# 7.2 - Expectations and the Law of Large Numbers

7.2 - Expectations and the Law of Large Numbers## Example 7.4

The most recent Life Tables for the U.S. Population indicates that the probability that a 63 year old man survives to his 64th birthday is 0.9861. An insurance company sells a \$100,000 one-year life insurance policy to 63 year old men for \$1600. How much money do they make on average per policy sold?

This is an illustration of the **Expected Value**, which is the long run average value of a measurement. Here the measurement is the profit made by the insurance company for a random male 63 year old customer. To compute the expected value, you sum over the possible values times the probability of getting that value.

## Example 7.5

Among passenger cars traveling on the Pennsylvania Turnpike on Thanksgiving weekend: 50% have only the driver; 30% have 2 people in the car; 10% have three people and 5% have 4 people, and 5% have 5 people. You stop along the turnpike. What is the expected number of people in the next car you see pass you?

The expected value denotes a long run average when the basic chance process is repeated over and over. The Law of Large Numbers indicates how this behavior works.

## The Law of Large Numbers:

Averages or proportions are likely to be more stable when there are more trials while sums or counts are likely to be more variable. This does not happen by compensation for a bad run of luck since independent trials have no memory.

## Example 7.6

Among passenger cars traveling on the Pennsylvania Turnpike, 30% use Fast-Pass.

* * Which is more likely?

- That between 25% and 35% of the next 100 cars pay with a Fast-pass, or
- That between 25% and 35% of the next 800 cars pay with a Fast-pass

* * Which is more likely?

- That more than 35% of the next 100 cars pay with a Fast-pass, or
- That more than 35% of the next 800 cars pay with a Fast-pass

* * Suppose that I see 10 cars in a row with a “Fast-pass” going through the toll area.

**True or False**: the Law of Averages says that we are now more likely to see cars without the Fast-pass in the next group of 10 cars.

## Example 7.7

Couples eating at a restaurant on Valentine’s Day spend an average of $50.

Which is more likely?

- The average amount spent by the next 10 couples is between \$40 and \$60, or
- The average amount spent by the next 40 couples is between \$40 and \$60