Page 1 of 5
Disjoint Events
BEFORE
Now
WHY?
You found the probability of a
single event.
You’ll find the probability that
either of two events occurs.
So you can analyze results of a
survey, as in Exs. 17–20.
Word Watch
disjoint events, p. 657
overlapping events, p. 657
complementary events,
p. 659
Disjoint events are events that have no outcomes in
common. Overlapping events are events that have
one or more outcomes in common. The Venn diagrams
below show how the events that involve rolling a number
cube are related.
Disjoint events
Overlapping events
Event A: Get an odd number.
Event A: Get a number less than 3.
Event B: Get a 4.
Event B: Get an even number.
Event A
5
1
6
1
1
4
3
EXAMPLE
Event A
Event B
2
Event B
6
4
2
5
3
Disjoint and Overlapping Events
Tell whether the events involving the spinner are
disjoint or overlapping.
Event P: Get an odd number.
2
7
9
15
6
8
4
Event Q: Get a prime number.
3
Solution
with
Notetaking
To help you understand
the difference between
disjoint and overlapping
events, you can make a
concept grid for each
term.
Make a list of the outcomes for each event. Then determine whether
the events have any outcomes in common.
Event P: 3, 7, 9, 15
List the odd numbers.
Event Q: 2, 3, 7
List the prime numbers.
ANSWER There are outcomes in common, so the events are overlapping.
Your turn now
Tell whether the events involving the spinner in
Example 1 are disjoint or overlapping.
1. Event J: Get an even number.
Event K: Get a number greater than 9.
Lesson 13.5
Disjoint Events
657
Page 2 of 5
Probability of Disjoint Events The Venn
diagram at the right shows two disjoint
events that involve rolling a number cube.
Event A: Get a number less than 3.
Event B
4
6
5
Event A
1
2
3
Event B: Get a number greater than 3.
5
6
The probability of event A or event B is because there are 5 favorable
outcomes out of 6 possible outcomes. Another way to find the
probability of event A or event B is to add the probabilities of each
2
6
3
6
5
6
event: .
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).
Sports
EXAMPLE
2
Probability of Disjoint Events
Arena Events The table shows the events
at an arena during one year. What is the
probability that a randomly chosen event is
an ice hockey game or an ice show?
Solution
The events are disjoint because two arena
events cannot occur at the same time.
Event
Basketball
Ice hockey
Concert
Ice show
Trade show
Other
Percent
24%
23%
16%
11%
8%
18%
P(ice hockey) P(ice show) 23% 11%
34%
■
ANSWER The probability that an arena event is either an ice hockey game
or an ice show is 34%.
Arena Events
To prepare for an ice hockey
game, the arena staff needs
to remove the 264 pieces of
the wooden floor used for a
basketball game. If each
piece of the floor weighs
180 pounds, how much does
the entire floor weigh?
658
Chapter 13
Your turn now
Probability
Refer to Example 2.
2. What is the probability that a randomly chosen event is a concert or
a trade show?
Page 3 of 5
Complementary Events Two disjoint events in which one or the other
must occur are called complementary events . If event A and event B
are complementary events and you know the probability of one event,
you can use the following rule to find the probability of the other event.
This rule comes from the fact that the sum of the
probabilities of two complementary events is 1.
P(B) 1 P(A)
EXAMPLE
3
Probability of Complementary Events
Sign Language At your school, 3% of the students know sign language.
What is the probability that a randomly chosen student at your school
does not know sign language?
Solution
P(does not know) 1 P(knows)
Write verbal model.
1 0.03
Substitute 3%, or 0.03,
for P(knows).
0.97
Subtract.
ANSWER The probability that a randomly chosen student does not know
sign language is 0.97, or 97%.
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 717
Getting Ready to Practice
1. Vocabulary Describe the difference between disjoint events and
overlapping events.
Events A and B are disjoint events. Find P(A or B).
2. P(A) 0.3
P(B) 0.2
3. P(A) 0.25
P(B) 0.35
4. P(A) 0.12
P(B) 0.3
Events A and B are complementary events. Find P(A).
