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Gambler's fallacy

The Gambler's fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a Monte Carlo Casino in 1913)[1] . Also referred to as the fallacy of the maturity of chances, which is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are then more likely. For example, if a fair coin is tossed repeatedly and tails comes up a larger number of times than is expected, a gambler may incorrectly believe that this means that heads is more likely in future tosses.[2] . Such an expectation could be mistakenly referred to as being due, and it probably arises from everyday experiences with nonrandom events (such as when a scheduled train is late, where it can be expected that it has a greater chance of arriving the later it gets). This is an informal fallacy. It is also known colloquially as the law of averages. What is true instead are the law of large numbers – in the long term, averages of independent trials will tend to approach the expected value, even though individual trials are independent – and regression toward the mean, namely that following a rare extreme event (say, a run of 10 heads), the next event is likely to be less extreme (the next run of heads is likely to be less than 10), simply because extreme events are rare. The gambler's fallacy implicitly involves an assertion of negative correlation between trials of the random process and therefore involves a denial of the exchangeability of outcomes of the random process. In other words, one implicitly assigns a higher chance of occurrence to an event even though from the point of view of "nature" or the "experiment", all such events are equally probable (or distributed in a known way). The reversal is also a fallacy, in which a gambler may instead decide that tails are more likely out of some mystical preconception that fate has thus far allowed for consistent results of tails; the false conclusion being: Why change if odds favor tails? Again, the fallacy is the belief that the "universe" somehow carries a memory of past results which tend to favor or disfavor future outcomes. The conclusion of this reversed gambler's fallacy may be correct, however, if the empirical evidence suggests that an initial assumption about the probability distribution is false. If a coin is tossed ten times and lands "heads" ten times, the gambler's fallacy would suggest an even-money bet on "tails", while the reverse gambler's fallacy (not to be confused with the inverse gambler's fallacy) would suggest an even-money bet on "heads". In this case, the smart bet is "heads" because the empirical evidence—ten "heads" in a row—suggests that the coin is likely to be biased toward "heads", contradicting the (general) assumption that the coin is fair.

Gambler's fallacy

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An example: coin-tossing

The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. With a fair coin, the outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is exactly 1⁄2 (one in two). It follows that the probability of getting two heads in two tosses is 1⁄4 (one in four) and the probability of getting three heads in three tosses is 1⁄8 (one in eight). In general, if we let Ai be the event that toss i of a fair coin comes up heads, then we have,

Simulation of coin tosses: Each frame, a coin is flipped which is red on one side and blue on the other. The result of each flip is added as a colored dot in the corresponding column. As the pie chart shows, the proportion of red versus blue approaches 50-50 (the Law of Large Numbers). But the difference between red and blue does not systematically decrease to zero.

. Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete...