Definition 1. An outcome is one particular possible result of the trial. When a die is rolled, the outcomes are \(1,2,3,4,5,6.\) The sample space, usually denoted by \(\Omega\), is the set of all possible outcomes:
\[\Omega=\lbrace 1,2,3,4,5,6\rbrace.\]An event is a collection of outcomes, so mathematically an event is a subset of the sample space. For example, if one die is rolled, the event that an even number appears is
\[A=\lbrace 2,4,6\rbrace \subseteq \Omega.\]The statement “event \(A\) occurs” means that the actual outcome of the trial belongs to \(A\). Thus, if the actual outcome is \(4\), then \(A\) occurs; if the actual outcome is \(5\), then \(A\) does not occur. Accordingly, the statement “event \(A\) can occur in \(m\) ways” means that \(A\) contains \(m\) possible outcomes:
\[|A|=m.\]For instance, the event \(A=\lbrace 2,4,6\rbrace\) can occur in three ways because there are three outcomes that make \(A\) occur.
Definition 2. Two events \(A\) and \(B\) are mutually exclusive, or disjoint, when they cannot occur in the same trial, which is denoted by
\[A\cap B=\varnothing.\]If \(|A|=m\), \(|B|=n\), and \(A\cap B=\varnothing\), then the event “\(A\) or \(B\) occurs” is denoted by \(A\cup B\), and
\[|A\cup B|=|A|+|B|=m+n.\]This formula is often called the addition rule.
Definition 3. The multiplication rule concerns a process that is completed in successive stages. We let \(A\) be the set of possible results of the first stage and \(B\) the set of possible results of the second stage, with
\[|A|=m,\qquad |B|=n.\]A complete outcome is then an ordered pair
\[(a,b)\in A\times B,\]where \(a\) records the first-stage result and \(b\) records the second-stage result. Since each of the \(m\) possible values of \(a\) can be paired with each of the \(n\) possible values of \(b\),
\[|A\times B|=mn.\]