Question

You work for a small insurance company. Your company is currently insuring 900 different cars for $10,000. If a car you are insuring gets in a crash you have to pay $10,000. Each car you insure has a 5% chance of crashing each year. If 50 cars you insure crash in a given year, you have to pay 50×$10, 000 = $500, 000 in insurance payouts that year. (a) What is the expected value of the amount your company would pay out? (b) What is the standard deviation? (c) What is the probability your company has to pay out more than $750,000? (d) If your company pays out an average of $750,000 per year over the next 4 years (or more) it will go bankrupt. What is the probability this will happen?

Answer 1

Solution

Let X = Number of cars that crash in a year. Then, X ~ B(900, 0.05) ……………………………………. (1)

[Given 900 cars are insured and ‘Each car you insure has a 5% chance of crashing each year.’]

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials

and p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ……………………………..…………...……..(2)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST………....(2a)

Mean (average) of X = E(X) = µ = np……………………………………………….…………….....………..(3)

Variance of X = V(X) = σ² = np(1 – p)………………………………………………………….....…………..(4)

Standard Deviation of X = SD(X) = σ = √{np(1 – p)} ……………………..…………………....…………...(5)

Now, to work out the solution,

Given insurance pay-out per crash = $10000 ……………………………………………………………… (6)

Part (a)

Expected value of the amount the company would pay out

= pay-out per crash x expected number of crashes per year

= 10000 x 900 x 0.05 [vide (6), (3) and (1)]

= $450000 Answer 1

Part (b)

Standard deviation

= pay-out per crash x standard deviation of number of crashes per year

= 10000 x √(900 x 0.05 x 0.95) [vide (6), (5) and (1)]

= $65383.48 Answer 2

Part (c)

Probability the company has to pay out more than $750,000

= P(X > 75) [because 750000 = number of crashes per year x 10000]

= 0 [vide (2a)] Answer 3

Part (d)

As per Answer 3, probability the company has to pay out more than $750,000 per year and hence the probability that the company will go bankrupt is zero. Answer 4

DONE