Question

Discrete Mathematics

Geico determines that for someone of your age, sex, and place of residence, the probability that you get into no accident at all in the next year is 0.95, the probability that you get into an accident where they must pay damages of $10,000 is 0.0495, and the probability that you get into an accident where they must pay damages of $1,000,000 is 0.0005. Use these probabilities to compute the expected value of the damages Geico will have to pay if they insure you. Assuming they insure many individuals with the same profile as you and they want to make a profit, should Geico charge you more, less, or equal to this expected value?

Answer 1

First we evaluate the expected value of damages.

We have 3 possibilities:

1) The insurance company has to pay no damages (no accident) with a probability of 0.95

2) Damages of $10,000 with a probability of 0.0495

3) Damages of $1,000,000 with a probability of 0.0005

The expected value is the sum of each possible outcome multiplied by its probability. We calculate this as

E = (0 × 0.95) (10.000 × 0.0495)+ (1,000,000 × 0.0005) 0 + 495 + 500 995

The expected value of damages Geico will have to pay is $995.

Now let us see how the company can make a profit. Assume that they sell policies to n people for the next year at a premium P dollars (n is assumed to be a large number).

Then the company makes an income nP for the year.

Now, we find the damages the company could expect to pay for the year. Using the given probabilities, the number of situations where they need to pay damages of $10,000 is n times 0.0495 and the number of situations where they need to pay $1,000,000 is n times 0.0005 (We can do this because n was taken to be a large number).

So the total amount paid in damages is

(10.000 × 0.0495 × n) + (1,000,000 × 0.0005 × n) 495n + 500n 995n

For the company to make a profit, the income has to be greater than the damages paid, that is

np 995n

If we cancel n on both sides, we find .

P 995

Therefore to make a profit, Geico should charge a premium more than the expected value of $995.