# 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

An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company's employees that choose coverages A, B, and C are 1?4 , 1?3 and 5?12 respectively.

Determine the probability that a randomly chosen employee will choose no supplementary coverage.

2

A device runs until either of two components fails, at which point the device stops running.  The joint density function of the lifetimes of the two components, both measured in hours, is

f (x,y)=x+y/8 for 0< x < 2 and 0< y < 2 .

What is the probability that the device fails during its first hour of operation?

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 insurance policy pays for a random loss X subject to a deductible of C, where 0 < C < 1. The loss amount is modeled as a continuous random variable with density function

Given a random loss X, the probability that the insurance payment is less than 0.5 is equal to 0.64 .

Calculate C.

5

An insurer's annual weather-related loss, X, is a random variable with density function

Calculate the difference between the 30th and 70th percentiles of X.