Lesson Explainer: Mutually Exclusive Events Mathematics

In this explainer, we will learn how to identify mutually exclusive events and find their probabilities.

We say that two events 𝐴 and 𝐡 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 𝐴 and 𝐡 are mutually exclusive if 𝐴∩𝐡=βˆ….

If 𝐴 and 𝐡 are mutually exclusive, then 𝑃(𝐴∩𝐡)=0.

We remember that the probability of an empty set is zero. By definition above, if 𝐴 and 𝐡 are mutually exclusive events, then 𝑃(𝐴∩𝐡)=0.

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 𝐴 be the event that an odd number is rolled, and let 𝐡 be the event that an even number is rolled. In formal notations, we can write 𝐴={1,3,5},𝐡={2,4,6}.

We notice that 𝐴∩𝐡=βˆ… since there is outcome in both 𝐴 and 𝐡. So they are mutually exclusive. The probability of getting an odd and an even number together is represented by 𝑃(𝐴∩𝐡). We remember that 𝑃(𝐴∩𝐡)=0 if the events 𝐴 and 𝐡 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 β€œor” statements.

Theorem: Addition Rule for Mutually Exclusive Events

If 𝐴 and 𝐡 are mutually exclusive events, then 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡).

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 𝑃()=,𝑃()=.selectingadognumberofdogsnumberofpetsselectingacatnumberofcatsnumberofpets

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 𝑃()=+=+=𝑃()+𝑃().selectingadogoracatnumberofdogsnumberofcatsnumberofpetsnumberofdogsnumberofpetsnumberofcatsnumberofpetsselectingadogselectingacat

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 𝑃()=1𝑃()=1.selectedstudentplayssoccerandselectedstudentplaysbaseball

Then, if we blindly applied the addition rule for mutually exclusive events, we would have 𝑃()=𝑃()+𝑃()=1+1=2.selectedstudentplayssoccerorbaseballselectedstudentplayssoccerselectedstudentplaysbaseball

But we remember that, by the rule of probability, a probability value cannot exceed 1. So the result cannot be true because 2>1. 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 𝐴 and 𝐡 have probabilities 𝑃(𝐴)=110 and 𝑃(𝐡)=15. Find 𝑃(𝐴βˆͺ𝐡).

Answer

We remember that if 𝐴 and 𝐡 are mutually exclusive events, then the addition rule states that 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡).

We are given that 𝑃(𝐴)=110 and 𝑃(𝐡)=15, so 𝑃(𝐴βˆͺ𝐡)=110+15=310.

So, 𝑃(𝐴βˆͺ𝐡)=310.

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 𝐴 be the event that the tenor singer’s name is chosen, and let 𝐡 be the event that a soprano singer’s name is chosen. The event that either the tenor’s or a soprano’s name is chosen is represented by 𝐴βˆͺ𝐡.

A singer cannot be both tenor and soprano, so it is not possible that 𝐴 and 𝐡 would occur at the same time. This implies that 𝐴 and 𝐡 are mutually exclusive events.

We remember that if 𝐴 and 𝐡 are mutually exclusive events, then the addition rule tells us that 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡).

There are total of 1+3+1+1=6.tenorsopranosbaritonemezzo-spranosingers

So, 𝑃(𝐴)==16,𝑃(𝐡)==36.numberoftenorsnumberofsingersnumberofsopranosnumberofsingers

Finally, using the additional rule for mutually exclusive events, we get 𝑃(𝐴βˆͺ𝐡)=16+36=46=23.

So the probability that the name of the tenor singer or soprano singer was chosen is 23.

We recall that the complement of an event 𝐴, denoted by 𝐴, is the event of all possible outcomes outside of 𝐴. 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, 𝑃(𝐴βˆͺ𝐴)=𝑃(𝐴)+𝑃(𝐴).

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, 𝑃(𝐴βˆͺ𝐴)=𝑃()=1.allpossibleoutcomes

Putting these two equations together, we have 𝑃(𝐴)+𝑃(𝐴)=1βŸΉπ‘ƒ(𝐴)=1βˆ’π‘ƒ(𝐴).

