Law of Large Numbers: Definition, Examples & Statistics

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  • 0:04 The Law of Large Numbers
  • 1:28 Example: Coin Tossing
  • 3:07 Statistics & Probability
  • 3:43 Lesson Summary
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Lesson Transcript
Instructor: Vanessa Botts
In this lesson, we'll learn about the law of large numbers and look at examples of how it works. We'll also see how businesses use the law of large numbers to do things like set insurance premiums. A short quiz will follow the lesson.

The Law of Large Numbers

Have you ever seen a contest where there is a jar full of jelly beans, along with a prize for the person who guesses how many jelly beans there are inside?

If you try to guess, your answer may not come too close to the total number of jelly beans in the jar. The same may be true if you average the guesses of ten people who give it a try, but what happens if 1,000 people each take a guess and we average their guesses? Interestingly, that average will likely be a lot closer to the actual number of jelly beans in the jar.

Taking it further, if 10,000 people take a guess and we average their guesses, that number will get even closer to the actual number of jelly beans in the jar. Which means the probability of guessing the correct amount of jelly beans is higher. As a matter of fact, as the number of guesses increases, the average of the guesses will come closer and closer to the actual number of jelly beans. This is the law of large numbers in action!

The theory of the law of large numbers describes the result of performing the same experiment a large number of times. Using the example from our jelly bean contest, how would we guess the expected value - in this case, the number of jelly beans in the jar?

We start with samples of n observations (where n represents the number of guesses). Next, we average all of the observations. Then, the sample mean (the average of all the guesses) will approach the expected value (real number of jelly beans in the jar) as the sample becomes larger and larger.

Example: Coin Tossing

Another example of the law of large numbers at work is found in predicting the outcome of a coin toss. If you toss a coin once, the probability of the coin landing on heads is 50% (which can also be written as ½ or 0.5) and the chance of it landing on tails is also 50%.

But what happens if you toss a coin ten consecutive times? Can you say with certainty that there it will land on heads half the time and on tails the other half? The answer is 'no' because each coin toss is an independent event. This means that the outcome of one event, in this case a coin toss, will not affect the outcome of the next event.

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