Actuarial Notes

Exam P-1

The Probability (P-1) Exam is a three-hour SOA exam of 30 multiple-choice questions, administered as a computer-based test (CBT). Its syllabus develops knowledge of the fundamental probability tools for quantitatively assessing risk, with emphasis on their application to problems encountered in actuarial science.

Exam format

  • Each question has five answer choices, A to E, only one of which is correct; answers for some questions have been rounded.
  • A few pilot questions are placed at random in the exam and are not scored; an unanswered question is scored incorrect.
  • A table of values for the normal distribution is provided during the exam under an Exhibit button, so candidates may not bring a copy of it into the exam.

Prerequisite knowledge

Other resources

  • Tables for Exam P — the normal distribution table provided in the exam
  • Exam P Sample Questions and Solutions: questions and solutions
  • Online Sample Exam P — a balanced yet randomized set of questions on each attempt, drawn from the sample questions and coded to the learning objectives

Learning Objectives

General Probability (23–30%) Understand basic concepts of Probability and Discrete Mathematics.

  1. Define set functions, Venn diagrams, sample space, and events. 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 Independence and calculate probabilities of Independent Events.
  4. Calculate probabilities of Mutually Exclusive Events.
  5. Calculate probabilities using addition and 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.

Univariate Random Variables (44–50%) Understand Discrete Univariate Distributions and Continuous Univariate Distributions and their applications.

  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.

Discrete Univariate Distributions

Continuous Univariate Distributions

Multivariate Random Variables (23–30%) 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.

  1. Determine Joint Probability Functions and Joint Cumulative Distribution Functions for discrete random variables.
  2. Determine Conditional Probability Function and Marginal Probability Function for discrete random variables.
  3. Calculate Moments for Joint Distributions for joint, conditional, and marginal discrete distributions.
  4. Calculate Variance and standard deviation for conditional and marginal probability distributions for discrete random variables.
  5. Calculate the Covariance and the Correlation Coefficient for discrete random variables.
  6. Determine the Joint Distribution of Order Statistics for a set of independent random variables.
  7. Calculate Probabilities for Linear Combinations of independent discrete random variables as well as for continuous normal random variables.
  8. Calculate Moments for Linear Combinations of independent random variables.
  9. Apply the Central Limit Theorem to calculate approximations of probabilities for linear combinations of independent and identically distributed random variables.

Source Material

Source Material

  • A First Course in Probability (Ross - 2019)
    • Chapter 1; Chapter 2; Chapter 3; Chapter 4 (exclude 4.8.4); Chapter 5 (exclude 5.6.2, 5.6.3, 5.6.5, 5.7); Chapter 6: 6.1, 6.2, 6.3.3, 6.3.4, 6.4, 6.6; Chapter 7 Discrete Only (exclude 7.2.1, 7.2.2, 7.3, 7.6, 7.7, 7.8, 7.9); Chapter 8: 8.1, 8.3
  • Mathematical Statistics with Applications (Wackerly, Mendenhall, & Scheaffer - 2008)
    • Chapter 1; Chapter 2 (exclude 2.12); Chapter 3: 3.1-3.8, 3.9 (exclude MGF); Chapter 4 (exclude 4.10); Chapter 5 (exclude continuous multivariate distributions, exclude 5.10); Chapter 6: 6.7; Chapter 7 (exclude 7.4)
  • Probability for Risk Management (Hassett - 2021)
    • Chapter 1; Chapter 2; Chapter 3; Chapter 4; Chapter 5; Chapter 6: 6.1, 6.2.1; Chapter 7; Chapter 8 (exclude 8.5, 8.6, 8.7); Chapter 9 (exclude 9.2, 9.3, 9.4, 9.6); Chapter 10 (exclude 10.2, 10.3.2, 10.3.3 continuous, 10.4.2); Chapter 11 (exclude 11.1.4, 11.1.5, 11.2.3 continuous, 11.2.5 continuous, 11.2.8, 11.3)
  • Probability and Statistics with Applications: A Problem-Solving Text (Asimow & Maxwell, Second Edition, 2015)
    • Chapter 1; Chapter 2; Chapter 3 (exclude 3.4.5, 3.7); Chapter 4 (exclude 4.6.2, 4.6.3, 4.6.5); Chapter 5 (exclude 5.6); Chapter 6 (exclude 6.3.4, 6.4.2, 6.4.6, 6.7); Chapter 7 (exclude 7.5, 7.6, 7.7, 7.9, 7.10); Chapter 8: 8.1.5, 8.3, 8.4 (exclude continuous), 8.6
  • Probability and Statistical Inference (Hogg - 2020)
    • Chapter 1; Chapter 2; Chapter 3 (exclude Chi-Square); Chapter 4 (exclude 4.4, 4.5); Chapter 5: 5.3 (discrete only), 5.5, 5.6, 5.7
  • Probability (Leemis - 2018)
    • Chapter 1 (exclude 1.1); Chapter 2; Chapter 3 (exclude moment generating functions in 3.4, exclude 3.5); Chapter 4 (exclude 4.7); Chapter 5 (include only beta distribution in 5.5); Chapter 6 (exclude continuous, exclude moment generating functions in 6.3, exclude 6.4); Chapter 7 (exclude 7.1, include only order statistics in 7.2, exclude 7.3); Chapter 8 (exclude 8.1, exclude 8.2)