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Sample Actuarial 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.


An insurance company determines that N, the number of claims received in a week, is a random variable with P[N = n] = 1/2n+1, where n > 0 . The company also determines that the number of claims received in a given week is independent of the number of claims received in any other week. Determine the probability that exactly seven claims will be received during a given two week period.


Let T1 be the time between a car accident and reporting a claim to the insurance company. Let T2 be the time between the report of the claim and payment of the claim. The joint density function of T1 and T2, f(t1, t2), is constant over the region 0 < t1 < 6, 0< t2 < 6, t1 + t2 < 10, and zero otherwise. Determine E[T1 + T2], the expected time between a car accident and payment of the claim.

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.


An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company’s policyholders, 50% are standard, 40% are preferred, and 10% are ultra-preferred. Each standard policyholder has probability 0.010 of dying in the next year, each preferred policyholder has probability 0.005 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year.

A policyholder dies in the next year.

What is the probability that the deceased policyholder was ultra-preferred?


The future lifetimes (in months) of two components of a machine have the following joint density function:

What is the probability that both components are still functioning 20 months from now?