LESSON
Page 1 of 6
11.8
Probabilities of Disjoint and
Overlapping Events
Now
Vocabulary
BEFORE
disjoint events, p. 627
mutually exclusive
events, p. 627
overlapping events,
p. 627
complementary
events, p. 630
You found the
probability of an event.
WHY?
You’ll find the probability that
event A or event B occurs.
So you can analyze data about
blood types, as in Ex. 23.
In this lesson, you will find the probability that an event A or an event B
occurs. To find the probability, first determine whether the events are
disjoint or overlapping. Disjoint events , or mutually exclusive events ,
are events that have no outcomes in common. Overlapping events are
events that have one or more outcomes in common.
For example, suppose you roll a number cube. The Venn diagrams below
illustrate examples of disjoint events and overlapping events.
Review Help
For help with using Venn
diagrams, see p. 784.
Disjoint Events
Overlapping Events
Event A: Roll a 2.
Event A: Roll an even number.
Event B: Roll an odd number.
Event B: Roll a prime number.
6
Event
A
2
4
Example 1
Event
B
1 3 5
1
Event
A
6
4
2
Event
B
5
3
Identifying Disjoint and Overlapping Events
Tell whether the events are disjoint or overlapping.
a. Roll a number cube.
b. Randomly select a student.
Event A: Roll a number less than 4.
Event A: Select a 7th grader.
Event B: Roll a 5.
Event B: Select a boy.
Solution
a. The outcomes for event A are 1,
b. Because some 7th graders
2, and 3. The outcome for event B
are boys, the events have
is 5. There are no outcomes in
outcomes in common.
common.
Answer The events are
Answer The events are disjoint.
overlapping.
Checkpoint
1. Suppose you choose a book to read. Are the events “choosing a hard
cover book” and “choosing a fiction book” disjoint or overlapping?
Lesson 11.8
Probabilities of Disjoint and Overlapping Events
627
Page 2 of 6
Probability of Disjoint Events The Venn
diagram shows two disjoint events that
involve rolling a number cube.
Event A: Roll a number less than 4.
Event
A
1 2 3
Event
B
5
6
4
Event B: Roll a number greater than 4.
There are 6 possible outcomes. There are 5 favorable outcomes for the
5
6
event A or B. So, P(A or B) . You can also find P(A or B) by finding the
sum of the probability of event A and the probability of event B.
3
6
2
6
5
6
P(A or B) P(A) P(B) This result suggests the following rule.
Probability of Disjoint Events
Words For two disjoint events, the probability that either of the
events occurs is the sum of the probabilities of the events.
Algebra If A and B are disjoint events, then
P(A or B) P(A) P(B).
Example 2
Finding the Probability of Disjoint Events
Raffle Fifty tickets are sold for a raffle. You buy 2 tickets, and your
friend buys 3 tickets. One ticket is randomly chosen as the winning
ticket. What is the probability that you or your friend wins the raffle?
Solution
The events are disjoint because you and your friend cannot both win.
Event A: You win the raffle.
Event B: Your friend wins the raffle.
P(A or B) P(A) P(B)
Probability of disjoint events
2
50
3
50
Substitute probabilities.
5
50
1
10
Add. Then simplify.
1
10
Answer The probability that you or your friend wins the raffle is .
Checkpoint
2. In an election, candidate A received 35% of the vote, candidate B
received 22% of the vote, and candidate C received 43% of the vote.
If you randomly select a person from all who voted, what is the
probability that the person voted for either candidate A or
candidate B?
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Chapter 11
Data Analysis and Probability
Page 3 of 6
Probability of Overlapping Events The
Venn diagram shows two overlapping
events that involve rolling a number cube.
Event
A
1
3
Event A: Roll a number less than 5.
Event
B
2
4
6
5
Event B: Roll an even number.
