Probability

Question

An insurance company knows that the average cost to build a home in a new California subdivision is $300,000, and that in any particular year there is a 1 in 200 chance of a wildfire destroying all the homes in the subdivision. Based on these data, how much should the company charge for its fire insurance policy if it wants a positive expected value for each policy sold in this subdivision?

a.less than $500 per year

 

 

b. more than $1,500 per year

 

c. between $500 and $1,000 per year

Answer 1

Let x be the amount charged. The expected profit for the insurance company is computed as follows: With probability 1/200, they lose $300,000, and with probability 1 – 1/200 = 199/200, they gain x dollars. The expected value is the sum of the products of the probabilities with the profits in each case, so the expected profit would be

E(Profit) = (1/200)(-300,000) + (199/200)(x)

= 199x/200 - 1500.

If we want this to be positive, we need this amount to be greater than 0, which gives this inequality:

199x/200 - 1500 > 0, which gives

199x/200 > 1500. Now multiply both sides by 200/199:

x > (200/199) * 1500, and this is approximately

x > $1,507.54. 

If we also want this to be less than $500, we solve this:

199x/200 – 1500 < 500, so

199x/200 < 2000, and so

x < (200/199) * 2000, and this is approximately

x < $1020.05

For the expected profit to be greater than $1,500, we get:

199x/200 – 1500 > 1500

Solving as above yields approximately

x > $3,015.08

For the expected profit to be between $500 and $1000, we solve these inequalities:

500 < 199x/200 – 1500 

and 

$199x/200 – 1500 < 1000.

Following the same procedure as above yields

$1,020.05 < x < $2,512.56