• 60 questions

SOA P Exam Practice Test

Unofficial probability questions in the three topics of the SOA's Exam P syllabus, each answer worked out.

P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. What is P(A ∩ B)?

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Exam mode: 30 questions, 180 minutes.

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Unofficial practice questions, not affiliated with the Society of Actuaries (SOA) or the Casualty Actuarial Society. Bank checked October 9, 2026.

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All 60 questions

General probability 16

  • 2% of parts are defective. A scanner flags 95% of defective parts and 10% of good parts. A part is flagged. What is the probability that it is defective? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome g (Bayes Theorem). Online Statistics Education, "Base Rates".
  • 30% of insured drivers are under 25. 12% of insured drivers are under 25 and filed a claim last year. What is the probability that a driver under 25 filed a claim? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome f (conditional probability). Online Statistics Education, "Basic Concepts".
  • A and B are independent with P(A) = 0.3 and P(B) = 0.6. What is the probability that neither happens? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome c (independent events). Wikipedia, "Independence (probability theory)".
  • A and B are mutually exclusive, with P(A) = 0.3 and P(B) = 0.5. What is the probability of A given that B does not happen? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome d (mutually exclusive events) and f (conditional probability).
Show all 60 questions (56 more)

General probability, continued

  • A committee of 3 is drawn at random from 5 actuaries and 4 underwriters. What is the probability that it holds exactly 2 actuaries? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome b (combinations). Online Statistics Education, "Permutations and Combinations".
  • A lot holds 5 defective parts and 15 good ones. Two parts are drawn without replacement. What is the probability that both are defective? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome e (multiplication rule). Online Statistics Education, "Basic Concepts".
  • An insurer's book is 60% low risk, 30% medium risk and 10% high risk. The yearly claim probabilities are 0.05, 0.10 and 0.30. What is the probability that a policyholder picked at random files a claim this year? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome g (law of total probability). Wikipedia, "Law of total probability".
  • Each of the 4 digits of a code is chosen independently and at random from 0 to 9. What is the probability that all 4 digits differ? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome b (permutations). Online Statistics Education, "Permutations and Combinations".
  • Events A and B both have positive probability. They are independent exactly when which statement holds? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome c (define independence). Wikipedia, "Independence (probability theory)".
  • Of 200 policyholders, 120 hold an auto policy, 90 hold a home policy and 40 hold both. One is picked at random. What is the probability that this person holds neither? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome a (sets, Venn diagrams) and e (addition rule).
  • P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. What is P(A ∩ B)? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome e (addition rule). Online Statistics Education, "Basic Concepts".
  • P(A) = 0.5, P(B) = 0.4, P(C) = 0.3, P(A ∩ B) = 0.2, P(A ∩ C) = 0.15, P(B ∩ C) = 0.1 and P(A ∩ B ∩ C) = 0.05. What is P(A ∪ B ∪ C)? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome e (addition rule). Wikipedia, "Inclusion–exclusion principle".
  • Six claim files are worked in a random order. What is the probability that two particular files are worked one right after the other? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome b (permutations). Online Statistics Education, "Permutations and Combinations".
  • Three components fail independently with probabilities 0.1, 0.2 and 0.3. What is the probability that at least one fails? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome c (probabilities of independent events).
  • Two fair dice are rolled and their sum is 8. What is the probability that at least one die shows a 6? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome f (conditional probability).
  • Which statement is one of the axioms of probability? SOA Exam P syllabus (November 2026), topic 1, General Probability, outcome a (axioms of probability). Wikipedia, "Probability axioms".

