Free SOA Exam P (Probability) Practice Questions

SOA Exam P tests your mastery of probability concepts essential to actuarial science. With over 1,100 practice questions spanning seven core topics, you can build the quantitative foundation needed to pass this first actuarial exam.

1116 Questions
7 Topics
24 Lessons
3 Difficulty Levels
2026 Syllabus
100% Free

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Sample Questions

Question 1 Easy
X ~ Beta(3, 2). Find E[X].
Solution
For Beta(α\alpha,β\beta): E[X]E[X] = α\alpha/(α\alpha+β\beta) = 33+2\frac{3}{3+2} = (3)5\frac{(3)}{5}.
Question 2 Medium
X ~ Bernoulli(p). Find E[X³].
Solution
Since X only takes values 0 and 1: X^3 = X. Therefore E[X3]E[X^3] = E[X]E[X] = p.
Question 3 Hard
X has PDF f(x) = 2/(1+x)³ for x > 0. What is E[X]?
Solution
Verify: ∫0∞2(1+x)3dx=2⋅[−12(1+x)2]0∞=2⋅12=1\int_0^\infty \frac{2}{(1+x)^3}dx = 2 \cdot \left[-\frac{1}{2(1+x)^2}\right]_0^\infty = 2 \cdot \frac{1}{2} = 1. ✓

E[X]=∫0∞2x(1+x)3dxE[X] = \int_0^\infty \frac{2x}{(1+x)^3}dx

Let u=1+xu = 1+x, du=dxdu = dx, x=u−1x = u-1:
=∫1∞2(u−1)u3du=2∫1∞(u−2−u−3)du=2[−u−1+u−22]1∞= \int_1^\infty \frac{2(u-1)}{u^3}du = 2\int_1^\infty (u^{-2} - u^{-3})du = 2\left[-u^{-1} + \frac{u^{-2}}{2}\right]_1^\infty
=2[(0−0)−(−1+1/2)]=2×1/2=1= 2\left[(0-0) - (-1 + 1/2)\right] = 2 \times 1/2 = 1

This is a Pareto II with α=2,θ=1\alpha = 2, \theta = 1: E[X]=θ/(α−1)=1/1=1E[X] = \theta/(\alpha-1) = 1/1 = 1.

Distractor analysis:
- 0.500: Computes θ/(α)=1/2\theta/(\alpha) = 1/2.
- 1.500: Computes αθ/(α−1)=2\alpha\theta/(\alpha-1) = 2... no. Uses different Pareto.
- 2.000: Uses α=2\alpha = 2 directly.
- ∞: Thinks the integral diverges (it would if α≤1\alpha \le 1).

The answer is 1.0001.000.

Sample Lesson

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Joint Probability Functions and Joint CDFs for Discrete Random Variables

Multivariate Random Variables · 12 min read

Pricing a joint-life policy requires a joint probability model for two correlated lifetimes. Get the joint distribution wrong and the premium is off by tens of thousands. Exam P tests whether you can build, read, and extract probabilities from joint PMF tables.

Read the Full Lesson + 23 More →

Topics

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