Question

The idea of insurance is that we all face risks that are unlikelybut carry high cost. Think of a fire destroying your home. So weform a group to share the risk: we all pay a small amount, and theinsurance policy pays a large amount to those few of us whose homesburn down. An insurance company looks at the records for millionsof homeowners and sees that the mean loss from fire in a year isstudent submitted image, transcription available belowper house and that the standard deviationof the loss isstudent submitted image, transcription available below. (The distribution of losses is extremelyright-skewed: most people have $0 loss, but a few have largelosses.) The company plans to sell fire insurance for $250 plusenough to cover its costs and profit.

(a) Explain clearly why it would be unwise to sell only 12policies. Then explain why selling many thousands of such policiesis a safe business.
(b) If the company sells 10,000 policies, what is the approximateprobability that the average loss in a year will be greater than$275

Answer 1

Insurance is a method that households and firms use to prevent any single event from having a significant detrimental financial effect. Selling only 12 policies would not provide a large enough pool of premiums to cover potential losses, while selling many thousands of policies spreads the risk and reduces the impact of individual claims. The approximate probability of the average loss being greater than $275 can be calculated using the Central Limit Theorem.

Step-by-step explanation:

Insurance is a method that households and firms use to prevent any single event from having a significant detrimental financial effect. Generally, households or firms with insurance make regular payments, called premiums. The insurance company prices these premiums based on the probability of certain events occurring among a pool of people. Members of the group who then suffer a specified bad experience receive payments from this pool of money.

In the case of selling fire insurance, it would be unwise to sell only 12 policies because the company would not have a large enough pool of premiums to cover the potential losses. Selling many thousands of policies is a safe business because it allows the insurance company to spread the risk among a larger group of people, reducing the impact of any individual claim.

If the company sells 10,000 policies, the approximate probability that the average loss in a year will be greater than $275 can be calculated using the Central Limit Theorem. Since the population mean loss is given as student submitted image, transcription available belowper house, and the standard deviation of the loss is student submitted image, transcription available below, we can use these values to calculate the standard error of the mean. This can then be used to find the z-score corresponding to a loss of $275 and calculate the probability using a standard normal distribution table.