SOA Exam P Practice Exam and Practice Problems
Work original Exam P problems on conditional probability, discrete and continuous distributions, deductibles and limits, joint distributions and the Central Limit Theorem, read the full solution, then sit a timed 30-question practice exam in the CBT format: 3 hours, five answer choices per question.
30 questions, five answer choices each, 3 hours, with a score by syllabus topic and a worked solution for every problem.
Or start with a free 10-question drill- 272 original Exam P problems with worked solutions
- 8 free drills: every syllabus topic plus a mixed set
- 46.7% of candidates passed in July 2026
Losses, in thousands, on a restaurant's spoilage policy have density f(x) = 3(10 − x)²/1,000 for 0 < x < 10, and 0 otherwise. The policy has an ordinary deductible of 2 (thousand). What is the expected payment per loss, in thousands?
Exam P exam format
- Questions
- 30 multiple-choice questions, 5 answer choices each (A to E), one correct. A few unscored pilot questions are mixed in; unanswered questions count as wrong
- Time
- 3 hours, about 6 minutes a question
- Where
- Computer-based at Prometric test centers
- 2027 windows
- January 14–25 · March 1–12 · May 7–18 · July 1–12 · September 16–27 · November 3–14
- Calculators
- Approved models only: BA-35, BA II Plus, BA II Plus Professional, TI-30Xa, TI-30X II (IIS solar or IIB battery) and TI-30XS MultiView (or XB battery)
- Pass mark
- Set before each exam by item response theory; 71% on recent computer-based sittings
- Pass rate
- 46.7% of 2,925 candidates in July 2026
- Fee and results
- $275 per sitting, paid to SOA. An unofficial pass or fail result by email within one hour
What’s on Exam P
The 3 topics of the current SOA syllabus with their weights, and the drills that cover them. Our practice exam draws its 30 questions in proportion to the middle of each weight range. Every drill opens with a study guide, then ten problems with a worked solution after every answer; each drill’s first attempt is free.
| Syllabus topic | Weight | Drills and study guides |
|---|---|---|
| 1. General Probability | 23–30%8 of 30 in our exam | |
| 2. Univariate Random Variables | 44–50%14 of 30 in our exam | |
| 3. Multivariate Random Variables | 23–30%8 of 30 in our exam |
Exam P study guide by topic
Sets, counting and probability rules
Venn diagrams and set algebra, the axioms of probability, permutations and combinations, mutually exclusive events and the addition and multiplication rules.
- Write every survey or policyholder question as a Venn diagram first: P(A ∪ B) = P(A) + P(B) − P(A ∩ B), and for three sets add back the triple overlap.
- Use complements for "at least one": P(at least one) = 1 − P(none).
Conditional probability and Bayes’ theorem
Independence, conditional probability, the law of total probability and Bayes’ theorem applied to risk classes, tests and claims.
- P(A | B) = P(A ∩ B) / P(B). Read the stem twice to see which event is given.
- For risk classes or test results, build a two-column table: prior × likelihood for each class, then divide by the total. That is the law of total probability and Bayes’ theorem in one step.
Discrete distributions
Binomial, geometric, negative binomial, hypergeometric, Poisson and discrete uniform variables: probabilities, means, variances and percentiles.
- Match the story to the family: a fixed number of trials is binomial, trials until the first success geometric, until the r-th success negative binomial, sampling without replacement hypergeometric, counts per period Poisson.
- Check which geometric convention the stem uses: number of failures before the first success, or number of trials including it. The means differ by 1.
Continuous distributions
Densities and CDFs, the uniform, exponential, gamma, beta and normal distributions, percentiles, the median and mode, and conditional probabilities.
- A density must integrate to 1, which fixes an unknown constant; the CDF is the integral of the density from the lower end.
- The exponential is memoryless: P(T > s + t | T > s) = P(T > t), and its median is θ·ln 2, below its mean θ.
Insurance: deductibles, limits and coinsurance
Payments under deductibles, policy limits, coinsurance and inflation, and the mean and variance of the loss and the payment.
- Write the payment as a function of the loss X first, for example Y = 0 for X ≤ d, X − d for d < X ≤ u, and u − d above, then take its expectation.
