Today, I will teach you about Mutually Exclusive Events

Many problems involve finding the probability of two or more events. For example, at a large political gathering, you might wish to know, for a person selected at random, the probability that the person is a female or is a Republican. In this case, there are three possibilities to consider:

- The person is a female.
- The person is a Republican.
- The person is both a female and a Republican.

Consider another example. At the same gathering there are Republicans, Democrats, and Independents. If a person is selected at random, what is the probability that the person is a Democrat or an Independent? In this case, there are only two possibilities:

- The person is a Democrat.
- The person is an Independent.

The difference between the two examples is that in the first case, the person selected can be a female and a Republican at the same time. In the second case, the person selected cannot be both a Democrat and an Independent at the same time. In the second case, the two events are said to be *mutually exclusive;* in the first case, they are not mutually exclusive.

Two events are mutually exclusive events or disjoint events if they cannot occur at the same time (i.e., they have no outcomes in common).

In another situation, the events of getting a 4 and getting a 6 when a single card is drawn from a deck are mutually exclusive events, since a single card cannot be both a 4 and a 6. On the other hand, the events of getting a 4 and getting a heart on a single draw are not mutually exclusive, since you can select the 4 of hearts when drawing a single card from an ordinary deck.**EXAMPLE 4–15****Determining Mutually Exclusive Events**

Determine whether the two events are mutually exclusive. Explain your answer.

- Randomly selecting a female student Randomly selecting a student who is a junior
- Randomly selecting a person with type A blood Randomly selecting a person with type O blood
- Rolling a die and getting an odd number Rolling a die and getting a number less than 3
- Randomly selecting a person who is under 21 years of age Randomly selecting a person who is over 30 years of age

SOLUTION

- These events are not mutually exclusive since a student can be both female and a junior.
- These events are mutually exclusive since a person cannot have type A blood and type O blood at the same time.
- These events are not mutually exclusive since the number 1 is both an odd number and a number less than 3.
- These events are mutually exclusive since a person cannot be both under 21 and over 30 years of age at the same time.

**EXAMPLE 4–16****Drawing a Card**

Determine which events are mutually exclusive and which events are not mutually exclusive when a single card is drawn at random from a deck.

- Getting a face card; getting a 6
- Getting a face card; getting a heart
- Getting a 7; getting a king
- Getting a queen; getting a spade

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SOLUTION

- These events are mutually exclusive since you cannot draw one card that is both a face card (jack, queen, or king) and a card that is a 6 at the same time.
- These events are not mutually exclusive since you can get one card that is a face card and is a heart: that is, a jack of hearts, a queen of hearts, or a king of hearts.
- These events are mutually exclusive since you cannot get a single card that is both a 7 and a king.
- These events are not mutually exclusive since you can get the queen of spades.

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The probability of two or more events can be determined by the *addition rules.* The first addition rule is used when the events are mutually exclusive.Addition Rule 1

Addition Rule 1

When two events *A* and *B* are mutually exclusive, the probability that *A* or *B* will occur is

Figure 4–6 shows a Venn diagram that represents two mutually exclusive events *A* and *B.* In this case, *P*(*A* or *B*) = *P*(*A*) + *P*(*B*), since these events are mutually exclusive and do not overlap. In other words, the probability of occurrence of event *A* or event *B* is the sum of the areas of the two circles.FIGURE 4–6Venn Diagram for Addition Rule 1 When the Events Are Mutually Exclusive

EXAMPLE 4–17Endangered Species

In the United States there are 59 different species of mammals that are endangered, 75 different species of birds that are endangered, and 68 species of fish that are endangered. If one animal is selected at random, find the probability that it is either a mammal or a fish.

*Source*: Based on information from the U.S. Fish and Wildlife Service.SOLUTION

Since there are 59 species of mammals and 68 species of fish that are endangered and a total of 202 endangered species, . The events are mutually exclusive.EXAMPLE 4–18Research and Development Employees

The corporate research and development centers for three local companies have the following numbers of employees:

U.S. Steel | 110 |

Alcoa | 750 |

Bayer Material Science | 250 |

If a research employee is selected at random, find the probability that the employee is employed by U.S. Steel or Alcoa.

*Source: Pittsburgh Tribune Review.*SOLUTION

Page 212EXAMPLE 4–19Favorite Ice Cream

In a survey, 8% of the respondents said that their favorite ice cream flavor is cookies and cream, and 6% like mint chocolate chip. If a person is selected at random, find the probability that her or his favorite ice cream flavor is either cookies and cream or mint chocolate chip.

*Source*: Rasmussen Report.SOLUTION

These events are mutually exclusive.Historical Note

Venn diagrams were developed by mathematician John Venn (1834–1923) and are used in set theory and symbolic logic. They have been adapted to probability theory also. In set theory, the symbol ∪ represents the *union* of two sets, and *A* ∪ *B* corresponds to *A* or *B*. The symbol ∩ represents the *intersection* of two sets, and *A ∩ B* corresponds to *A* and *B.* Venn diagrams show only a general picture of the probability rules and do not portray all situations, such as *P*(*A*) = 0, accurately.

The probability rules can be extended to three or more events. For three mutually exclusive events *A, B,* and *C,*

When events are not mutually exclusive, addition rule 2 can be used to find the probability of the events.