The Law of Large Numbers
Date: 02/08/2001 at 17:30:41 From: Garrick Rothstein Subject: Laws of large numbers Can you explain the law of large numbers to me?
Date: 02/09/2001 at 11:31:54 From: Doctor Jordi Subject: Re: Laws of large numbers Hi Garrick - thank you for writing to Dr. Math. You want to know about the law of large numbers? Sure, I can explain it to you. It really is not something you don't already know from common experience, but perhaps its mathematical formulation is not quite as simple. Simply stated, the law of large numbers states that if you repeat a random experiment, such as tossing a coin or rolling a die, many, many, many times, your outcomes should on average be equal to the theoretical average. For example, say you have a fair coin, such that the odds of getting heads are the same as the odds of getting tails. Say you toss this coin 100 times. How many heads and how many tails would you expect to get? Well, you can't predict exactly how many tails and heads you'll get. But you can probably give me an estimate: about 50 heads and about 50 tails. Maybe a little more on one side or the other, say maybe 48 heads and 52 tails, but somewhere in that range. What would you think if you got 90 heads and only 10 tails? Even worse, what if you got all heads and no tails at all? Is that impossible? Well, certainly not - it is conceivable that by a great coincidence you could toss the coin 100 times and get all heads, but then something very fishy would be afoot. You would have a very strong reason to suspect that your coin was not fair at all. Basically, that's the law of large numbers. By the law of large numbers, if you toss the coin many times, say 100, you should expect to get about the same number of heads and tails. Furthermore, if you tossed it many more times than 100, say 10,000, you would expect to get even closer to a 1:1 ratio of heads and tails (closer and closer to 50% heads and 50% tails). The more times you repeat the experiment, the closer you should get to the true theoretical average of heads and tails. On the Web, see Philip B. Stark's Glossary: Law of Large Numbers: http://www.stat.berkeley.edu/~stark/SticiGui/Text/gloss.htm#l and his Java applet, which lets you change the number of trials and the probability of success in each trial, and toggle between viewing either the difference between the number of successes and the expected number of successes, or the difference between the percentage of successes and the probability of success in each trial: http://www.stat.berkeley.edu/~stark/Java/lln.htm If this is still confusing, or if you need a more solid mathematical formulation of the law of large numbers, please write back. - Doctor Jordi, The Math Forum http://mathforum.org/dr.math/
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