An insurance company found that 2.5% of male drivers between the ages of 18 and

Question

An insurance company found that 2.5% of male drivers between the ages of 18 and 25 are involved in serious accidents annually. Assume that every such accident costs the company $65,000 and that a driver can only have one of these accidents in a year.

A. If the company charges $2,500 for such coverage, what is the probability that it loses money on a single policy?

B. Suppose that company writes 1,000 such policies to a collection of drivers. What is the probability that the company makes a profit on these policies? Assume that the drivers don't run into each other and behave independently.

C. Does the difference between the probabilities of parts (a) and (b) explain how insurance companies stay in business? Large auto insurers are certainly profitable. One report for example, claims that Allstate pays out less than $0.50 in accident claims for every dollar collected in premiums.

Explanation / Answer

Probability of accident = 0.025

Probability of no accident = 0.975

Thus,

for insurance company, average payout

= 2500(0.975) + (0.025 * -65000)

= 812.5 (thus company makes 812.5 dollars on an average)

For, 1000 such policies, it will as a binomial distribution with mean number of cases = 0.025 * 1000 = 25 cases

thus,

Profits = 25(-65000) + 975(2500)

= 812500

The insurance company stays in business by application of law of large numbers. The total expected loss per policy is replaced by average loss a person might face. Thus, the losses are pooled together. A large pool of premium is made which is then used to pay for any claims that occur.

Essentially, the insured losses are things with a low chance of occurence.

Hope this helps.

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