Lecture Notes in Actuarial Mathematics A Probability Course for the Actuaries A Preparation for Exam P/1 Marcel B. Finan In memory of my parents August 1, 2008 January 7, 2009 Preface The present manuscript is designed mainly to help students prepare for the Probability Exam (known as Exam P/1), the first actuarial examination administered by the Society of Actuaries. This examination tests a student’s knowledge of the fundamental probability tools for quantitatively assessing risk. A thorough command of calculus is assumed. More information about the exam can be found on the webpage of the Society of Actuaries www.soa.org. Problems taken from previous exams provided by the Society of Actuaries will be indicated by the symbol ‡. This manuscript can be used for personal use or class use, but not for commercial purposes. If you find any errors, I would appreciate hearing from you: mfinan@atu.edu This manuscript is also suitable for a one semester course in an undergraduate course in probability theory. Answer keys to text problems are found at the end of the book. Marcel B. Finan Russellville, AR May, 2008 i ii PREFACE Contents Preface i Set Theory Prerequisite 5 1 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Counting and Combinatorics 3 The Fundamental Principle of Counting . . . . . . . . . . . . . . 4 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 38 44 Probability: Definitions and Properties 6 Sample Space, Events, Probability Measure . . . . . . . . . . . . 7 Probability of Intersection, Union, and Complementary Event . . 8 Probability and Counting Techniques . . . . . . . . . . . . . . . . 55 55 67 78 Conditional Probability and Independence 87 9 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . 87 10 Posterior Probabil