FREE Exam P Study Guide 2026: Probability for the SOA/CAS Exam

This free Exam P study guide teaches the probability you need for the Society of Actuaries (SOA) / Casualty Actuarial Society (CAS) Probability Exam — every topic the official syllabus tests, organized the way the exam is built.^[2] Exam P is the first exam on the path to actuarial credentials: a three-hour, calculus-based, 30-question multiple-choice test of how you model and quantify risk.^[1]

The guide covers the three official topics — General Probability, Univariate Random Variables, and Multivariate Random Variables— broken into eight study modules. It’s interactive, not a wall of text: every module has a built-in checkpoint quiz, hover-able glossary terms, worked examples, and concept questions, so you learn by doing. Because Exam P assumes calculus, you’ll integrate densities and differentiate moment generating functions throughout.^[2]

Already comfortable with probability and looking ahead to financial mathematics? When you’re ready, move on to our Exam FM study guide. Read this guide module by module, test yourself at each checkpoint, then round out your free Exam P prep with our practice questions and flashcards.

Exam P at a Glance

The two random-variable topics together are roughly 85% of the exam, so most of your study time belongs in distributions.^[2] But General Probability is the language everything else is written in — get fluent with conditional probability and Bayes first:

The SOA reports topic shares as ranges, so the exact mix shifts slightly from form to form.^[2] This guide teaches all three topics as eight modules — the General Probability foundation first, then the univariate distributions, then the multivariate material.

1 · Foundations of Probability

Part of General Probability (23–30%). Every Exam P problem rests on the same foundation: a sample space of outcomes, events as subsets, and the three axioms of probability.^[2]

Sample Spaces, Events & Axioms

A probability is a function on events satisfying three axioms: $P(A) \ge 0$, the whole sample space has $P(S) = 1$, and for mutually exclusive events probabilities add. From these come the complement rule $P(A^c) = 1 - P(A)$ and the bounds $0 \le P(A) \le 1$.

Addition Rule & Combinatorics

The general addition rule is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$; the last term is 0 when the events are mutually exclusive. To count equally-likely outcomes, use a permutation when order matters and a combination when it does not:

Venn Diagrams & De Morgan’s Laws

A Venn diagram turns set algebra into pictures. De Morgan’s laws let you switch between unions and intersections: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$. For three events the inclusion-exclusion principle adds the singles, subtracts the pairs, and adds back the triple.

2 · Conditional Probability & Bayes’ Theorem

The heart of General Probability (23–30%).Conditional probability, independence, the law of total probability, and Bayes’ theorem appear on nearly every Exam P form.^[2]

Conditional Probability & Independence

Conditional probability is $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, which rearranges to the multiplication rule $P(A \cap B) = P(A \mid B)\,P(B)$. Events are independent exactly when $P(A \cap B) = P(A)\,P(B)$ — equivalently when $P(A \mid B) = P(A)$.

Law of Total Probability

When an event A can happen through several mutually exclusive causes, the law of total probability averages over them: $P(A) = \sum_i P(A \mid B_i)\,P(B_i)$. Picture a tree: each branch is a cause with its probability, and A occurs along that branch with a conditional probability.

Bayes’ Theorem

Bayes’ theorem reverses the conditioning to update a cause given the evidence: $P(B_k \mid A) = \dfrac{P(A \mid B_k)\,P(B_k)}{P(A)}$. The denominator is just $P(A)$ from the law of total probability.

3 · Random Variables & Their Functions

The start of Univariate Random Variables (44–50%) — the biggest topic. A random variable assigns a number to each outcome. Master the functions that describe it, then the named distributions fall into place.^[2]

pmf, pdf & CDF

A discrete variable is described by a probability mass function $p(x) = P(X = x)$, whose values sum to 1. A continuous variable uses a probability density function $f(x)$, where probability is area: $P(a \le X \le b) = \int_a^b f(x)\,dx$. Both share a cumulative distribution function $F(x) = P(X \le x)$, and for a continuous variable $f(x) = F'(x)$.