There are 6 possible outcomes. There are 5 favorable outcomes for the
5
6
event A or B. So, P(A or B) . There are 2 favorable outcomes for the
2
6
event A and B. So, P(A and B) . These outcomes are counted twice
when you find the sum of P(A) and P(B). In order to find P(A or B)
using the sum of P(A) and P(B), you must subtract P(A and B) once.
4
6
3
6
2
6
5
6
P(A or B) P(A) P(B) P(A and B) Probability of Overlapping Events
Words For two overlapping events, the probability that either of
the events occurs is the sum of the probabilities of the events
minus the probability of both events.
Algebra If A and B are overlapping events, then
P(A or B) P(A) P(B) P(A and B).
Example 3
Finding the Probability of Overlapping Events
You roll two number cubes, one red and one blue. What is the
probability that you roll a 4 on at least one of the number cubes?
Review Help
In Example 2, a table is used
to display all the possible
outcomes. For help with
making tables, see p. 799.
Solution
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
The table lists all the possible
outcomes of rolling the two
number cubes.
Event A: The red number
cube shows 4.
Event B: The blue number
cube shows 4.
6
36
P(B) P(A or B) P(A) P(B) P(A and B)
6
36
6
36
1
36
11
36
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
6
36
P(A) 4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
1
36
P(A and B) Probability of overlapping
events
Substitute probabilities.
Simplify.
11
36
Answer The probability that you roll a 4 is .
Lesson 11.8
Probabilities of Disjoint and Overlapping Events
629
Page 4 of 6
Complementary Events Two events are complementary events if they
are disjoint events and one event or the other must occur. The sum of the
probabilities of complementary events is always 1. If you know the
probability of an event A, then the probability of the complementary
event, not A, is given by the following rule.
P(not A) 1 P(A)
Example 4
Finding the Probability of Complementary Events
Weather The forecast claims that there is a 40% probability of snow
tomorrow. What is the probability that it will not snow tomorrow?
Solution
The events snow and no snow are complementary events because one
or the other must occur.
P(no snow) 1 P(snow)
Probability of complementary events
1 0.4
Substitute 40%, or 0.4, for P(snow).
0.6
Subtract.
Answer The probability that it will not snow tomorrow is 0.6, or 60%.
11.8
Exercises
INTERNET
More Practice, p. 813
CLASSZONE.COM
eWorkbook Plus
Guided Practice
Vocabulary Check
1. Explain what it means for two events to be disjoint.
2. What is the complement of rolling an odd number on a number cube?
Skill Check
You roll a number cube. Tell whether the events are disjoint or overlapping.
Then find P(A or B).
3. Event A: Roll an even number.
Event B: Roll a 3.
5. Event A: Roll a multiple of 3.
Event B: Roll a 5.
Guided Problem Solving
4. Event A: Roll a number less than 2.
Event B: Roll an odd number.
6. Event A: Roll an odd number.
Event B: Roll a 1.
7. Marbles A bag contains 3 red marbles, 4 black marbles, 4 blue marbles,
and 3 yellow marbles. You randomly draw a marble from the bag. What
is the probability that you draw a red or a blue marble?
630
Chapter 11
1
Are the events disjoint or overlapping?
2
Find the probability that you draw a red marble.
3
Find the probability that you draw a blue marble.
4
Find the probability that you draw a red or a blue marble.
Data Analysis and Probability
Page 5 of 6
Practice and Problem Solving
Homework Help
Example
1
2
3
4
Exercises
8–11,
8–15, 23
8–11,16–18, 24
19–23, 25
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
The spinner is divided into equal parts. For the specified events A and B,
tell whether the events are disjoint or overlapping. Then find P(A or B).
8. Event A: Stops on an even number.
6
Event B: Stops on green.
9. Event A: Stops on an odd number.
1
5
2
Event B: Stops on a multiple of 3.
4
10. Event A: Stops on red.
3
Event B: Stops on blue.
11. Event A: Stops on blue.
Event B: Stops on a multiple of 3.
Events A and B are disjoint. Find P(A or B).