Univariate random variables 28

  • 4 of 12 claims are fraudulent. An auditor picks 3 of the 12 at random without replacement. What is the probability that exactly 1 of the 3 is fraudulent? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (hypergeometric distribution). Online Statistics Education, "Hypergeometric Distribution".
  • A claim size is exponential with mean 10. What is its median? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome c (median). NIST/SEMATECH e-Handbook of Statistical Methods, "Exponential Distribution".
  • A loss is 0 with probability 0.7, 1,000 with probability 0.2 and 5,000 with probability 0.1. The policy has a deductible of 400. What is the standard deviation of the payment, to the nearest whole number? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome f (standard deviation of the payment random variable). SOA study note P-21-05, "Risk and Insurance": Deductibles.
  • A loss is exponential with mean 1,000. The policy pays the loss up to a benefit limit of 2,000. What is the expected payment, to the nearest whole number? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome e and f (benefit limits). SOA study note P-21-05, "Risk and Insurance": Benefit Limits.
  • A loss is gamma distributed with shape 4. What is its coefficient of variation? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (coefficient of variation). Wikipedia, "Coefficient of variation".
  • A loss is uniform on (0, 1000). Given that the loss is more than 250, what is the probability that it is more than 400? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome b (conditional probabilities, uniform distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Uniform Distribution".
  • A loss is uniform on (0, 1000). The policy has a deductible of 200. What is the expected payment per loss? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome e and f (deductibles). SOA study note P-21-05, "Risk and Insurance": Deductibles.
  • A loss of 4,000 this year grows 10% with inflation next year. Next year's policy has a deductible of 600 and pays 80% of the loss above it. What does the insurer pay? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome e (deductibles, coinsurance, inflation). SOA study note P-21-05, "Risk and Insurance".
  • Claim sizes are normal with mean 2,000 and standard deviation 500. Using z = 1.645, what is the 95th percentile? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome c (percentiles, normal distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Normal Distribution".
  • Claims in a month follow a Poisson distribution with mean 2. What is the probability of at least 2 claims in a month? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (Poisson distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Poisson Distribution".
  • E[X] = 3 and E[X²] = 13. What is X's standard deviation? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (variance and standard deviation). Wikipedia, "Variance".
  • Each of 10 independent policies produces a claim with probability 0.2. What is the probability of exactly 2 claims? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (binomial distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Binomial Distribution".
  • Each sales call ends in a sale with probability 0.25, independently. What is the probability that the first sale comes on the 4th call? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (geometric distribution). Wikipedia, "Geometric distribution".
  • Independent trials each succeed with probability 0.3. What is the probability that the third success comes on the 6th trial? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (negative binomial distribution). Wikipedia, "Negative binomial distribution".
  • The life of a part is exponential with a mean of 4 years. What is the probability that it fails within 2 years? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (exponential distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Exponential Distribution".
  • N is Poisson and P(N = 0) = 0.2. What is the variance of N? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (variance). Online Statistics Education, "Poisson Distribution".
  • The number of claims N is binomial with n = 50 and p = 0.1. What is the variance of N? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (variance). Online Statistics Education, "Binomial Distribution".
  • The wait until the third claim is gamma distributed with shape 3 and scale 2 months. What is its standard deviation, in months? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (standard deviation, gamma distribution). Wikipedia, "Gamma distribution".
  • X counts independent trials through the first success, with success probability 0.2 on each trial. What is P(X > 5 | X > 2)? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome b (conditional probabilities). Wikipedia, "Geometric distribution".
  • X has cumulative distribution function F(x) = 1 − e^(−x/5) for x > 0. What is P(2 < X < 6)? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (cumulative distribution functions).
  • X has density f(x) = 2x for 0 < x < 1. What is the 90th percentile of X? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome c (percentiles).
  • X has density f(x) = 3x² for 0 < x < 1. What is the variance of X? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (variance of a continuous random variable).
  • X has density f(x) = 12x(1 − x)² for 0 < x < 1. What is E[X]? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome c (expected value, beta distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Beta Distribution".
  • X has density f(x) = 20x³(1 − x) for 0 < x < 1. What is the mode of X? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome c (mode, beta distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Beta Distribution".
  • X has density f(x) = c·x² for 0 < x < 3 and 0 elsewhere. What is c? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (probability density functions).
  • X is equally likely to be any of 1, 2, 3, 4, 5 or 6. What is the variance of X? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome d (variance, discrete uniform distribution). Wikipedia, "Discrete uniform distribution".
  • X is normal with mean 100 and standard deviation 15. What is P(X > 130)? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (normal distribution). NIST/SEMATECH e-Handbook of Statistical Methods, "Normal Distribution".
  • X takes the values 1, 2, 3 and 4 with probability p(x) = x/10. What is P(X ≥ 3)? SOA Exam P syllabus (November 2026), topic 2, Univariate Random Variables, outcome a (random variables and probability functions).

