Free SOA Exam P (Probability) Discrete Distributions Practice Questions

Discrete probability distributions tested on SOA Exam P include binomial, Poisson, geometric, negative binomial, and hypergeometric distributions. Questions require identifying the correct distribution from a problem description and computing exact probabilities.

112 Questions

51 Easy

36 Medium

25 Hard

2026 Syllabus

Sample Questions

Question 1 Easy

X ~ Binomial(n=10, p=0.3). Find P(X=3).

Solution

C is correct.

P(X=3) = C(10,3)(0.3)³(0.7)⁷ = 120(0.027)(0.0824) ≈ 0.267.

Question 2 Medium

X follows a Poisson distribution with λ = 3. What is E[X | X ≥ 1]?

Solution

B is correct.

E[X | X \ge 1] = \frac{E[X \cdot \mathbf{1}_{X \ge 1}]}{P(X \ge 1)} = \frac{E[X] - 0 \cdot P(X=0)}{P(X \ge 1)} = \frac{E[X]}{1 - e^{-\lambda}} = \frac{3}{1 - e^{-3}}

e^{-3} = 0.04979

1 - e^{-3} = 0.95021

E[X|X \ge 1] = \frac{3}{0.95021} = 3.1573 \approx 3.158

The answer is

3.158

Question 3 Hard

N₁ ~ Poisson(3) and N₂ ~ Poisson(5) are independent. Find P(N₁ = 2 | N₁ + N₂ = 7).

Solution

B is correct.

Given

N_1+N_2=7

N_1|N_1+N_2=7 \sim \text{Binomial}(7, \frac{3}{8})

P(N_1=2) = \binom{7}{2} (\frac{3}{8})^2 (\frac{5}{8})^5 \approx 0.282

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