This lesson explains how to determine whether two events are mutually exclusive. A quick review of the sample space and events in probability may be helpful before proceeding.
Two events are said to be mutually exclusive if they cannot occur at the same time.
Using set notation, two mutually exclusive events \(E_1\) and \(E_2\) satisfy:
\[ E_1 \cap E_2 = \varnothing \]This means that the two events have no outcomes in common.
By contrast, events that are not mutually exclusive share at least one common outcome:
\[ E_1 \cap E_2 \neq \varnothing \]
A fair die is rolled. Let \(E_1\) be the event that the number is even, and \(E_2\) be the event that the number is odd. Are \(E_1\) and \(E_2\) mutually exclusive?
Since \(E_1 \cap E_2 = \varnothing\), the two events have no outcomes in common. Therefore, \(E_1\) and \(E_2\) are mutually exclusive.
A die result cannot be even and odd at the same time.
A die is rolled. Let \(E_1\) be the event that the number is even, and \(E_2\) be the event that the number is greater than 3. Are the events mutually exclusive?
The intersection is:
\[ E_1 \cap E_2 = \{4,6\} \]Since the intersection is not empty, the events are not mutually exclusive.
A card is drawn from a standard deck. Define the events:
Determine whether the following pairs of events are mutually exclusive.
Two dice are rolled. Define the events:
