[[Actuarial Notes Wiki|Wiki]] / [[Actuarial Certifications]] / [[Society of Actuaries (SOA)]] / **Exam P-1 (SOA)** <div class="exam-nav" data-color="#2563eb" data-current="P-1|Probability" data-next="FM-2|Financial Mathematics|Exam FM-2 (SOA).md" </div> ## Exam P-1 Syllabus [March 2026 Exam P-1 Syllabus](https://www.soa.org/globalassets/assets/files/edu/2026/spring/syllabi/2026-03-exam-p-syllabus.pdf) The **Probability (P-1) Exam** is a 3 hour exam with 30 multiple choice questions about probability theory concepts and their application to measuring risk. ### Prerequisite Knowledge - Concepts in [[calculus]], including series, differentiation, and integration. - Concepts introduced in [Risk and Insurance](https://www.soa.org/49355c/globalassets/assets/files/edu/p-21-05.pdf) ### Learning Objectives > [!example]- 1 General Probability {23-30%} {19 Concepts} > Understand basic concepts of probability and discrete mathematics. > 1. Define [[Set Function]], [[Venn Diagram]], [[Sample Space]], and [[Event]]. Define [[Probability]] as a set function on a collection of events and state the basic [[Axioms of Probability]]. > 2. Calculate probabilities using [[Combinatorics]], such as [[Combination]] and [[Permutation]]. > 3. Define [[Independent Events|independence]] and calculate probabilities of independent events. > 4. Calculate probabilities of [[Mutually Exclusive Events]]. > 5. Calculate probabilities using [[Probability Addition Rule]] and [[Probability Multiplication Rules]]. > 6. Define and calculate [[Conditional Probability]]. > 7. State [[Bayes Theorem]] and [[The Law of Total Probability]] and use them to calculate conditional probabilities. > [!example]- 2 Univariate Random Variables {44-50%} {20 Concepts} > Understand discrete univariate distributions and continuous univariate distributions and their applications. > > **Discrete Distributions** > - binomial > - geometric > - hypergeometric > - negative binomial > - Poisson > - uniform > > **Continuous Distributions** > - beta > - exponential > - gamma > - lognormal > - normal > - uniform > > 1. Explain and apply the concepts of probability, random variables, probability density functions, and cumulative distribution functions. > 2. Calculate conditional probabilities. > 3. Explain and calculate expected values, including moments, mode, median, and percentiles. > 4. Explain and calculate variance, standard deviation, and coefficient of variation. > 5. Calculate the amount that an insurance company pays to a policyholder for a claim given policy information, including deductibles, coinsurance percentages, and benefit limits, as well as other factors, such as inflation. > 6. Calculate the expected value, variance, and standard deviation of both the loss random variable and the corresponding payment amount random variable. > [!example]- 3 Multivariate Random Variables {23-30%} {11 Concepts} > Understand key concepts in the discrete and continuous settings concerning multivariate distributions, the distribution of order statistics for independent random variables, and linear combinations of independent random variables, along with associated applications. ### Concepts #### 1. General Probability ==Probability== is the mathematical framework that quantifies uncertainty by assigning numerical values (between 0 and 1) to the likelihood of events occurring. The mathematics of probability is expressed naturally in terms of sets. ###### Set Theory ![[Set Theory#Definition]] ###### Venn Diagram ![[Venn Diagram#Definition]] ###### Sample Space ![[Sample Space#Definition]] ###### Event ![[Event#Definition]] ###### Set Operations ![[Set Operations#Definition]] ###### Set Function ![[Set Function#Definition]] ###### Probability ![[Probability#Definition]] ###### Axioms of Probability ![[Axioms of Probability#Definition]] ###### Inclusion-Exclusion Principle ![[Inclusion-Exclusion