Question

An insurance company will insure a $75,000 particular automobile make and model for its full value against theft at a premium of $1500 per year. Suppose that the probability that this particular make and model will be stolen is. 75. Find the premium that the insurance company should charge if it wants its expected net profit to be $2000.

Answer 1

Let X be the random variable representing whether the car is stolen or not, where X = 1 if the car is stolen and X = 0 if it is not stolen. Then, we have:

P(X = 1) = 0.75 (probability that the car is stolen)
P(X = 0) = 0.25 (probability that the car is not stolen)

Let Y be the random variable representing the insurance company's net profit. Then, we have:

Y = 1500X - P (where P is the premium charged)

The expected net profit is given by:

E(Y) = E(1500X - P)
= 1500E(X) - P

To find the premium that the insurance company should charge to have an expected net profit of $2000, we set E(Y) equal to 2000 and solve for P:

2000 = 1500(0.75) - P
P = 750

Therefore, the insurance company should charge a premium of $750 per year to have an expected net profit of $2000.