In this explainer, we will learn how to identify mutually exclusive events and find their probabilities.
We say that two events and are mutually exclusive if they cannot occur at the same time. For example, an animal cannot be both a dog and a cat. Say that we are picking at random a pet from a store. In this context, the event of picking a dog for a pet and the event of picking a cat for a pet are mutually exclusive. We can also visualize mutually exclusive events using the following Venn diagram.
As seen in the Venn diagram above, mutually exclusive events do not have any overlap.
Definition: Mutually Exclusive
Events and are mutually exclusive if
If and are mutually exclusive, then
We remember that the probability of an empty set is zero. By definition above, if and are mutually exclusive events, then
Let us consider an example to put this principle in context.
Example 1: Finding the Probability of Intersection of Two Mutually Exclusive Events
If a die is rolled once, then what is the probability of getting an odd and an even number together?
Answer
Let be the event that an odd number is rolled, and let be the event that an even number is rolled. In formal notations, we can write
We notice that since there is no comon outcome in both and . So they are mutually exclusive. The probability of getting an odd and an even number together is represented by . We remember that if the events and are mutually exclusive.
So the probability of getting an odd and an even number together is zero.
For mutually exclusive events, we have the addition rule for computing the probability of βorβ statements.
Theorem: Addition Rule for Mutually Exclusive Events
If and are mutually exclusive events, then
To understand the addition rule above more clearly, let us consider the pet store example mentioned earlier. We randomly select one pet from this store. The event of selecting a dog and the event of selecting a cat cannot occur together, so they are mutually exclusive. Since a pet is randomly selected, we know that
Now we consider the union of these two events, which is the event that we select either a dog or a cat. Then we know
This verifies the addition rule for mutually exclusive events.
The addition rule stated above only works when the events are mutually exclusive. We demonstrate how this version of addition rule fails when two events are not mutually exclusive.
Say that we select one student from a classroom at random. We consider the event that the selected student plays soccer and the event that the selected student plays baseball.
We note that a student who plays soccer may also play baseball. In other words, it is possible that the selected student plays soccer and also plays baseball. So the two events are not mutually exclusive.
Let us consider an extreme case where every student in the classroom plays soccer and also baseball so that
Then, if we blindly applied the addition rule for mutually exclusive events, we would have
But we remember that, by the rule of probability, a probability value cannot exceed 1. So the result cannot be true because . This extreme example serves to remind us that the addition rule stated above does not hold when the two events are not mutually exclusive. It is very important that we check for mutual exclusivity before applying the addition rule.
Let us examine a few examples to familiarize ourselves with different contexts.
Example 2: Determining the Probability of Union of Two Mutually Exclusive Events
Two mutually exclusive events and have probabilities and . Find .
Answer
We remember that if and are mutually exclusive events, then the addition rule states that
We are given that and , so
So, .
Example 3: Using the Addition Rule to Determine the Probability of Union of Two Events
A small choir has a tenor singer, 3 soprano singers, a baritone singer, and a mezzo-soprano singer. If one of their names was randomly chosen, determine the probability that it was the name of the tenor singer or soprano singer.
Answer
Let be the event that the tenor singerβs name is chosen, and let be the event that a soprano singerβs name is chosen. The event that either the tenorβs or a sopranoβs name is chosen is represented by .
A singer cannot be both tenor and soprano, so it is not possible that and would occur at the same time. This implies that and are mutually exclusive events.
We remember that if and are mutually exclusive events, then the addition rule tells us that
There are total of
So,
Finally, using the additional rule for mutually exclusive events, we get
So the probability that the name of the tenor singer or soprano singer was chosen is .
We recall that the complement of an event , denoted by , is the event of all possible outcomes outside of . For example, the complement of selecting a dog is the event of selecting any pet other than a dog. An event and its complement can never occur together, so they are mutually exclusive. So, by the addition rule for mutually exclusive events,
On the other hand, the union of an event and its complement contains all possible outcomes. This is because any outcome that is not in the event must then belong to its complement. We recall that the probability of all possible outcome equals to 1. Then,
Putting these two equations together, we have
This is known as the complement rule.
Theorem: Complement Rule
Let and be an event and its complement. Then,
Let us consider a few examples using the complement rule to become familiar with different contexts.
Example 4: Determining the Probability of an Event Involving Mutually Exclusive Events
Suppose and are two mutually exclusive events. Given that and , determine .
Answer
We are given that . We remember that for any event , the probability of the complement is equal to . So we have
Solving this equation for , we get .
Next, we remember the addition rule that tells us that if and are mutually exclusive events, then
Since we know and , we have
Solving this equation for , we get .
So, .
Example 5: Finding the Probability of Union of Complement of Two Mutually Exclusive Events
If and are two mutually exclusive events from a sample space of a random experiment, find .
Answer
For this problem, it is helpful to draw a Venn diagram to visualize the context. First, we begin with the diagram for the mutually exclusive events and , shown below.
We note that there is no overlap between and . This is because they are assumed to be mutually exclusive. Next, we draw the complements and separately below.
We take the union of the red and the blue using a Venn diagram.
In the graph above, the overlapping region is colored purple. We note that the union constitutes the whole sample space, which is the set of all possible outcomes. So .
We remember that the probability of all possible outcomes is equal to 1. So,
So, .
When the events and are mutually exclusive, set differences do not result in any change in the original set. In other word, . This fact immediately leads to the difference rule.
Theorem: Difference Rule for Mutually Exclusive Events
If event and event are mutually exclusive, then
As it was the case for the addition rule, it is very important that we check for mutual exclusivity before applying the difference rule.
Example 6: Determining the Probability of an Event Involving Mutually Exclusive Events
Suppose and are two mutually exclusive events. Given that , find .
Answer
We remember that if and are mutually exclusive events, then the difference rule states that
It is helpful to remind ourselves of why this rule is true. When and are mutually exclusive, then there is no overlap between and . In other words . So the set difference does not take anything away from the set . Naturally, this leads to the identity .
We are given and also that and are mutually exclusive.
So, by the difference rule, .
We have discussed three different rules of probability in this explainer: the addition rule, the difference rule, and the complement rule. We summarize these concepts in the following key points.
Key Points
- Mutually exclusive events are two events that cannot occur at the same time.
- If and are mutually exclusive events, then
- ,
- (the addition rule for mutually exclusive events),
- (the difference rule for mutually exclusive events).
- For any event , the complement rule states that