Mean, Median, Mode & Percentiles

The expected value is the long-run average, $E[X] = \int x\,f(x)\,dx$ for a continuous variable. The median solves $F(m) = 0.5$, the mode maximizes the density, and the p-th percentile solves $F(x) = p$.

4 · Discrete Distributions

Univariate Random Variables (44–50%). The named discrete families — and knowing when each applies — are bread-and-butter Exam P points.^[2]

Bernoulli, Binomial & Hypergeometric

A single success/failure trial is Bernoulli (mean $p$, variance $p(1-p)$). The binomial counts successes in $n$ independent trials: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$, mean $np$, variance $np(1-p)$. Use the hypergeometric instead when you sample without replacement from a finite population.

Poisson

The Poisson models counts of rare events at rate $\lambda$: $P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!}$. Uniquely, its mean and variance are both $\lambda$. Sums of independent Poissons are Poisson, with rates adding.

Geometric & Negative Binomial

The geometric counts trials until the first success (mean $1/p$) and is the discrete memoryless distribution. The negative binomial generalizes it to the trials needed for $r$ successes.

5 · Continuous Distributions

Univariate Random Variables (44–50%). The continuous families you integrate — uniform, exponential, gamma, beta, and the all-important normal.^[2]

Uniform

The continuous uniform on $[a, b]$ has constant density $\dfrac{1}{b-a}$, mean $\dfrac{a+b}{2}$, and variance $\dfrac{(b-a)^2}{12}$.

Exponential & Gamma

The exponential models a waiting time with rate $\lambda$: density $\lambda e^{-\lambda x}$, mean and standard deviation both $1/\lambda$, variance $1/\lambda^2$. It is memoryless and has a constant hazard rate. A gamma is a sum of independent exponential waits.

Normal & Beta

The normal is the symmetric bell curve. Standardize any normal value with $z = \dfrac{x - \mu}{\sigma}$, then read the standard-normal table the exam provides.^[2] The empirical rule is the fast check:

The beta distribution lives on $[0, 1]$ and is used to model proportions and probabilities.

6 · Expectation, Variance & Insurance Applications

Univariate Random Variables (44–50%). Moments summarize a distribution, and the syllabus explicitly tests turning a loss random variable into a payment random variable.^[2]

Moments, Variance & Transformations

Variance measures spread: $\text{Var}(X) = E[X^2] - (E[X])^2$. Under a linear transform $Y = aX + b$, the mean scales and shifts ($E[Y] = aE[X] + b$) while the variance ignores the shift and scales by the square: $\text{Var}(Y) = a^2\,\text{Var}(X)$.

Skewness, MGFs & the Coefficient of Variation

The third central moment measures skewness: positive means a long right tail (mode < median < mean). The coefficient of variation is $\sigma/\mu$. The moment generating function $M(t) = E[e^{tX}]$ generates moments by differentiation — $M'(0) = E[X]$, $M''(0) = E[X^2]$ — and uniquely identifies a distribution.

Deductibles, Coinsurance & Policy Limits

Exam P turns a loss $X$ into a payment through policy provisions. With an ordinary deductible $d$, the payment per loss is $\max(X - d,\ 0)$; coinsurance multiplies by a fraction $\alpha$; and a policy limit caps the payment.

7 · Joint, Marginal & Conditional Distributions

The start of Multivariate Random Variables (23–30%). Now two or more variables interact. Master the joint distribution and you can extract everything else from it.^[2]

Joint & Marginal Distributions

A joint distribution gives $p(x, y) = P(X = x, Y = y)$ (discrete) or a joint density $f(x, y)$ (continuous). The marginal distribution of one variable comes from summing or integrating out the other: $f_X(x) = \int f(x, y)\,dy$. Variables are independent exactly when the joint factors into the marginals, $f(x, y) = f_X(x)\,f_Y(y)$.

Conditional Distributions & Expectation

The conditional density of $Y$ given $X = x$ is the joint over the marginal, $f_{Y \mid X}(y \mid x) = \dfrac{f(x, y)}{f_X(x)}$. The conditional expectation $E[X \mid Y = y]$ is the mean of that conditional distribution. The double-expectation rule gives $E[X] = E[\,E[X \mid Y]\,]$.