4
17
12. P(A) , P(B) 25
25
7
2
13. P(A) , P(B) 15
15
7
3
14. P(A) , P(B) 20
20
13
22
15. P(A) , P(B) 50
50
Events A and B are overlapping. Find P(A or B).
7
3
1
16. P(A) , P(B) , P(A and B) 10
10
10
8
3
13
17. P(A) , P(B) , P(A and B) 25
25
25
3
1
1
18. P(A) , P(B) , P(A and B) 36
6
9
Given P(A), find P(not A).
19. P(A) 45%
20. P(A) 82%
11
21. P(A) 23
23. Blood Types A city surveyed its population
16
22. P(A) 33
Blood Types in a City
to find the blood types of its residents. Each
resident has only one type of blood. The results
of the survey are given in the circle graph.
In the
Real World
Blood Cells The colorenhanced photo of red and
white blood cells above was
taken through a microscope.
Red blood cells have a
diameter of about
7 106 cm. White blood
cells have a diameter as
large as 1.2 105 cm.
How many times wider are
white blood cells than red
blood cells?
a. Individuals with type A blood can
accept type A or type O blood in a blood
transfusion. Find the probability that blood
from a randomly selected resident will be
either type A or type O.
O 45%
A 40%
B 11%
AB 4%
b. Individuals with type B blood can accept type B or type O blood in a
blood transfusion. Find the probability that blood from a randomly
selected resident will be either type B or type O.
c. What is the probability that a randomly selected resident does not
have type O blood?
d. What is the probability that a randomly selected resident does not
have type A or type B blood?
Lesson 11.8
Probabilities of Disjoint and Overlapping Events
631
Page 6 of 6
24. Pets A survey found that in a city, 19% of the population owned dogs,
14% of the population owned cats, and 6% of the population owned
both a cat and a dog. Find the probability that a randomly chosen
member of the population owns a cat or a dog.
25. Speech You and your friend are among 12 students who will give
speeches in front of the class. The teacher will randomly choose one
of the 12 students to give the first speech. Each student has an equal
probability of being chosen. Find the probability that neither you nor
your friend will be chosen to give the first speech.
26.
Writing
The probability that a coin shows heads when you flip it is
0.5. Explain why, when you flip two coins at the same time, the
probability that at least one coin shows heads is not 0.5 0.5 1.0.
27. Critical Thinking You know that P(A and B) 0. What can you conclude
about event A and event B? Explain.
28. Critical Thinking You roll a number cube. What is the probability that
the number cube shows a number that is not even or that is not a
multiple of 3?
29. Challenge The numbers 1 through 20 are written on separate slips
of paper and placed in a hat. One of the slips is randomly drawn.
a. Draw a Venn diagram that shows the possible outcomes for three
events: the number drawn is less than 5, the number drawn is
prime, and the number drawn is a multiple of 3.
b. Find the probability that the number drawn is less than 5 or a
prime number or a multiple of 3.
c. Write a general rule for finding P(A or B or C) for overlapping events
A, B, and C.
Mixed Review
Find the product. (Lesson 5.4)
4 5
30. p 7 9
5
3
31. p 8
4
9
5
32. p 10 12
13
15
33. p 18
48
34. Map A cartographer is creating a map of 5 states. Each state is to be a
different color. The cartographer will use red, green, blue, brown, and
yellow. How many different ways can the cartographer color the map?
(Lesson 11.6)
Find the number of combinations. (Lesson 11.7)
35.
Standardized Test
Practice
C
11 5
36. 9C4
37.
C
10 6
38.
C
15 8
39. Multiple Choice A meteorologist forecasts that there is a 30%
probability of rain tomorrow. What is the probability that it will
not rain tomorrow?
A. 30%
B. 60%
C. 70%
D. 100%
40. Short Response The numbers 1 through 10 are written on separate
slips of paper and placed in a hat. One of the slips is randomly drawn.
Find the probability that the slip shows a number that is neither odd
nor greater than 5. Explain your reasoning.
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Chapter 11
Data Analysis and Probability