Law of Large Numbers
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[edit] Definition
The Law of Large Numbers is the statement that the number of times with which a chance event has occurred, divided by the number of trials made, will tend more and more closely to approximate the actual probability of that event as the number of trials gets larger. For example, if we toss a fair coin repeatedly, the proportion of the tosses which have come up heads will tend to get closer and closer to 50% the more tosses we have made.
This law is sometimes called the Law of Averages, especially by non-mathematicians, but as this phrase is also used to refer to the error known as the Gambler's Fallacy, it can be misleading.
[edit] Discussion
The law of large numbers should not be confused with the Gambler's Fallacy. The fallacious gambler would have you believe that if you toss a fair coin a hundred times, and the first five trials all come up heads, then the coin will behave as if it knows that it has skimped on tails, and will have a tendency to make up for this on the remaining 95 trials.
This is not the case. But it is true that the statistical effect of the first five tosses will tend to be swamped by the data from the remaining 95 tosses. Even if the last 95 tosses came out, (for example) 55 heads to only 40 tails, which is a greater absolute difference in heads to tails than was produced by the first five tosses, then nonetheless this would mean that over the series of a hundred tosses, we would have gotten a 60:40 ratio of heads to tails. This is closer to being 50:50 than was produced by the first five tosses, which gave us a ratio of 100:0.
Indeed, in this example, the only way we could not get closer to a 50:50 ratio is if all of the 95 remaining trials came up heads. The chances of this are 39614081257132168796771975167 to 1 against.
[edit] Statistical Sampling
The law of large numbers is the basis of statistical sampling. Suppose, for example, we wish to know what proportion of the population likes mustard. It would be impractical to ask them all, so instead we ask a randomly chosen sample of them. If our sample is truly random, then the law of large numbers gives us confidence that as the sample size becomes large, so the proportion of pro-mustard votes we get from the people we've polled will better and better approximate the actual probability that a randomly selected member of the population is pro-mustard; and this actual probability is, of course, equal to the proportion of people in the population who like mustard.
