Free SOA Exam P (Probability) Insurance Applications Practice Questions

Apply probability concepts to insurance settings as tested on Exam P. Questions involve deductibles, policy limits, loss distributions, and expected claim payments — bridging pure math to actuarial practice.

131 Questions
65 Easy
38 Medium
28 Hard
2026 Syllabus
100% Free

Sample Questions

Question 1 Easy
Losses are lognormal with ln(X) ~ N(7, 4). Calculate E[X].
Solution
E[X]E[X] = e^{(μ\mu + σ\sigma^22\frac{2}{2})} = e^{(7 + 2)} = e^{9.}
Question 2 Medium
X has PDF f(x) = 3/x⁴ for x > 1 and S(x) = 1/x³. Policy deductible is 2. Calculate E[(X−2)₊].
Solution
E[(X−2)+]E[(X−2)₊] = ∫\int S(x) dx = ∫\int x^{-}^3 dx = [−x^{-}^22\frac{2}{2}]_2^∞\infty = (1)8\frac{(1)}{8}.
Question 3 Hard
X ~ Uniform(0, 500). Policy: deductible 100, max payment 300. Calculate Var(payment).
Solution
Y = min(max(X−100,0), 300). Cap at X=400. E[Y]E[Y] = ((1)500\frac{(1)}{500})(121^{2}/2) + (300)5\frac{(300)}{5} = 90 + 60 = 150. E[Y2]E[Y^2] = ((1)500\frac{(1)}{500})(131^{3}/3) + 121^{2}/5 = 18000 + 18000 = 36000.
Var = 36000 − 22500 = 13500.
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More Exam P Topics

Probability Fundamentals Conditional Probability & Bayes Discrete Distributions Continuous Distributions Joint & Marginal Distributions Expectation & Variance

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