7.2 - Expectations and the Law of Large Numbers

Example 7.4 Section

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?

A 63 year old man will die during that year with a probability of 1 - 0.9861 = 0.0139. Thus, for every 10,000 policies sold, you would expect them to collect \$1600 from all 10,000 men (totaling \$16,000,000) and pay out \$100,000 to 139 of them (totaling \$13,900,000). That is an average profit of (\$16,000,000 - \$13,900,000) / 10,000 = \$210.

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 Section

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?

To sum the values times the probabilities we get 1(0.5) + 2(0.3) + 3(0.1) + 4(0.05) + 5(0.05) = 1.85 people is the expected value.

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: Section

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 Section

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

b. The Law of Large Numbers says you are more likely to be within 5% of what you expect for the larger number of trials.

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

a. The Law of Large Numbers says you are more likely to be more than 5% away from what you expect for the smaller number of trials.

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.

False (the cars are independent so the chance process is still the same). The Law of Large Numbers does not work by compensation.

Example 7.7 Section

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

b. The Law of Large Numbers says you are more likely to be close to what you expect (here within \$10) for the larger number of trials