Chapter 2 The Language of Probability

A phenomenon is random if there are multiple potential outcomes, and there is uncertainty about which outcome will occur. This chapter introduces the fundamental terminology and objects of random phenomena, including

• Possible outcomes of the random phenomenon
• Related events that could occur
• Random variables which measure numeric quantities based on outcomes
• Probability measures which assign likelihoods to events in a logically coherent way
• Probability spaces which put it all together

Full disclosure: many of the examples in this chapter involve rather dry tasks like discussing mathematical notation or listing elements of sets. Also, some of the things we do in these examples are rarely done in practice. So why bother? Many common mistakes in solving probability problems arise from misunderstanding these foundational objects. We hope that concrete — though sometimes uninteresting — examples foster understanding of fundamental concepts.

This chapter introduces what the fundamental objects of probability are, but not yet how to solve probability problems. Don’t worry; we’ll solve many interesting problems in the remaining chapters. Think of this chapter as introducing the “language” or “grammar” of probability. When first learning to write, we learn the basic elements of sentences: subjects, predicates, clauses, modifiers, etc. Understanding these fundamental building blocks is essential to learning how to write well, even if we don’t explicitly identify the subject, the verb, etc., in every sentence we write. Likewise, understanding the language of probability is crucial to learning how to solve probability problems, even if the language is sometimes unspoken.