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
E is correct.

P(X=3) = C(10,3)(0.3)Ā³(0.7)ā· = 120(0.027)(0.0824) ā‰ˆ 0.267.
Question 2 Medium
X follows a Poisson distribution with Ī» = 3. What is E[X | X ā‰„ 1]?
Solution
D is correct.

E[Xāˆ£Xā‰„1]=E[Xā‹…1Xā‰„1]P(Xā‰„1)=E[X]āˆ’0ā‹…P(X=0)P(Xā‰„1)=E[X]1āˆ’eāˆ’Ī»=31āˆ’eāˆ’3E[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}}

eāˆ’3=0.04979e^{-3} = 0.04979
1āˆ’eāˆ’3=0.950211 - e^{-3} = 0.95021

E[Xāˆ£Xā‰„1]=30.95021=3.1573ā‰ˆ3.158E[X|X \ge 1] = \frac{3}{0.95021} = 3.1573 \approx 3.158

The answer is 3.1583.158.
Question 3 Hard
Nā‚ ~ Poisson(3) and Nā‚‚ ~ Poisson(5) are independent. Find P(Nā‚ = 2 | Nā‚ + Nā‚‚ = 7).
Solution
B is correct.

Given N1+N2=7N_1+N_2=7, N1āˆ£N1+N2=7āˆ¼Binomial(7,38)N_1|N_1+N_2=7 \sim \text{Binomial}(7, \frac{3}{8}). P(N1=2)=(72)(38)2(58)5ā‰ˆ0.282P(N_1=2) = \binom{7}{2} (\frac{3}{8})^2 (\frac{5}{8})^5 \approx 0.282.
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