- For a nonnegative loss, E[(X − d)₊] = ∫ from d to ∞ of S(x) dx, and E[min(X, u)] = ∫ from 0 to u of S(x) dx.
Joint distributions, covariance and correlation
Joint, marginal and conditional probability functions of discrete variables, conditional means and variances, covariance and correlation.
- Marginals are row and column sums of the joint table; a conditional distribution is one row or column divided by its total.
- Cov(X, Y) = E[XY] − E[X]E[Y], and ρ = Cov(X, Y) / (σ_X σ_Y).
Order statistics, sums and the Central Limit Theorem
The minimum and maximum of independent variables, probabilities and moments of linear combinations, and normal approximations by the Central Limit Theorem.
- For independent variables, P(max ≤ x) = ∏ F_i(x) and P(min > x) = ∏ S_i(x); the minimum of independent exponentials is exponential with the rates added.
- Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y); the covariance term drops out only for independent variables.
What Exam P tests
- Univariate random variables carry 44% to 50% of the exam: the binomial, geometric, negative binomial, hypergeometric, Poisson, uniform, exponential, gamma, beta and normal distributions, and insurance payments with deductibles, coinsurance and limits.
- General probability (23% to 30%) covers sets and Venn diagrams, counting, independence, conditional probability and Bayes’ theorem.
- Multivariate random variables (23% to 30%) are discrete joint distributions with covariance and correlation, plus order statistics, linear combinations of independent variables and the Central Limit Theorem.
Common questions
How many questions are on SOA Exam P?
30 multiple-choice questions in 3 hours, taken on a computer at a Prometric test center. Each question has five answer choices, A to E, with one correct answer, and a few unscored pilot questions are mixed in. A normal distribution table is available on screen.
What is the passing score for Exam P?
SOA sets the pass mark before each exam using item response theory and publishes it with the results. Recent computer-based sittings list a pass mark of 71%, about 21 to 22 of the 30 questions, though a few unscored pilot questions sit among the 30. Unofficial pass or fail results are emailed within one hour of the exam.
What calculators can I use on Exam P?
Only the SOA-approved models: BA-35, BA II Plus, BA II Plus Professional, TI-30Xa, TI-30X II (IIS solar or IIB battery) and TI-30XS MultiView (or XB battery). You may bring more than one approved calculator; for the BA II Plus models and the TI-30X IIS, TI-30XB and TI-30XS MultiView, you show the supervisor that the memory is cleared.
How long is Exam P?
3 hours for 30 multiple-choice questions, about 6 minutes a question. Each question has five answer choices, A to E, and unanswered questions are scored as wrong, so answer every one.
Are these official SOA questions?
No. These are original practice problems written to the current SOA syllabus, with every answer recomputed independently in code and a worked solution for every problem. They are not SOA sample questions.
When is Exam P offered?
Six times a year in odd months. The 2027 windows are January 14–25, March 1–12, May 7–18, July 1–12, September 16–27 and November 3–14; registration closes about a month before each window. The fee is $275.
What is the Exam P pass rate?
46.7% of the 2,925 candidates passed the July 2026 sitting, with a 71% pass mark, according to SOA’s result sheet.
Is the lognormal distribution on Exam P?
No. Since the May 2026 sitting the syllabus lists the beta, exponential, gamma, normal and uniform as its continuous distributions, and the lognormal is gone; the January 2027 syllabus is the same. Multivariate questions use discrete joint distributions, plus order statistics, linear combinations of independent variables and the Central Limit Theorem.
Ready for exam day?
30 questions, five answer choices each, 3 hours, with a score by syllabus topic and a worked solution for every problem.
Or start with a free 10-question drillMore SOA exam practice
One Exam P and FM pass covers both timed practice exams and unlimited drills.
Exam format, fees, syllabus, dates and pass rates checked November 2026: SOA Probability Exam syllabus (January 2027) · SOA July 2026 Exam P results · SOA: Probability (P) Exam · SOA: Financial Mathematics (FM) Exam · SOA exam schedule · SOA: identification and approved calculators.
Original independent practice problems written to the published SOA syllabus. Not affiliated with or endorsed by the Society of Actuaries (SOA).