The law of averages is usually mentioned in reference to situations without enough outcomes to bring the law of large numbers into effect.
A common example of how the law of averages can mislead involves the tossing of a fair coin (a coin equally likely to come up heads or tails on any given toss). If someone tosses a fair coin and gets several heads in a row, that person might think that the next toss is more likely to come up tails than heads in order to "even things out." But the true probabilities of the two outcomes are still equal for the next coin toss and any coin toss that might follow. Past results have no effect whatsoever: Each toss is an independent event.
The law of large numbers is often confused with the law of averages, and many texts use the two terms interchangeably.
However, the law of averages, strictly defined, is not a law at all, but a logic error that is sometimes referred to as the gambler’s fallacy.
The law of large numbers is a principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances.
For example, if a fair coin (where heads and tails come up equally often) is tossed 1,000,000 times, about half of the tosses will come up heads, and half will come up tails. The heads-to-tails ratio will be extremely close to 1:1. However, if the same coin is tossed only 10 times, the ratio will likely not be 1:1, and in fact might come out far different, say 3:7 or even 0:10.
The law of large numbers is sometimes referred to as the law of averages and generalized, mistakenly, to situations with too few trials or instances to illustrate the law of large numbers. This error in logic is known as the gambler’s fallacy.
Gambling.
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