Gambler's fallacy

The gambler's fallacy is a logical fallacy, the false idea is based on a random event will more likely if it will not occur, or less likely, if it is recently / frequently occurred.

This mistake is common in everyday life also in the evaluation of such probabilities that are already carefully analyzed. Many people squander money on his account. The refutation of this idea can be summed up in the phrase: "Chance has no memory. "

The gambler's fallacy is sometimes seen as a thinking error that is generated by a psychological heuristic process called representativeness heuristic.

Example: coin toss

The gambler's fallacy can be illustrated by considering the repeated toss of a coin. For a flawless coin the chances for " head " or " number" exactly 0.5 ( half). The chance for head twice in succession is 0.5 × 0.5 = 0.25 (one quarter ). The probability of head three times in succession is 0.5 × 0.5 × 0.5 = 0.125 (one-eighth ), etc.

Suppose it were just four times in succession head thrown. A player might say, " If the next coin toss results in head again, the head would be five times already after the other. The probability of such a series is 0.55 = 0.03125 "So you think that the chance that the coin comes up heads the next time, 1: . Amounts to 32.

Here is the problem. When the coin is error free, the probability for " count " should always be 0.5, no more or less, and the probability of the "head" must be always 0.5, no more or less. The probability of 1:32 ( 0.03125 ) for a series of five heads is only valid before one has thrown the first time. The same probability 1: 32 also applies to four times " head " followed by one " number" - and every other possible combination. After each throw its result is known and no longer count. Each of the two options "head" or " number" has the same probability, no matter how many times the coin was tossed and what came of it. The error is based on the assumption that previous litters may make the coin more likely to fall on the "head" than " number"; that is, that a lucky streak past could somehow affect the odds of the future.

Sometimes players argue, with regard to the law of large numbers, like this: "I've just lost four times. The coin is fair, so everything is balanced in the long run. If I just keep playing, I 'll win back my money. " However, it is irrational, the " " to begin the long term to the point at which the player began to play. Just as well he might in the long run expected to land back to its present position ( four losses).

Mathematically, the probability is 1 that is gains and losses cancel at some point and that a player reaches his starting balance again. However, the expected value of the necessary games is infinite, and also those for the capital outlay. A similar argument shows that the popular doubling strategy ( start with € 1, and if you lose, put € 2, then € 4 and so on until you win ) does not necessarily work ( cf. Martingale, St. Petersburg Paradox ). Such situations are in the mathematical theory of random walks (literally, random walks) explored. The doubling and similar strategies either replace many small gains against some big losses, or vice versa. With working capital to an unlimited amount if they were successful. In practice, however, it is sensible to only put a fixed amount because the loss per day or hour is then easier to estimate.

Apparent player fallacies

There are many scenarios in which the gambler's fallacy is present only at first glance.

• If the probabilities of successive random events are not independent, the chance for future events of past events can be changed. An example of this are playing cards that are drawn without replacement from a deck. If the first card drawn a jack, is the probability, nor to go with the second one, smaller than if the first card was an ace. The reason for this are only three existing boys.

• If the probability of the possible events is not as high, such as in a loaded dice, a frequent event in the past continue to occur frequently ( autocorrelation ): the distortion of the cube favors it. This variant - to believe in the fairness of the cube and to the honor of the players, although both missing - was as Nerd's Gullibility Fallacy (such as " credulity of the nerds " ) titled. It is also an example of Hume's principle: Twenty times " number" one after the other rather suggest that the coin was dovetailed, as for a fair coin, the next litter 50: 50 " head " or " number" will result.

• The probabilities of future events can be influenced by external factors; For example, rule changes in the sport might affect the chances of success of a given team.

• Some puzzles are reflected in front of the reader, they were an example of the gambler's fallacy; For example, the Monty Hall problem ( Monty Hall problem).

Conditions of a random experiment

It should be noted that the gambler's fallacy of the following train of thought is different: An event occurs frequently, therefore the probability distribution is assumed to doubt. This consideration leads to the opposite conclusion, that frequently occurring event is more likely. You may be correct, what with unknown random conditions (as they almost always present in reality), however, can only be decided with a certain probability always.