Multivariate random variables 16

  • 100 independent policies each have a claim amount with mean 250 and standard deviation 400. By the Central Limit Theorem, what is the approximate probability that total claims exceed 31,000? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome i (Central Limit Theorem). Wikipedia, "Central limit theorem".
  • A portfolio holds 25 independent claims, each with mean 400 and standard deviation 100. What is the standard deviation of the total? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome h (moments of linear combinations of independent random variables).
  • Five independent losses are each uniform on (0, 1). What is the probability that the largest is at most 0.8? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome f (order statistics). Wikipedia, "Order statistic".
  • Three machines have independent exponential lifetimes, each with mean 6 years. What is the expected time until the first failure? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome f (order statistics). Wikipedia, "Exponential distribution": distribution of the minimum.
  • Var(X) = 4, Var(Y) = 9 and Var(X + Y) = 19. What is the correlation coefficient of X and Y? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome e (correlation coefficient). Wikipedia, "Covariance".
  • X and Y are independent Poisson variables with means 2 and 3. What is P(X + Y = 4)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome g (linear combinations of independent discrete random variables). Wikipedia, "Poisson distribution": sums.
  • X and Y are independent with Var(X) = 2 and Var(Y) = 5. What is Var(3X − 2Y + 5)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome h (moments of linear combinations). Wikipedia, "Variance": properties.
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is Cov(X, Y)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome e (covariance). Wikipedia, "Covariance".
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is E[XY]? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome c (moments of joint distributions).
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is E[Y | X = 0]? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome c (moments of conditional distributions). Wikipedia, "Conditional expectation".
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is P(X + Y ≥ 2)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome a (joint probability functions).
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is P(X = 1 | Y = 2)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome b (conditional probability functions).
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is the marginal probability P(Y = 1)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome b (marginal probability functions). Wikipedia, "Marginal distribution".
  • X and Y have joint probabilities p(0,0) = 0.10, p(0,1) = 0.20, p(0,2) = 0.10, p(1,0) = 0.15, p(1,1) = 0.25, p(1,2) = 0.20. What is Var(Y | X = 1)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome d (variance of conditional distributions).
  • X is normal with mean 50 and variance 9. Y is normal with mean 40 and variance 16. They are independent. What is P(X − Y > 15)? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome g (linear combinations of normal random variables). Wikipedia, "Sum of normally distributed random variables".
  • X₁ and X₂ are independent and uniform on (0, 10). What is the larger one's expected value? SOA Exam P syllabus (November 2026), topic 3, Multivariate Random Variables, outcome f (order statistics). Wikipedia, "Order statistic".

How this list was compiled: Our own questions, written to the Society of Actuaries' published syllabus of the Probability Exam for November 2026 (https://www.soa.org/globalassets/assets/files/edu/2026/fall/syllabi/2026-11-exam-p-syllabus.pdf, read 2026-10-09): General Probability 23-30%, Univariate Random Variables 44-50%, Multivariate Random Variables 23-30%. The bank holds 16, 28 and 16 questions. Every answer is worked from the numbers the question prints and was worked out a second time on 2026-10-09. The rule each answer uses was read on 2026-10-09 in the public page named under the question: the syllabus itself, the SOA study note P-21-05 "Risk and Insurance", Online Statistics Education (D. M. Lane, ed., Rice University), the NIST/SEMATECH e-Handbook of Statistical Methods, or Wikipedia. Licence: Questions, choices and explanations are original work of this site.

How it works

This free SOA P exam practice test has 60 probability questions of our own, sorted by the three topics of the syllabus. It is unofficial and is not affiliated with, endorsed by or sponsored by the Society of Actuaries (SOA).

Read the full guide

Three topics, weighted

The SOA's syllabus for November 2026 weights General Probability at 23–30%, Univariate Random Variables at 44–50% and Multivariate Random Variables at 23–30%. This bank holds 16, 28 and 16 questions.

Full length

Exam mode draws 30 of our questions on a three-hour clock, the real exam's length. Ours have four choices, not the real five.

What the syllabus assumes

Calculus (series, differentiation and integration) and the SOA's study note Risk and Insurance, which defines deductibles, coinsurance and benefit limits.

  • The exam supplies a normal table, and you may not bring your own. Practice turning a z value into a probability by table.
  • An unanswered question is scored as incorrect, so answer every one.
  • For a deductible, coinsurance or benefit limit, write the payment as a function of the loss before you integrate.

Questions and answers

How many questions are on the SOA P exam?

30 multiple-choice questions in three hours on computer, by the SOA's syllabus for November 2026. Each has five choices, and a few pilot questions are not scored.

What score do you need to pass Exam P?

The syllabus publishes no percent. Each candidate gets a form drawn from a pool of questions, forms are scaled so their passing criteria are comparable, and an unofficial pass or fail result is emailed within one hour.

Is there an SOA P exam study guide?

The SOA publishes a syllabus, not a study guide: three topics with their learning outcomes, six suggested texts with none required, a normal table and the study note Risk and Insurance. The questions here are sorted by the same three topics.

Is this a practice test for SOA Exam P and CAS Exam 1?

Yes, in subject. The Casualty Actuarial Society calls its probability requirement Exam 1 and labels its study links Exam 1/P. Its page, read October 9, 2026, sends candidates to the SOA and other societies to sit the exam.

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