Question

In Florida, an insurance company charges an annual premium of $6,500 to insure a home worth $325,000 for its full value against damage from a hurricane. The insurance company figures that the probability of total destruction of the house from a hurricane is 0.02.

The same insurance company charges an annual premium of $1,800 to insure a rebuilt classic motorcycle worth $28,000 for its full value against theft. The insurance company figures that the probability of the motorcycle being stolen is 0.05.  

These two insurance policies are owned independently by different homeowners, in different locations in the state of Florida.

Calculate the company’s expected profit for insuring the house and the motorcycle for one year against complete loss. Let X = a random variable that denotes the insurance company’s profit for the year that the two policies are in place.

1. Fill out the table below that defines the probability distribution of the random variable X.

Possible outcomes	X =insurance company profit ($)	p(x)	X*p(x)

2. Calculate the insurance company’s expected profit, E(x), for these two policies for one year.

                                                                                                                                 ____________

3. Assume that the motorcycle annual premium remains at $1,800. What should the insurance company charge as a premium for the home hurricane insurance, if they want the expected profit, E(x), to equal a total of $1,000 for these two policies for one year?

Answer 1

a)

Possible outcomes	Profit (X)	p(x)	X*P(x)
house insurance	6500	0.98	6370
	-325000	0.02	-6500
Motorcycle insurance	1800	0.95	1710
	-28000	0.05	-1400
		Sum	180

b)

The expected profit from the two insurance policies is,

c)

Let the new insurance premium for the home insurance policy is Y, then then the expected profit is $1000,

Possible outcomes	Profit (X)	p(x)	X*P(x)
house insurance	Y	0.98	0.98Y
	-325000	0.02	-6500
Motorcycle insurance	1800	0.95	1710
	-28000	0.05	-1400
			-6190+0.98Y

Hence the insurance company should charge $7336.735 for home insurance policy.

Possible outcomes	Profit (X)	p(x)	X*P(x)
house insurance	7336.735	0.98	7190
	-325000	0.02	-6500
Motorcycle insurance	1800	0.95	1710
	-28000	0.05	-1400
			1000