10.6 PROJECT

EXPECTED VALUE AND INSURANCE PREMIUMS

When you purchase a homeowners insurance policy, you pay a certain amount of money (the premium) to the insurance company. If nothing happens to your home, the insurance company keeps the money. If you make a claim, the company will pay to fix your home, usually spending a lot more than what your premium was. Due to this set up, the insurance company makes money on some policies and loses money on others.

In this activity, we will investigate the way an insurance company computes insurance premiums.

Suppose that in a certain neighborhood, an insurance company has used historical data to determine that the probability of a house fire occurring at a home over the course of one year is $0.02\%$.

1. What is the probability that there will not be a fire at a house in the neighborhood over the course of one year?

Assume that the insurance company charges a $\mathrm{\$}300$ annual premium for fire insurance. If there is a fire, the insurance company will pay out $\mathrm{\$}\mathrm{200,000}$ to the homeowner.

2. Fill in the table below. What happens when there is no fire? Explain why this is.

   Probability and Payout for Insurance Company

   Event	Probability	Payout − Premium
   Fire
   No Fire

3. Determine the expected value that the insurance company will pay out.

4. What does the expected value say about the average loss or gain for the insurance company on each policy they sell?

5. How much would the insurance company expect to earn on average if it sold $\mathrm{10,000}$ policies in a year?