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Gambler's fallacy
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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 (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future (or vice versa). 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.
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 1/2 (one in two). 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 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 1/32 (one in thirty-two), 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 1/32. 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 1/32 = 0.03125 (a little over 3%), 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 (100%). 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.
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 1/2. Assuming a fair coin:
The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. When flipping a fair coin 21 times, the outcome is equally likely to be 21 heads as 20 heads and then 1 tail. These two outcomes are equally as likely as any of the other combinations that can be obtained from 21 flips of a coin. All of the 21-flip combinations will have probabilities equal to 0.521, or 1 in 2,097,152. Assuming that a change in the probability will occur as a result of the outcome of prior flips is incorrect because every outcome of a 21-flip sequence is as likely as the other outcomes. In accordance with Bayes' theorem, the likely outcome of each flip is the probability of the fair coin, which is 1/2.
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 1/16 (6.25%). If a win is defined as rolling a 1, the probability of a 1 occurring at least once in 16 rolls is:
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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 (whose occurrences are independent and identically distributed) has occurred less frequently than expected, it is more likely to happen again in the future (or vice versa). 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.
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 1/2 (one in two). 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 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 1/32 (one in thirty-two), 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 1/32. 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 1/32 = 0.03125 (a little over 3%), 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 (100%). 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.
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 1/2. Assuming a fair coin:
The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. When flipping a fair coin 21 times, the outcome is equally likely to be 21 heads as 20 heads and then 1 tail. These two outcomes are equally as likely as any of the other combinations that can be obtained from 21 flips of a coin. All of the 21-flip combinations will have probabilities equal to 0.521, or 1 in 2,097,152. Assuming that a change in the probability will occur as a result of the outcome of prior flips is incorrect because every outcome of a 21-flip sequence is as likely as the other outcomes. In accordance with Bayes' theorem, the likely outcome of each flip is the probability of the fair coin, which is 1/2.
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 1/16 (6.25%). If a win is defined as rolling a 1, the probability of a 1 occurring at least once in 16 rolls is: