3.2: Compound Events
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 the two (or more) events.
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.
 Definition

The union of two events
A andB occurs if either eventA or eventB or both occur on a single performance of an experiment. We denote the union of the two events by the symbolA∪B . You can say this symbol with either “A unionB ” or “A orB ”.  The word “or” is used with union:

A∪B means everything that is in setA OR in setB OR in both sets.
 Definition

The intersection of two events
A andB occurs if both eventA and eventB occur on a single performance of an experiment. We denote the intersection of two events by the symbolA∩B . The most common way to say this symbol is “A andB ”.  The word “and” is used with intersection:

A∩B means everything that is in setA AND in setB .
Example:
Consider the throw of a die experiment. Assume we define the following events:
 Describe
A∪B for this experiment.  Describe
B∩B for this experiment.  Calculate
P(A∪B) andP(A∩B) , assuming the die is fair.
Solution:
The sample space of a fair die is
1. The union of
2. The intersection of
In other words, the intersection of
3. Remember the probability of an event is the sum of the probabilities of the simple events,
Similarly,
Intersections and unions can also be defined for more than two events. For example, the union
Example:
Refer to the above example and define the new events
 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*}
Solution:
1. Event \begin{align*}C\end{align*} corresponds to finding the simple event \begin{align*}S(C) = C = \left \{6 \right \}\end{align*}. So
\begin{align*}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. Event \begin{align*}D\end{align*} corresponds to finding the simple event \begin{align*}S(D) = D = \left \{5 \right \}\end{align*}. So
\begin{align*}A \cap D & = \left \{2, 4, 6 \right \} \cap \left \{5 \right \}\\ &= \phi\end{align*}
Where \begin{align*}\phi\end{align*} is the empty set. This says that there are no elements in the set \begin{align*}A \cap D\end{align*}. This means that you will not observe any events that combine sets \begin{align*}A\end{align*} and \begin{align*}D\end{align*}.
3. Here, we need to be a little careful. We need to find the intersection of three sets. To do so, it is a good idea to use the associativity property by finding first 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*}. Here is how:
\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 \})\\ & = \phi\end{align*}
Again, we get the empty set.
Lesson Summary
 The union of two events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, \begin{align*}A \cup B\end{align*}, occurs if either event \begin{align*}A\end{align*} or 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 two events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, \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.
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