2. An insurance company found that 2.5% of male drivers between the ages of 18 t

Question

2. An insurance company found that 2.5% of male drivers between the ages of 18 to 25 are involved in serious accidents annually. To simplify the analysis, assume (artificially) the following: 1) every such accident costs the insurance company $50,000; 2) a driver can only have one of these accidents in a year; 3) the insurer charges $2,000 for a policy; 4) the accidents occur independently of each other.

a) Define a random variable X that describes the payment of an individual policy from the point of view of the insurer. What are the expected value and the SD of X?

b) Suppose that the company writes 1,000 such policies to a collection of drivers as described above. Describe the total payments from these policies by the random variable Y. What are the expected value and SD of Y?

c) i) What is the probability that the insurance company loses money?

ii) What is the probability that they make more than $500,000?

Explanation / Answer

c)

i)

ii)

n= 1000 p= 2.5/1000 = .0025 a) money that insurance company will be making = 2000*1000 = 2000000 per accident cost to company = 50000 break even point = =2000000/50000 40 b) n=1000 x=40 p=.025 mean = np = 1000*.025 = 25 sd = sqrt(npq)= sqrt(1000*.025*(1-.025)) = 4.94

c)

i)

p(y<0) =NORMDIST(0,25,4.94,TRUE) 2.08829E-07

ii)

P(Y>40) = 1-p(y<40) =1-NORMDIST(40,25,4.94,TRUE) 0.00119696

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