Free SOA Exam P (Probability) Discrete Distributions Practice Questions

Master discrete distributions tested on SOA Exam P, including binomial, Poisson, geometric, negative binomial, and hypergeometric. These questions require you to identify the right distribution and compute exact probabilities.

82 Questions
49 Easy
20 Medium
13 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
X ~ Poisson(7). Find P(X = 7).
Solution
P(X=7)P(X=7) = e^{-}⁷ ×\times7⁷ / 1!≈1! \approx0.149.
Question 2 Medium
N ~ Poisson(λ) and P(N=1) = 2·P(N=2). Find λ.
Solution
P(N=1)P(N=1) = λ\lambdae^{-}λ\lambda. P(N=2)P(N=2) = λ\lambda^2e^{-}λ\lambda/2. Setting λ\lambdae^{-}λ\lambda = 2⋅λ\cdot \lambda^2e^{-}λ\lambda/2 = λ\lambda^2e^{-}λ\lambda gives 1 = λ\lambda, so λ\lambda = 1.
Question 3 Hard
N₁ ~ Poisson(3) and N₂ ~ Poisson(5) are independent. Find P(N₁ = 2 | N₁ + N₂ = 7).
Solution
Given N_1+N_2=7, N_1|N_1+N_2=7 ~ Binomial(7, (3)8\frac{(3)}{8}). P(N1=2)P(N_1=2) = (72)\binom{7}{2} ((3)8\frac{(3)}{8})^2 ((5)8\frac{(5)}{8})^5 ≈\approx0.282.
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More Exam P Topics

Probability Fundamentals Conditional Probability & Bayes Continuous Distributions Joint & Marginal Distributions Expectation & Variance Insurance Applications

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