In this explainer, we will learn how to identify mutually exclusive events and non-mutually exclusive events and find their probabilities.
Before we discuss mutually exclusive events, letβs recap compound events and the addition rule of probability.
Key Terms: Compound Events and the Addition Rule of Probability
The intersection of events and , denoted by , is the collection of all outcomes that are elements of both of the sets and ; this is equivalent to both events occurring.
The union of events and , denoted by , is the collection of all outcomes that are elements of either set , set , or both sets; this is equivalent to either of the events occurring.
If an event in a sample space cannot occur, then its probability is 0. Since , we must have . In other words, there are no elements in . We call the set with no elements the empty set and denote this set by .
The addition rule for probability states that
We can use this as motivation for a definition. If , then the addition rule for probability would simplify to
We call events where Β mutually exclusive events since both events cannot occur at the same time. We can write this formally as follows:
Definition: Mutually Exclusive Events and the Additive Rule for Mutually Exclusive Events
We say that and are mutually exclusive events if . This is equivalent to saying the events cannot occur at the same time since .
We say that a list of events is mutually exclusive if they are pairwise mutually exclusive, so for any and .
If and are mutually exclusive, then
To see an example of mutually exclusive events, we can recall that the intersection of two events and can be shown in a Venn diagram as the part of the diagram in the overlap of and . So, in a Venn diagram, mutually exclusive events have an empty intersection.
In the first diagram, there is no intersection between the events, so the events are mutually exclusive. However, in the second diagram, there is an intersection between liking cats and dogs, so the events are not mutually exclusive. Of course, even though there is an intersection, we should still check that there is at least one member in the intersection to be sure.
In our first example, we will determine whether given pairs of events are mutually exclusive.
Example 1: Deciding If Events Are Mutually Exclusive
Farida has a deck of 52 cards. She randomly selects one card and considers the following events:
Event A: picking a card that is a heart
Event B: picking a card that is black
Event C: picking a card that is not a spade
- Are events A and B mutually exclusive?
- Are events A and C mutually exclusive?
- Are events B and C mutually exclusive?
Answer
We start by recalling that two events and are mutually exclusive if they cannot occur at the same time. In other words, .
Therefore, to determine if the given pairs of events are mutually exclusive, we need to check whether there are any cards from the 52-card deck that satisfy both events.
Part 1
There are two ways of checking whether the events of βpicking a card that is a heartβ and βpicking a card that is blackβ are mutually exclusive.
First, we can note that all hearts are red, so there are no cards that are both black and a heart. Therefore, both events cannot happen, and so the events are mutually exclusive.
Second, we can mark all of the cards for each event and check whether there is any overlap in the events.
We mark all of the hearts in red (event A) and all of the black cards in green (event B). We can see that there is no overlap in these events, so they cannot occur simultaneously.
Hence, we can say that yes, the events A and B are mutually exclusive.
Part 2
As in part 1, there are two ways of checking whether the events of βpicking a card that is a heartβ and βpicking a card that is not a spadeβ are mutually exclusive.
First, we can note that all of the hearts are not spades, so choosing any heart will satisfy both events. Then, since the events can both occur at the same time, we can conclude they are not mutually exclusive.
Second, we can mark all of the cards for each event and check whether there is any overlap in the events.
We mark the hearts with red (event A) and all of the cards that are not spades in blue (event C). We can see that all hearts satisfy both events A and C, so the events are not mutually exclusive.
Hence, we can say that no, the events A and C are not mutually exclusive.
Part 3
There are two ways of checking whether the events of βpicking a card that is blackβ and βpicking a card that is not a spadeβ are mutually exclusive.
First, we could note that all clubs are black and not a spade, so choosing any of these cards satisfies both events.
Second, we can mark all of the cards for each event and check whether there is any overlap in the events.
We mark all of the black cards with purple (event B) and all of the cards that are not spades in blue (event C). We can see that all clubs satisfy both events B and C, so the events are not mutually exclusive.
Hence, we can say that no, events B and C are not mutually exclusive.
In our second example, we will use the mutual exclusivity of two events and their probabilities to determine the probability of either event occurring.
Example 2: Determining the Probability of Union of Two Mutually Exclusive Events
Two mutually exclusive events and have probabilities and . Find .