This is known as the complement rule.

Theorem: Complement Rule

Let 𝐴 and 𝐴 be an event and its complement. Then, 𝑃(𝐴)=1βˆ’π‘ƒ(𝐴).

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 𝐴 and 𝐡 are two mutually exclusive events. Given that 𝑃(𝐴)=0.61 and 𝑃(𝐴βˆͺ𝐡)=0.76, determine 𝑃(𝐡).

Answer

We are given that 𝑃(𝐴)=0.61. We remember that for any event 𝐴, the probability of the complement 𝐴 is equal to 1βˆ’π‘ƒ(𝐴). So we have 0.61=1βˆ’π‘ƒ(𝐴).

Solving this equation for 𝑃(𝐴), we get 𝑃(𝐴)=1βˆ’0.61=0.39.

Next, we remember the addition rule that tells us that if 𝐴 and 𝐡 are mutually exclusive events, then 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡).

Since we know 𝑃(𝐴βˆͺ𝐡)=0.76 and 𝑃(𝐴)=0.39, we have 0.76=0.39+𝑃(𝐡).

Solving this equation for 𝑃(𝐡), we get 𝑃(𝐡)=0.76βˆ’0.39=0.37.

So, 𝑃(𝐡)=0.37.

Example 5: Finding the Probability of Union of Complement of Two Mutually Exclusive Events

If 𝐴 and 𝐡 are two mutually exclusive events from a sample space of a random experiment, find 𝑃(𝐴βˆͺ𝐡).

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 𝐴 and 𝐡, shown below.

We note that there is no overlap between 𝐴 and 𝐡. This is because they are assumed to be mutually exclusive. Next, we draw the complements 𝐴 and 𝐡 separately below.

We take the union of the red 𝐴 and the blue 𝐡 using a Venn diagram.

In the graph above, the overlapping region is colored purple. We note that the union 𝐴βˆͺ𝐡 constitutes the whole sample space, which is the set of all possible outcomes. So 𝐴βˆͺ𝐡=setofallpossibleoutcomes.

We remember that the probability of all possible outcomes is equal to 1. So, 𝑃(𝐴βˆͺ𝐡)=𝑃()=1.allpossibleoutcomes

So, 𝑃(𝐴βˆͺ𝐡)=1.

When the events 𝐴 and 𝐡 are mutually exclusive, set differences do not result in any change in the original set. In other word, π΄βˆ’π΅=𝐴. This fact immediately leads to the difference rule.

Theorem: Difference Rule for Mutually Exclusive Events

If event 𝐴 and event 𝐡 are mutually exclusive, then 𝑃(π΄βˆ’π΅)=𝑃(𝐴).

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 𝐴 and 𝐡 are two mutually exclusive events. Given that 𝑃(π΄βˆ’π΅)=0.52, find 𝑃(𝐴).

Answer

We remember that if 𝐴 and 𝐡 are mutually exclusive events, then the difference rule states that 𝑃(π΄βˆ’π΅)=𝑃(𝐴).

It is helpful to remind ourselves of why this rule is true. When 𝐴 and 𝐡 are mutually exclusive, then there is no overlap between 𝐴 and 𝐡. In other words 𝐴∩𝐡=βˆ…. So the set difference π΄βˆ’π΅ does not take anything away from the set 𝐴. Naturally, this leads to the identity 𝑃(π΄βˆ’π΅)=𝑃(𝐴).

We are given 𝑃(π΄βˆ’π΅)=0.52 and also that 𝐴 and 𝐡 are mutually exclusive.

So, by the difference rule, 𝑃(𝐴)=0.52.

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 𝐴 and 𝐡 are mutually exclusive events, then
    • 𝑃(𝐴∩𝐡)=0,
    • 𝑃(𝐴βˆͺ𝐡)=𝑃(𝐴)+𝑃(𝐡) (the addition rule for mutually exclusive events),
    • 𝑃(π΄βˆ’π΅)=𝑃(𝐴) (the difference rule for mutually exclusive events).
  • For any event 𝐴, the complement rule states that 𝑃(𝐴)=1βˆ’π‘ƒ(𝐴).

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