Discrete Mathematics: An Active Approach to Mathematical Reasoning

Let $S$ be a set of possible outcomes, called a sample space. Let $E$ be a subset of $S$ with some property. We call $E$ an event.

The probability of event $E\text{,}$ $P(E)\text{,}$ is the number of outcomes in $E$ divided by the total number of outcomes in $S\text{.}$

Let $N(E)$ be the number of elements in $E\text{.}$ Similarly, $N(S)\text{,}$ is the number of outcomes in $S\text{.}$

In notation, $P(E)=\frac{N(E)}{N(S)}\text{.}$

In order to calculate a probability, we need to be able to count equally-likely events. Just because there are, say, 3 outcomes, that doesn’t mean all the outcomes are equally-likely. For example, the Linfied football team could win, lose, or tie a game. Given the team’s winning streak, the three outcomes are not equally-likely. They have a much higher probability of winning than just 1/3.

Thus, the focus for the rest of the chapter will be on counting events. Once we can count outcomes, it is straight-forward to find the probability.

Before moving on, though, we will state a few useful facts about probabilities.