Question

Suppose a computer company sells service plans for their laptop computers. Each service plan costs $250 and lasts for three years. The service plan covers any genuine hardware malfunctions during the warranty period but will only allow for up to one claim. Suppose that the average cost of repair is $850. Suppose also that the likelihood of a warranty claim is 22%. Let x denote the discrete random variable that represents the amounts the company (i) makes from a service plan for a person that makes no claims and (ii) loses for a service plan for a person that makes one claim. Please complete the following tasks.

1. Construct a probability distribution for x.

2. Use the probability distribution to calculate the expected value μ.

3. Write an explanation for what the expected value tells us as it relates to the company’s profits on service plans. Make sure you are very specific and include units.

4. By how much does the company need to raise the price of a service plan to increase their expected value by $50? Show your work.

Answer 1

Solution:-

According to question,

The computer company sells service plans each costs $250 that lasts for three years. This service plan covers any genuine hardware malfunctions during the warranty period but will only allow for up to one claim.

The average cost of repair is $850 and the warranty claim is about 22% and x denote the discrete random variable that represents

(i) The amount of money that company makes from a service plan for a person that makes no claims = $250 .

(ii) The loses for a service plan for a person that makes one claim = $250- $850 = -$600 (loss of $600).

Now,

Since 22% of customer makes 1 clams.

So, probability that company loss $600 is

P(x= -$600) = 22/100 = 0.22

And probability that company makes $ 250 is

P(x = $250) = 1-0.22 = 0.78

So, the probability distribution of x is

x(dollars)	probability P(x)	xP(x)
250	0.78	195
-600	0.22	-132

Now, the expected value is the summation if xP(x)

E(x) = 195 -132 = 63 dollars

The expected value ($63) tells us that the company makes a profits of about $63 on each service plans.

It tell us that on an average company makes $63 on evah service plan.

Let the company raise the price of a service plan to $y to increase their expected value by $50 .

So, now new probability distribution table will be

is

x(dollars)	probability P(x)	xP(x)
250+y	0.78	(250+y)0.98=195+0.78y
(250+y-850) = y-600	0.22	(y-600)0.22=0.22y -132

Now, the expected value is the summation if xP(x)

E(x) = 63 + 50 dollars ( $50 increase in prevevious value)

So, 195 +0.78y +0.22y -132 = 63 +50

Or 63 + 1y = 63 +50

Or y = 63 +50 -63

Or y = 50 dollars.

Hence, to increase expected value by $50 , the company need to raise the price of service plan also by $50.