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Solve Real Problems

Apply your math skills to actuarial exam questions.

Actuaries earn professional credentials by passing a series of examinations. This online exam is designed to give you an idea of the types of questions you might encounter on the preliminary actuarial examinations administered by the Casualty Actuarial Society and Society of Actuaries. The sample problems are actual questions from prior exams, but they do not cover all the topics or all levels of difficulty.

Answer the five multiple choice questions below, then click submit to see your results.

1

A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is 3/5.

The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months.

Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.

2

A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold.
P(X = 0, Y = 0) = 1 / 6
P(X = 1, Y = 0) = 1/12
P(X = 1, Y = 1) = 1 /6
P(X = 2, Y = 0) = 1 /12
P(X = 2, Y = 1) = 1 /3
P(X = 2, Y = 2) = 1/6

What is the variance of X?

3

A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 (ticket cost + 50 penalty) to the tourist. What is the expected revenue of the tour operator?

4

An auto insurance company insures an automobile worth 15,000 for one year under a policy with a 1,000 deductible. During the policy year there is a 0.04 chance of partial damage to the car and a 0.02 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function

What is the expected claim payment?

5

Claim amounts for wind damage to insured homes are independent random variables with common density function

where x is the amount of a claim in thousands.

Suppose 3 such claims will be made.

What is the expected value of the largest of the three claims?