Player fallacy

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The gambler's fallacy ( English Gambler's Fallacy ) is a logical fallacy , which is the misconception based on a random event will if it is recently / occurred more frequently likely if it did not happen a long time, or less likely.

This mistake of reasoning is also common in everyday life when assessing probabilities that have already been carefully analyzed. Many people gamble away money because of it. The refutation of this consideration can be summarized in the sentence: "Chance has no memory."

The gambler's fallacy is sometimes viewed as a mistake in reasoning generated by a psychological, heuristic process called the representativity heuristic .

Example: tossing a coin

The gambler's fallacy can be illustrated by looking at the repeated tossing of a coin. If the coin is faultless, the chances of “heads” or “tails” are exactly 0.5 (half). The chance for two heads in a row is 0.5 × 0.5 = 0.25 (a quarter). The probability of three heads in a row is 0.5 × 0.5 × 0.5 = 0.125 (one eighth), etc.

Suppose it had just been thrown head four times in a row. A player might say to himself: “If the next coin toss were heads again, that would be heads five times in a row. The probability for such a row is 0.5 5  = 0.03125. ”So you think that the next time the coin will be heads is 1:32 (= 0.03125).

Here lies the fault. If the coin is flawless, the probability of “tails” must always be 0.5, never more or less, and the probability of “heads” must always be 0.5, never more or less. The 1:32 probability for a series of 5 heads only applies before you have thrown the first time. The same probability of 1 in 32 also applies to four consecutive “heads”, followed by “tails” once - and any other possible combination. After each throw, the result is known and no longer counts. Each of the two possibilities “heads” or “tails” has the same probability, regardless of how often the coin has already been tossed and what came out of it. The flaw is based on the assumption that previous tosses could cause the coin to fall on heads rather than tails; That is, a past streak of luck could somehow affect future betting odds.

Sometimes, referring to the Law of Large Numbers , players will argue, “I've just lost four times. The coin is fair so everything will balance out in the long run. If I just keep playing, I'll get my money back. ”However, it is irrational to start the“ long term ”from where the player started playing. He might as well expect to land back at his current position (four losses) in the long run.

Mathematically speaking, the probability is 1 that winnings and losses will cancel each other out and that a player will reach his starting balance again. However, the expected value of the games necessary for this is infinite , and also that for the capital to be used. A similar argument shows that the popular doubling strategy (start with € 1; if you lose, bet € 2; then € 4, etc. until you win) does not necessarily work (see Martingale game , Saint Petersburg Paradox ). Such situations are explored in the mathematical theory of random walks (literally: random walks ). The doubling down and similar strategies either trade many small wins for some large losses, or vice versa. With unlimited working capital, they would be successful. In practice, however, it makes more sense to only bet a fixed amount, because the loss per day or hour is then easier to estimate.

Apparent player mistakes

There are many scenarios in which the player's fallacy only appears at first glance.

  • If the probabilities of successive random events are not independent , the chance of future events can be altered by past events. An example of this are playing cards that are drawn from a pile without being replaced. If a jack was drawn with the first card, the probability of drawing another with the second is lower than if the first card was an ace. The reason for this is that there are only three jacks left.
  • If the probability of the possible events is not equally high, for example with a marked die, an event that has occurred frequently in the past can continue to occur more frequently ( autocorrelation ): the falsification of the die favors it. This variant - to believe in the fairness of the dice and in the honor of the fellow players, although neither is missing - was dubbed Nerd's Gullibility Fallacy ( something like "gullibility of the technical idiot "). It is also an example of Hume's principle: Twenty "tails" in a row are more likely to indicate that the coin has been marked than for a fair coin, the next toss of which is equally likely to result in "heads" or "tails".
  • The probabilities of future events can be influenced by external factors; z. B. Rule changes in sport could affect the chances of success of a certain team.
  • Some puzzles pretend to be an example of the gambler's fallacy, such as the goat problem (Monty Hall problem) .

Conditions of a random experiment

One event does not affect the next, but a correlation can point to underlying conditions of the experiment (e.g. unequal mass distribution of a coin).

It should be noted that the gambler's fallacy differs from the following train of thought: an event occurs frequently, so the assumed probability distribution is to be questioned. This consideration leads to the opposite conclusion, that the frequently occurring event is more likely. It can be correct, which in the case of unknown random conditions (as they practically always exist in reality) can only be decided with a certain probability.

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See also