Lesson 2

Let \(U\) be a finite set, which we will call the set of possible outcomes, and \(\Pr \colon U \to [0,1]\) be a function such that \[\sum_{u \in U}\Pr[u]=1.\] The pair \((U, \Pr)\) is called a probability space, and the function \(\Pr\) is called a probability distribution. The probability of an outcome \(u \in U\) is defined as \(\Pr[u]\). An event is any subset \(A \subseteq U\), and its probability is defined as \[\Pr[A]=\sum_{u \in A}\Pr[u].\] Events of the form \(\{u\}\) (i.e., consisting of a single possible outcome) are called elementary events. For this reason, the set \(U\) is also often called the sample space of elementary events.