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 \begin{align*}A\end{align*} and \begin{align*}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. We denote the union of the two events by the symbol \begin{align*}A \cup B\end{align*}. You can say this symbol with either “\begin{align*}A\end{align*} union \begin{align*}B\end{align*}” or “\begin{align*}A\end{align*} or \begin{align*}B\end{align*}”.
- The word “or” is used with union:
- \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.
- Definition
- The intersection of two 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 on a single performance of an experiment. We denote the intersection of two events by the symbol \begin{align*}A \cap B\end{align*}. The most common way to say this symbol is “\begin{align*}A\end{align*} and \begin{align*}B\end{align*}”.
- The word “and” is used with intersection:
- \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*}.
Example:
Consider the throw of a die experiment. Assume we define the following events:
\begin{align*}& \text{A:} \left \{\text{observe an even number}\right \}\\ & \text{B:} \left \{\text{observe a number less than or equal to 3}\right \}\end{align*}
- Describe \begin{align*}A \cup B\end{align*} for this experiment.
- Describe \begin{align*}B \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.
Solution:
The sample space of a fair die is \begin{align*}S = \left \{1, 2, 3, 4, 5, 6 \right \}\end{align*}. The sample spaces of the events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} above are \begin{align*}S(A) = A = \left \{2, 4, 6 \right \}\end{align*} and \begin{align*}S(B) = B = \left \{1, 2, 3 \right \}. \end{align*}
1. The union of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} is the event if we observe either an even number, a number that is equal to \begin{align*}3\end{align*} or less, or both on a single toss of the die. 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. The intersection of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} is the event that occurs if we observe both an even number and a number that is equal to or less than \begin{align*}3\end{align*} on a single toss of a die.
\begin{align*}A \cap B & = \left \{2, 4, 6 \right \} \cap \left \{1, 2, 3 \right \}\\ & = \left \{2 \right \}\end{align*}
In other words, the intersection of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} is the simple event to observe a \begin{align*}2\end{align*}.
3. Remember the probability of an event is the sum of the probabilities of the simple events,
\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,
\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, the union \begin{align*}A \cup B \cup C\end{align*} represents the union of three events.
Example:
Refer to the above example and define the new events
\begin{align*}& \text{C:} \left \{\text{observe a number that is greater than 5}\right \}\\ & \text{D:} \left \{\text{observe a number that is exactly 5}\right \}\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*}
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.