Principle#Definition]] ###### Probability Addition Rule ![[Probability Addition Rule#Definition]] ###### Probability Multiplication Rules ![[Probability Multiplication Rules#Definition]] ###### Bayes Theorem ![[Bayes Theorem#Definition]] ###### Independent Events ![[Independent Events#Definition]] ###### Mutually Exclusive Events ![[Mutually Exclusive Events#Definition]] ###### Conditional Probability ![[Conditional Probability#Definition]] ###### The Law of Total Probability ![[The Law of Total Probability#Definition]] ###### Combinatorics ![[Combinatorics#Definition]] ###### Combination ![[Combination#Definition]] ###### Permutation ![[Permutation#Definition]] --- #### 2. Univariate Random Variables A ==Univariate Random Variable== is a single random variable that represents uncertain outcomes of an experiment, taking values in one dimension. <figure> <img src="Media/Attachments/Symmetric distribution for continuous probability distribution.png" alt="Standard Normal Distribution"> </figure> ###### Random Variable ![[Random Variable#Definition]] ###### Probability Mass Function (PMF) ![[Probability Mass Function (PMF)#Definition]] ###### Expected Value ![[Expected Value#Definition]] ###### Probability Density Function (PDF) ![[Probability Density Function (PDF)#Definition]] ###### Cumulative Distribution Function (CDF) ![[Cumulative Distribution Function (CDF)#Definition]] ###### Variance ![[Variance#Definition]] ###### Standard Deviation (SD) ![[Standard Deviation (SD)#Definition]] ###### Moments ![[Moments#Definition]] ###### Mode ![[Mode#Definition]] ###### Median ![[Median#Definition]] ###### Percentile ![[Percentile#Definition]] ###### Coefficient of Variation ![[Coefficient of Variation#Definition]] ###### Policy Adjustments ![[Policy Adjustments#Definition]] ###### Deductible ![[Deductible#Definition]] ###### Coinsurance ![[Coinsurance#Definition]] ###### Benefit Limits ![[Benefit Limits#Definition]] ###### Payment ![[Payment#Definition]] ###### Inflation ![[Inflation#Definition]] ###### Loss Random Variable ![[Loss Random Variable#Definition]] ###### Payment Random Variable ![[Payment Random Variable#Definition]] --- #### 3. Multivariate Random Variables A ==Multivariate Random Variable== is a collection of two or more random variables considered jointly, capturing the relationships and dependencies among them. <figure> <img src="Media/Attachments/Multivariate Gaussian.png"> </figure> ###### Joint Probability Function ![[Joint Probability Function#Definition]] ###### Joint Cumulative Distribution Function ![[Joint Cumulative Distribution Function#Definition]] ###### Conditional Probability Function ![[Conditional Probability Function#Definition]] ###### Marginal Probability Function ![[Marginal Probability Function#Definition]] ###### Joint Moments ![[Joint Moments#Definition]] ###### Covariance ![[Covariance#Definition]] ###### Correlation Coefficient ![[Correlation Coefficient#Definition]] ###### Linear Combination ![[Linear Combination#Definition]] ###### Moments ![[Moments#Definition]] ###### Order Statistics ![[Order Statistics#Definition]] ###### Central Limit Theorem (CLT) ![[Central Limit Theorem (CLT)#Definition]] ## Sources | Source | Coverage | | ----------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------- | | [[A First Course in Probability (Ross - 2019)]] | Chapters 1-8^[Excluding 4.8.4, 5.6.2, 5.6.3, 5.6.5, 5.7, 7.2.1, 7.2.2, 7.3, 7.6, 7.7, 7.8, 7.9]<br> | | [[Mathematical Statistics with Applications - 2008]] | Chapters 1-8^[Exlcuding 2.12, MGF, 4.10, Continuous Multivariate Distributions, 5.10, 7.4] | | [[Probability for Risk Management (Hassett - 2021)]] | Chapters 1-11 | | [[Probability and Statistics with Applications-- A Problem-Solving Text (Asimow - 2021)]] | Chapters 1-8 | | [[Probability and Statistical Inference (Hogg - 2020)]] | Chapters 1-5 | | [[Probability (Leemis - 2018)]] | Chapters 1-8 |