Gambler's Fallacy
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[edit] Definition
The gambler's fallacy is sometimes called The Law of Averages by people who think that it's true. This term is also, however, also used to refer to the perfectly valid statistical principle which mathematicians prefer to call the Law of Large Numbers.
[edit] Example 1: the "law of averages"
The most usual form of the gambler's fallacy is to believe that after a sequence of one result (let's say tossing five heads in a row) makes it more likely that the coin will come up tails the next time.
But unlike the gambler, the coin has no recollection of how it has fallen on the previous occasions; it does not share the gambler's conviction that he is "due" to get tails.
If the gambler's fallacy was correct, then it would be possible to cheat at heads and tails by privately "pre-tossing" a coin, as it were, until you had produced a sequence of, again, let us say five heads. You could then approach some suitable sucker and bet him at even odds that the coin would come up tails, knowing that in fact the coin is biased in favor of this result.
[edit] Example 2: the "lucky streak"
Another form of the gambler's fallacy is the belief in a "lucky streak"; that one's "luck is in", that one has "hot hands".
In a sense, this is the opposite fallacy to the one in the previous example. In our first example, the gambler believes that five heads in a row is likely to be corrected by tails coming up. But the believer in the "winning streak" supposes that five wins in a row is likely to be succeeded by another win.
However, it is the same fallacy in that the gambler believes that his past experience at the gaming table can lead him to conclusions about what chance event is likely to happen next. Again, the answer to this is that coins, dice, cards, or the other usual implements of gambling have no memory. The gambler may think that he's on a winning streak, but the dice do not share his opinion.
[edit] A Note on the Term "Gambler's Fallacy"
In writing this article, we have talked of gamblers and of games of chance. The "gambler's fallacy" is by no means confined to gamblers. However, it reveals a lot about human psychology to note that this fallacy is committed by gamblers, because the gambler has many advantages over the man in the street when it comes to matters of probability.
- The habitual gambler has a lot of experience of the game of chance he's playing. Experience alone should have taught him the laws of probability, at least as they apply to his particular chosen game. But they haven't.
- The habitual gambler's judgements are not casual; he has money riding on his judgement. He has a great deal to concentrate his mind on the actual laws of probability, but he doesn't learn them.
- The underlying probabilities on which a gambler gambles are nakedly apparent. The operation of a fair coin is easy to understand; the odds on a roulette wheel are laid out, in visual form, as clearly as on a pie chart.
And yet the gambler gambles. Research by psychologists has shown that the average man or woman is just as confused about probabilities as the gamblers in our examples, and in the same way: the difference between the man in the street and the gambler in the casino is merely that the "man in the street" has not been persuaded to bet money on his false intuitions.
[edit] Some Things Which Are NOT Examples of the Gambler's Fallacy
By long observation of a roulette wheel, you might discover that it is mechanically biased towards one section of the wheel, and if the bias is sufficient, you might make money betting that way. Superficially, this might seem like a case where the gambler's fallacy works, since in this case you are using your knowledge of past events to make statistical judgements about the future. But this is, in fact, a sensible approach to the data, since in this case you do not believe that the past spins of the wheel affect its future behavior; rather, you believe that your collection of data allows you to understand the underlying statistics of the wheel better than the house: this is a sensible use of statistical analysis.
The practice of "counting cards" when playing blackjack might also seem at first glance to be a successful application of the gambler's fallacy. It is not, because the events on which one is gambling are not truly independent. There are only so many cards in a blackjack dealer's "shoe", and so the fact that he has dealt a king (for example) does indeed decrease the chances that he will deal another king subsequently.
