1.3: Mutually Inclusive and Mutually Exclusive Events
When determining the probabilities of events, we must also look at 2 additional terms. These terms are mutually inclusive and mutually exclusive. When we add probability calculations of events described by these terms, we can apply the words and
2 events
To calculate the probability of picking a number from 1 to 10 that is even or picking a number from 1 to 10 that is odd, you would follow the steps below:
The probability of picking a number from 1 to 10 that is even and picking a number from 1 to 10 that is odd would just be 0, since these are mutually exclusive events. In other words,
If events
The reason why
When representing this on the Venn diagram, we would see something like the following:
Mutually exclusive events, remember, cannot occur at the same time. Mutually inclusive events can. Look at the Venn diagram below. What do you think we need to do in order to calculate the probability of
If you look at the diagram, you see that the calculation involves not only
where
This is known as the Addition Principle (Rule).
Addition Principle
Think about the idea of rolling a die. Suppose event
Event
Event
Notice that the sets containing the possible outcomes of the events have 2 elements in common. Therefore, the events are mutually inclusive.
What if we said that we were choosing a card from a deck of cards? Suppose event
Notice that the sets containing the possible outcomes of the events have no elements in common. Therefore, the events are mutually exclusive.
Now take a look at the example below to understand the concept of double counting.
Example 8
What is the probability of choosing a card from a deck of cards that is a club or a ten?
Solution:
Example 9
What is the probability of choosing a number from 1 to 10 that is less than 5 or odd?
Solution:
Notice in the previous 2 examples how the concept of double counting was incorporated into the calculation by subtracting the
Example 10
2 fair dice are rolled. What is the probability of getting a sum less than 7 or a sum equal to 10?
Solution:
There are no elements that are common, so the events are mutually exclusive.
Example 11
2 fair dice are rolled. What is the probability of getting a sum less than 7 or a sum less than 4?
Solution:
Notice that there are 3 elements in common. Therefore, the events are not mutually exclusive, and we must account for the double counting.
Points to Consider
 What is the difference between the probabilities calculated with the Multiplication Rule versus the Addition Rule?
 Can mutually exclusive events be independent? Can they be dependent?
Vocabulary
 Addition Principle (Rule)

With 2 events, the probability of one event occurring or another is given by:
P(A∪B)=P(A)+P(B)−P(A∩B) .
 Dependent events
 2 or more events whose outcomes affect each other. The probability of occurrence of one event depends on the occurrence of the other.
 Independent events
 2 or more events whose outcomes do not affect each other.
 Multiplication Rule

For 2 events (
A andB ), the probability ofA andB is given by:P(A and B)=P(A)×P(B) .
 Mutually exclusive events
 2 events are mutually exclusive when they cannot both occur simultaneously.
 Mutually inclusive events
 2 events are mutually inclusive when they can both occur simultaneously.
 Outcomes
 The possible results of 1 trial of a probability experiment.
 Probability
 The chance that something will happen.
 Random
 When everyone or everything in a population has an equal chance of being selected.
 Sample space
 The set of all possible outcomes of an event or group of events.
 Venn diagram
 A diagram of overlapping circles that shows the relationships among members of different sets.

∪  The union of 2 events, where the sample space contains 2 events, \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, and each member of the set belongs to \begin{align*}A\end{align*} or \begin{align*}B\end{align*}.
 \begin{align*}\cap\end{align*}
 The intersection of 2 events, where the sample space contains 2 events, \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, and each member of the set belongs to \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.
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