8 · Covariance, the CLT & Order Statistics

Multivariate Random Variables (23–30%). How variables co-vary, how their linear combinations behave, and the theorem that makes the normal universal.^[2]

Covariance & Correlation

Covariance measures co-movement: $\text{Cov}(X, Y) = E[XY] - E[X]\,E[Y]$; it is 0 for independent variables (though 0 covariance alone does not prove independence). The correlation coefficient $\rho = \dfrac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$ standardizes it to $[-1, 1]$. Covariance is bilinear: $\text{Cov}(aX, bY) = ab\,\text{Cov}(X, Y)$.

Linear Combinations & the CLT

For any two variables, $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\,\text{Cov}(X, Y)$; when they are independent the covariance term vanishes and the variances simply add. The Central Limit Theorem says the standardized sum of many i.i.d. variables is approximately standard normal: $\dfrac{S_n - n\mu}{\sigma\sqrt{n}} \to N(0, 1)$.

Order Statistics

Order statistics are a sample sorted smallest to largest. For $n$ i.i.d. observations with CDF $F$, the maximum has CDF $[F(t)]^n$ and the minimum has $1 - [1 - F(t)]^n$. The sample range is the largest minus the smallest order statistic.

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside official SOA sample questions and our free tools.^[3] Exam P is heavily computational, so practice solving problems against the clock, not just reading. Spend the bulk of your time on Univariate Random Variables (nearly half the exam), but never skip General Probability — Bayes and conditional probability thread through everything, including the multivariate material.

Exam P Glossary

Quick definitions for the terms you’ll see most across SOA/CAS Exam P:

Axioms of probability

The three rules every probability obeys: $P(A) \ge 0$; $P(S) = 1$; and for mutually exclusive events, the probability of their union is the sum of their probabilities.

Bayes' theorem

Reverses a conditional probability: $P(B_k \mid A) = \dfrac{P(A \mid B_k)\,P(B_k)}{P(A)}$, where the denominator comes from the law of total probability.

Beta distribution

A continuous distribution on the interval $[0,1]$, often used to model proportions and probabilities.

Binomial distribution

Counts successes in $n$ independent trials with success probability $p$: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, mean $np$, variance $np(1-p)$.

Central Limit Theorem

The sum or average of many independent, identically distributed variables is approximately normal for large $n$, whatever the original shape.

Coefficient of variation

The standard deviation divided by the mean, $\sigma / \mu$ — a relative measure of spread.

Coinsurance

A provision under which the insurer pays a fraction $\alpha$ of the covered loss above the deductible, e.g. 80%.

Combination

A selection of objects where order does NOT matter: $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$.

Conditional probability

The probability of A given that B has occurred: $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, defined when $P(B) > 0$.

Correlation coefficient

$\rho = \dfrac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$, a standardized covariance ranging from $-1$ to $1$.

Covariance

Measures how two variables move together: $\text{Cov}(X,Y) = E[XY] - E[X]\,E[Y]$; positive, negative, or 0 (always 0 if independent).

Cumulative distribution function

$F(x) = P(X \le x)$. It is nondecreasing from 0 to 1; for a continuous variable $f(x) = F'(x)$.

Deductible

An amount the policyholder absorbs before the insurer pays. With an ordinary deductible $d$, the payment per loss is $\max(X - d,\,0)$.

Empirical rule

For a normal distribution, about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean.

Expected value

The long-run average of a random variable: $E[X] = \sum_x x\,p(x)$ (discrete) or $\int x\,f(x)\,dx$ (continuous).

Exponential distribution

Models waiting time with rate $\lambda$; density $\lambda e^{-\lambda x}$, mean and standard deviation $1/\lambda$, variance $1/\lambda^2$. It is memoryless.

Gamma distribution

A continuous distribution generalizing the exponential; a sum of $\alpha$ independent exponential waits has a gamma distribution.

Geometric distribution

Counts trials (or failures) until the first success, with mean $1/p$. It is the discrete memoryless distribution.

Hypergeometric distribution

Models successes when sampling a fixed number of items WITHOUT replacement from a finite two-type population.

Independent events

Events whose occurrence does not affect each other, so $P(A \cap B) = P(A)\,P(B)$. Independence concerns influence; mutual exclusivity concerns overlap — they are not the same.

Joint distribution

Describes two (or more) random variables together: $p(x,y) = P(X=x,\,Y=y)$ (discrete) or a joint density $f(x,y)$ (continuous).

Law of total probability

For a partition $B_1,\dots,B_n$ of the sample space, $P(A) = \sum_i P(A \mid B_i)\,P(B_i)$ — the weighted average of conditional probabilities.

Marginal distribution

The distribution of one variable alone, found by summing or integrating the joint distribution over the other variable.

Memoryless property

$P(X > s + t \mid X > s) = P(X > t)$: a used item that has survived $s$ has the same remaining-life distribution as a new one. Held by the exponential and geometric.

Moment generating function

$M(t) = E[e^{tX}]$. Its derivatives at 0 give the moments ($M'(0)=E[X]$); it uniquely identifies a distribution.

Mutually exclusive

Two events that cannot both occur, so $P(A \cap B) = 0$. For mutually exclusive events the addition rule simplifies to $P(A \cup B) = P(A) + P(B)$.

Negative binomial distribution

Counts the trials needed to reach a fixed number $r$ of successes — a generalization of the geometric distribution.

Normal distribution

The symmetric bell curve with mean $\mu$ and standard deviation $\sigma$. Standardize with $z = \dfrac{x-\mu}{\sigma}$ to use the standard-normal table.

Order statistics

A sample sorted from smallest to largest. For $n$ i.i.d. variables with CDF $F$, the maximum has CDF $[F(t)]^n$ and the minimum has $1 - [1 - F(t)]^n$.

Permutation

An arrangement of objects where order matters: $\,_nP_r = \dfrac{n!}{(n-r)!}$.

Poisson distribution

Counts rare events at rate $\lambda$: $P(X=k) = \dfrac{e^{-\lambda}\lambda^k}{k!}$, with mean and variance both equal to $\lambda$.

Policy limit

The maximum amount an insurer will pay on a loss; the payment is capped at this value.

Probability density function

For a continuous variable, $f(x) \ge 0$ whose integral over the support is 1; probability is the area under it, $P(a \le X \le b) = \int_a^b f(x)\,dx$.

Probability mass function

For a discrete variable, $p(x) = P(X = x)$; the values are nonnegative and sum to 1.

Random variable

A function that assigns a number to each outcome of an experiment. It is discrete if it takes countable values, continuous if it takes values on an interval.

Sample space

The set of all possible outcomes of a random experiment, usually written S or $\Omega$. An event is any subset of the sample space.

Uniform distribution

All outcomes equally likely. Continuous on $[a,b]$ it has density $\dfrac{1}{b-a}$, mean $\dfrac{a+b}{2}$, and variance $\dfrac{(b-a)^2}{12}$.

Variance

A measure of spread: $\text{Var}(X) = E[(X - \mu)^2] = E[X^2] - (E[X])^2$. Its square root is the standard deviation $\sigma$.

Free Exam P Study Materials & Resources

Everything you need to prepare for Exam P is free here — no paywall, no credit card. This guide is the foundation; pair it with the rest of our free Exam P study materials for active recall, timed practice, and last-minute review:

• Exam P Practice Test — exam-style probability questions across all three topics, with worked explanations.
• Exam P Flashcards — active-recall decks for the named distributions, their means and variances, and the key theorems.
• Exam FM Study Guide — the next preliminary exam, on financial mathematics, for when you move past Exam P.

Exam P Study Guide FAQ

References

1. 1.Society of Actuaries. “Probability (P) Exam.” Society of Actuaries. ↑
2. 2.Society of Actuaries. “Probability Exam Syllabus.” Society of Actuaries. ↑
3. 3.Society of Actuaries. “Studying for the Probability (P) Exam.” Society of Actuaries. ↑
4. 4.Casualty Actuarial Society. “Exam P — Probability.” Casualty Actuarial Society. ↑

Sources for the concept answers

Every answer in the Exam P concept questions above is drawn from an official primary source:

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