# The Law of Large Numbers

## Law of Large Numbers

After a large number of trials the average value should be close to the expected value and gets closer to the expected value as more trials are performed. This law is the foundation for the ___.

## Example

After rolling two six sided dice, you take their sum. What is the probability that the sum is at most 1.

Below in black is a relative frequency histogram showing the percentile chance of rolling each sum in the long run. (Note the vertical axis is written as a percent.) What is the expected (average) value?

After simulating a dice roll the empirical relative frequencies will be shown as a histogram in red. After one roll what do you expect the relative frequency histogram to look like? What do you guess the empirical average value to be?

## Beyond the Law of Large Numbers

We see that the mean value draws closer to the expected value of 7 as we make many trials. What else do you notice about these two histograms as the number of trials increase?

## Deviations from the Mean

A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital 15 babies are born each day. Although the overall proportion of boys is about 50 percent, the actual proportion at either hospital may be more or less than 50 percent on any day. At the end of a year, which hospital will have the greater number of days on which more than 60 percent of the babies born were boys? Assume that the probability that a baby is a boy is .5 (actual estimates make this more like .513).

Explain

## Example: Valentine's Day

On Valentine’s Day a local store sold 9 gifts for pets and 133 gifts for humans. What is the probability that a random gift in the checkout line is for a pet next Valentine’s Day? Round to three significant digits.

What method did you use?

You are trying to find the probability of rolling a 6 sided die and getting a number less than 5. Call this event A. What is event A? A={1, 2, 3, 4 }. What is the sample space?

Find the probability of event A.

What method did you use?