be unlikely to get a double six on their first attempt.
Retrospective gambler's fallacy
Researchers have examined whether a similar bias exists for inferences about unknown past events based upon known subsequent events, calling this the "retrospective gambler's fallacy".An example of a retrospective gambler's fallacy would be to observe multiple successive "heads" on a coin toss and conclude from this that the previously unknown flip was "tails". Real world examples of retrospective gambler's fallacy have been argued to exist in events such as the origin of the Universe. In his book Universes, John Leslie argues that "the presence of vastly many universes very different in their characters might be our best explanation for why at least one universe has a life-permitting character". Daniel M. Oppenheimer and Benoît Monin argue that "In other words, the 'best explanation' for a low-probability event is that it is only one in a multiple of trials, which is the core intuition of the reverse gambler's fallacy." Philosophical arguments are ongoing about whether such arguments are or are not a fallacy, arguing that the occurrence of our universe says nothing about the existence of other universes or trials of universes. Three studies involving Stanford University students tested the existence of a retrospective gamblers' fallacy. All three studies concluded that people have a gamblers' fallacy retrospectively as well as to future events. The authors of all three studies concluded their findings have significant "methodological implications" but may also have "important theoretical implications" that need investigation and research, saying " thorough understanding of such reasoning processes requires that we not only examine how they influence our predictions of the future, but also our perceptions of the past."
Childbirth
In 1796, Pierre-Simon Laplace described in A Philosophical Essay on Probabilities the ways in which men calculated their probability of having sons: "I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls ought to be the same at the end of each month, they judged that the boys already born would render more probable the births next of girls." The expectant fathers feared that if more sons were born in the surrounding community, then they themselves would be more likely to have a daughter. This essay by Laplace is regarded as one of the earliest descriptions of the fallacy. Likewise, after having multiple children of the same sex, some parents may erroneously believe that they are due to have a child of the opposite sex.Monte Carlo Casino
An example of the gambler's fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. This was an extremely unlikely occurrence: for any given sequence of 26 spins, the probability of either red or black occurring 26 times in a row on a single zero roulette wheel is or around 1 in 68.4 million, assuming the mechanism is unbiased. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an imbalance in the randomness of the wheel, and that it had to be followed by a long streak of red.Non-examples
Non-independent events
The gambler's fallacy does not apply when the probability of different events is not independent. In such cases, the probability of future events can change based on the outcome of past events, such as the statistical permutation of events. An example is when cards are drawn from a deck without replacement. If an ace is drawn from a deck and not reinserted, the next card drawn is less likely to be an ace and more likely to be of another rank. The probability of drawing another ace, assuming that it was the first card drawn and that there are no jokers, has decreased from to , while the probability for each other rank has increased from to . This effect allows card counting systems to work in games such as blackjack.Bias
In most illustrations of the gambler's fallacy and the reverse gambler's fallacy, the trial is assumed to be fair. In practice, this assumption may not hold. For example, if a coin is flipped 21 times, the probability of 21 heads with a fair coin is 1 in 2,097,152. Since this probability is so small, if it happens, it may well be that the coin is somehow biased towards landing on heads, or that it is being controlled by hidden magnets, or similar. In this case, the smart bet is "heads" because Bayesian inference from the empirical evidence — 21 heads in a row — suggests that the coin is likely to be biased toward heads. Bayesian inference can be used to show that when the long-run proportion of different outcomes is unknown but exchangeable and that previous observations demonstrate the likely direction of the bias, the outcome which has occurred the most in the observed data is the most likely to occur again.For example, if the a priori probability of a biased coin is say 1%, and assuming that such a biased coin would come down heads say 60% of the time, then after 21 heads the probability of a biased coin has increased to about 32%.
The opening scene of the play Rosencrantz and Guildenstern Are Dead by Tom Stoppard discusses these issues as one man continually flips heads and the other considers various possible explanations.