Question

A cell phone company offers a simple extended warranty plan. If your phone is damaged, they will repair it for up to $50. If you lose or destroy your phone, they will give you a $200 voucher towards a new phone. The company believes that 5% of customers will need the replacement voucher and 10% will request a repair. 1. If the company charges $25 for this extended warranty, what is the expected value of the profit they will earn? 2. What is the standard deviation of their profit? 3. Suppose the company collects 10 warranty plans on one day. What is the mean of the company's total profit? 4. What is the standard deviation of the 10 total warranty plans? What assumption does this calculation require? Do you think this assumption is reasonable? 5. What are the mean and standard deviation for the profit on a 1000 plans? 6. What do your answers to the previous question tell you about the company's likelihood of making a profit? 7. Is the $25 warranty a wise purchase for you? Given that you will probably buy dozens of devices over the next decade, are these types of warranties a wise purchase for you?

Answer 1

ANSWER::

1. Let X denotes the amount of claim customer makes.

Since, X can takes values $50 and $200 with probability 10% and 5% respectively.

So, X : $50. $200

P(X): 10%. 5%

E(X)=$[( 50*10/100)+(200*5/100)]

So, expected loss = $15

And premium money as per the question is $25.

Thus, expected profit = $(25-15) =$10

2. Standard deviation of expected profit=(variance of expected profit)^(1/2) =[V(X)^(1/2)]= [ E(X^2) - E(X)^2]^(1/2)

Now, E(X^2)=[(50)2*(10/100)+(200)2*(5/100)]

E(X^2)= 250+2000=2250

From solution for question 1 we have E(X)=15

So, standard deviation of expected profit=

[2250 - (15)2](1/2) = ​​​​​$45

3.given in question, no. Of plans purchased=10

So, total earnings= $(25*10)=$250

No. Of person claims for voucher among 10 who purchased plan= (5/100)*10= 0.5 (which is not possible as 0.5 person doesn't exist)

So,Amount from voucher of mobile=0

Now, no.of people who are asking for repair among 10 who bought plan = 10*(10/100)=1

Amount money of repair= $50

Total profit= $(250-50) =$200

Mean of profit=$(200/10)=$20

5.total no.of plans = 1000

Total earnings=$( 25*1000)=$25000

No. Of people claiming for voucher=(5%) of 1000=50

Total amount of money company losses on voucher=$(200*50)=$1000

No.of people claiming for repair= 10% of 1000=100

So, amount that company losses due to repairment

= $(50*100)=$5000

Total amount that company losses =$(1000+5000)=$6000

Total profit of company= $(25000-6000)=$19000

Mean profit of company=$(19000/1000)=$19

6. There are 1000 plans out of which claims for repair=10% of 1000=100

Claims for voucher=5% of 1000=50

Companies likelihood for making profit=

(5/100)50*(10/100)100=( 550)/1051=1/(250*10)

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