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law of large numbers

Contributor(s): Stan Gibilisco

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. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes.

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.

If, for example, 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 because they expect frequencies of outcomes to become equal. But, because each coin toss is an independent event, the true probabilities of the two outcomes are still equal for the next coin toss and any coin toss that might follow.

Nevertheless, if the coin is tossed enough times, because the probability of the either outcome is the same, the law of large numbers comes into play and the number of heads and tails will be close to equal.

This was last updated in December 2012

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The last paragraph isn't quite right. The difference between the number of heads and the of tails tends to diverge.  It's the frequency, that is, the fraction of heads that approaches a constant, 0.5 for a fair coin.
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The expected value of the difference between the number of heads and the number of tails is sqrt(N), where N is the number of tosses.
For 1 million tosses the difference between heads and tails would be generally 1 000, for 100 million tosses this difference must be around 10 000.
The relative difference as compared to the tosses is getting to zero.
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