Answer
We recall that since and are mutually exclusive, . Therefore, the addition rule for probability tells us that when and are mutually exclusive,
We can then substitute and into this equation to get
In our next example, we will use the addition rule for probability on mutually exclusive events to determine the probability of either event occurring.
Example 3: Determining the Probability of an Event Using A Given relationship between Mutually Exclusive Events
Suppose that and are two mutually exclusive events. The probability of the event occurring is five times that of the event occurring. Given that the probability that one of the two events occurs is 0.18, find the probability of event occurring.
Answer
We know that the probability of either or occurring is 0.18. We know that and are mutually exclusive, and we recall that this means that . We also know that the probability of occurring is five times that of occurring, so . We can then recall that since and are mutually exclusive, the addition rule for probability tells us
Substituting and into the formula gives
We can then divide the equation by 6 to get
In our next example, we will use the context of a word problem to deduce whether the given events are mutually exclusive and then use this to determine the probability of either event occurring.
Example 4: Using the Addition Rule to Determine the Probability of Union of Mutually Exclusive 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
We can start by sketching the information given to get a clear idea of the composition of the choir.
If we assume that the singers stick to their parts so that, for example, a soprano singer does not sing tenor or baritone parts and vice versa, then the events of choosing a soprano, tenor, baritone, or mezzo-soprano singer are mutually exclusive, since no two events can occur at the same time.
This being the case, to find the probability that a randomly chosen singer is either a tenor or a soprano singer, we can use the probability rule , since, by mutual exclusivity of and , .
As there are 6 singers in total and only one is a tenor singer, the probability that a randomly chosen singer is a tenor singer is
Similarly, there are 3 soprano singers; hence,
Applying the rule that says that for mutually exclusive events, , we have
In our next example, we will use the mutual exclusivity of three events and properties of probability to determine the probability of a compound event occurring in a word problem.
Example 5: Determining the Union of Two Mutually Exclusive Events
A bag contains red, blue, and green balls, and one is to be selected without looking. The probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue. The probability that the chosen ball is blue is the same as the probability that the chosen ball is green.
Find the probability that the chosen ball is red or green.
Answer
We can name the events of choosing a red ball, a blue ball, and a green ball , , and respectively. Since a chosen ball can only be one of these three colors, we can conclude that the events are mutually exclusive. We want to determine the probability that the ball chosen is red or green, that is, .
The additive rule for probability tells us
Since the events are mutually exclusive, we know that , so
We are told in the question that the probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue, so
We are also told that the probability that the chosen ball is blue is the same as the probability that the chosen ball is green, so
Finally, since there are only red, blue, and green balls in the bag and these are mutually exclusive events, we have
Substituting and into this equation gives
Dividing the equation by 9 gives
Since the probability that the chosen ball is blue is the same as the probability that the chosen ball is green, we have
Since the probability that the chosen ball is red is equal to seven times the probability that the chosen ball is blue, we have
Substituting these values into the additive rule for probability for mutually exclusive events, we get
In our final example, we will note that two events are not mutually exclusive and then use the given probabilities along with the addition rule for probability to determine the probability of an event.
Example 6: Finding the Probability of a Difference of Two Events given the Probability of Each Event as well as Their Intersection
The probability that a student passes their physics exam is 0.71. The probability that they pass their mathematics exam is 0.81. The probability that they pass both exams is 0.68. What is the probability that the student only passes their mathematics exam?
Answer
To find the probability that the student passes mathematics but not physics, letβs illustrate the events in a Venn diagram. We note that since there is an overlap, the events are not mutually exclusive and can occur together. This gives us the following:
Now, if we highlight the probabilities that we know concerning the event βpasses mathβ on the diagram, that is, and , we have
The event βpasses mathβ is everything inside the red oval, which has a probability of 0.81. The overlap in the center of the diagram covers βpasses both math and physicsβ and has a probability of 0.68. But we want to find the probability of passing mathematics but not physics, which covers the dark purple section in the diagram below.
Since the probability of passing math is made up of the probability of passing math but not physics and the probability of passing both, we have
Rearranging this gives us
Hence, the probability that a student passes mathematics but not physics is 0.13.
Letβs finish by recapping some of the important points from this explainer.
Key Points
- We say that and are mutually exclusive events if . This is equivalent to saying the events cannot occur at the same time, since .
- We say that a list of events is mutually exclusive if they are pairwise mutually exclusive.
- If events and are mutually exclusive, then
- If events and are not mutually exclusive, then where
