# 3.2: Compound Events

**At Grade**Created by: CK-12

## Learning Objectives

- Know basic operations of unions and intersections.
- Calculate the probability of occurrence of two (or more) simultaneous events.
- Calculate the probability of occurrence of either of two (or more) events.

## Introduction

In this lesson, you will learn how to combine two or more events by finding the union of the two events or the intersection of the two events. You will also learn how to calculate probabilities related to unions and intersections.

### Union and Intersection

Sometimes we need to combine two or more events into one *compound event*. This compound event can be formed in two ways.

The *union of events* \begin{align*}A\end{align*} and \begin{align*}B\end{align*} occurs if either event \begin{align*}A\end{align*}, event \begin{align*}B\end{align*}, or both occur in a single performance of an experiment. We denote the union of the two events by the symbol \begin{align*}A \cup B\end{align*}. You read this as either “\begin{align*}A\end{align*} union \begin{align*}B\end{align*}” or “\begin{align*}A\end{align*} or \begin{align*}B\end{align*}.” \begin{align*}A \cup B\end{align*} means everything that is in set \begin{align*}A\end{align*} or in set \begin{align*}B\end{align*} or in both sets.

The *intersection of events* \begin{align*}A\end{align*} and \begin{align*}B\end{align*} occurs if both event \begin{align*}A\end{align*} and event \begin{align*}B\end{align*} occur in a single performance of an experiment. It is where the two events overlap. We denote the intersection of two events by the symbol \begin{align*}A \cap B\end{align*}. You read this as either “\begin{align*}A\end{align*} intersection \begin{align*}B\end{align*}” or “\begin{align*}A\end{align*} and \begin{align*}B\end{align*}.” \begin{align*}A \cap B\end{align*} means everything that is in set \begin{align*}A\end{align*} and in set \begin{align*}B\end{align*}. That is, when looking at the intersection of two sets, we are looking for where the sets overlap.

*Example*: Consider the throw of a die experiment. Assume we define the following events:

\begin{align*}& A: {\text{observe an even number}}\end{align*}

\begin{align*}& B: {\text{observe a number less than or equal to } 3}\end{align*}

- Describe \begin{align*}A \cup B\end{align*} for this experiment.
- Describe \begin{align*}A \cap B\end{align*} for this experiment.
- Calculate \begin{align*}P(A \cup B)\end{align*} and \begin{align*}P(A \cap B)\end{align*}, assuming the die is fair.

The sample space of a fair die is \begin{align*}S = \left \{1,2,3,4,5,6\right \}\end{align*}, and the sample spaces of the events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} above are \begin{align*}A =\left \{2,4,6\right \}\end{align*} and \begin{align*}B=\left \{1,2,3\right \}\end{align*}.

1. An observation on a single toss of the die is an element of the union of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} if it is either an even number, a number that is less than or equal to 3, or a number that is both even and less than or equal to 3. In other words, the simple events of \begin{align*}A \cup B\end{align*} are those for which \begin{align*}A\end{align*} occurs, \begin{align*}B\end{align*} occurs, or both occur:

\begin{align*}A \cup B = \left \{2,4,6\right \} \cup \left \{1,2,3\right \} = \left \{1,2,3,4,6\right \}\end{align*}

2. An observation on a single toss of the die is an element of the intersection of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} if it is a number that is both even and less than 3. In other words, the simple events of \begin{align*}A \cap B\end{align*} are those for which both \begin{align*}A\end{align*} and \begin{align*}B\end{align*} occur:

\begin{align*}A \cap B = \left \{2,4,6\right \} \cap \left \{1,2,3\right \} = \left \{2\right \}\end{align*}

3. Remember, the probability of an event is the sum of the probabilities of its simple events. This is shown for \begin{align*}A \cup B\end{align*} as follows:

\begin{align*}P(A \cup B) & = P(1)+P(2)+P(3)+P(4)+P(6)\\ & =\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\\ & =\frac{5}{6}\end{align*}

Similarly, this can also be shown for \begin{align*}A \cap B\end{align*}:

\begin{align*}P(A \cap B) = P(2) = \frac{1}{6}\end{align*}

Intersections and unions can also be defined for more than two events. For example, \begin{align*}A \cup B \cup C\end{align*} represents the union of three events.

*Example*: Refer to the above example and answer the following questions based on the definitions of the new events \begin{align*}C\end{align*} and \begin{align*}D\end{align*}.

\begin{align*}& C: {\text{observe a number that is greater than } 5}\end{align*}

\begin{align*}& D: {\text{observe a number that is exactly } 5}\end{align*}

- Find the simple events in \begin{align*}A \cup B \cup C\end{align*}.
- Find the simple events in \begin{align*}A \cap D\end{align*}.
- Find the simple events in \begin{align*}A \cap B \cap C\end{align*}.

1. Since \begin{align*}C = \left \{ 6 \right \}, A \cup B \cup C = \left \{ 2,4,6 \right \} \cup \left \{ 1,2,3 \right \} \cup \left \{ 6 \right \} = \left \{ 1,2,3,4,6 \right \}\end{align*}.

2. Since \begin{align*}D = \left \{ 5 \right \}, A \cap D = \left \{ 2,3,6 \right \} \cap \left \{ 5 \right \} = \varnothing\end{align*},

where \begin{align*}\varnothing\end{align*} is the empty set. This means that there are no elements in the set \begin{align*}A \cap D\end{align*}.

3. Here, we need to be a little careful. We need to find the intersection of the three sets. To do so, it is a good idea to use the associative property by first finding the intersection of sets \begin{align*}A\end{align*} and \begin{align*}B\end{align*} and then intersecting the resulting set with \begin{align*}C\end{align*}.

\begin{align*}(A \cap B)\cap C = (\left \{2,4,6\right \} \cap \left \{1,2,3\right \})\cap \left \{6\right \} = \left \{2\right \} \cap \left \{6\right \} = \varnothing\end{align*}

Again, we get the empty set.

## Lesson Summary

The union of the two events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, written \begin{align*}A \cup B\end{align*}, occurs if either event \begin{align*}A\end{align*}, event \begin{align*}B\end{align*}, or both occur on a single performance of an experiment. A union is an 'or' relationship.

The intersection of the two events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, written \begin{align*}A \cap B\end{align*}, occurs only if both event \begin{align*}A\end{align*} and event \begin{align*}B\end{align*} occur on a single performance of an experiment. An intersection is an 'and' relationship. Intersections and unions can be used to combine more than two events.