5. P(B) 0.4
6. P(B) 0.75
7. P(B) 0.23
8. Pets Are the events “dog owner” and “cat owner” disjoint or
overlapping? Explain.
Lesson 13.5
Disjoint Events
659
Page 4 of 5
Practice and Problem Solving
with
Homework
Tell whether the events are disjoint or overlapping.
Example Exercises
1
9–10
2
11–13, 17–20,
24–27
3
14–20
9. Event A: A student knows how to play a musical instrument.
Event B: A student doesn’t know how to play a musical instrument.
10. Event A: A student knows how to play the clarinet.
Event B: A student knows how to play the guitar.
Online Resources
Events A and B are disjoint events. Find P(A or B).
CLASSZONE.COM
11. P(A) 0.24
• More Examples
• eTutorial Plus
P(B) 0.37
12. P(A) 33%
P(B) 8%
13. P(A) 16.1%
P(B) 28.2%
Events A and B are complementary events. Find P(A).
14. P(B) 0.51
2
15. P(B) 5
History The circle graph shows the
results of a survey. Find the probability
that a randomly chosen student who
participated in the survey responded
as indicated.
16. P(B) 64%
Which ancient civilization
would you visit?
Ancient
Greece 29%
Aztecs 11%
17. Chose Ancient Egypt or the Aztecs
18. Chose Ancient Greece or the Incas
19. Didn’t choose Ancient Egypt
Incas 8%
Ancient Egypt 52%
20. Didn’t choose the Aztecs
Extended Problem Solving In Exercises 21–23, use the following
information. You and 19 other students volunteer to clean up the
outside of a community center. The table shows the number of
volunteers who will be randomly assigned to work in specific areas.
Machu Picchu is an
ancient Incan fortress
in Peru.
21. Calculate Find the probability that you
will be assigned to work in the front
yard, the back yard, or the side yard by
dividing the sum of the numbers of
volunteers assigned to these areas by
the total number of volunteers.
Area
Front yard
Back yard
Playground
Side yard
Volunteers
6
8
4
2
22. Calculate Find the probability of the events in Exercise 21 by adding
the probabilities of being assigned to the front yard, of being
assigned to the back yard, and of being assigned to the side yard.
23. Conjecture Can you find the probability that one of three disjoint
events will occur by adding the probabilities of each event? Explain.
660
Chapter 13
Probability
Page 5 of 5
You randomly choose a letter from the word PINEAPPLE. Find the
probability of choosing either of the given letters.
24. P or N
25. E or A
26. I or P
27. N or E
Critical Thinking In Exercises 28–30, tell whether the statement is
always, sometimes, or never true.
28. Two disjoint events are complementary.
29. Two overlapping events are disjoint.
30. Two complementary events are overlapping.
31. Writing Events A and B involve rolling two number cubes. If the
probability of event A is 0.75 and the probability of event B is 0.5, are
events A and B disjoint events? Explain.
32. Challenge There are red, blue, and green marbles in a bag. The
probability of randomly choosing a blue marble is 0.3, and the
probability of randomly choosing a blue or red marble is 0.7. If there
are a total of 20 marbles in the bag, how many marbles of each color
are in the bag?
Mixed Review
Solve the equation. (Lesson 11.1)
33. x 2 3 39
34. 20 y 2 24
35. 55 z 2 9
36. Find the number of ways to arrange the letters in the word EQUATION.
(Lesson 13.4)
Basic Skills Find the mean, median, mode, and range of the data.
37. 8, 10, 10, 13, 9, 7, 12, 11, 10
38. 23, 15, 23, 27, 28, 25, 26, 17
39. 3.6, 2.7, 3.8, 3.6, 2.9, 3.3, 2.5
40. 8.4, 8.5, 7.3, 7.5, 7.9, 8.9, 8.5, 6.2
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
41. Multiple Choice The spinner shown is divided
into equal parts. What is the probability that
the spinner lands on green or an odd number?
1
6
A. 1
3
B. 1
2
5
6
C. D. 6
5 11
2
1
8
3
7
10
12
9
4
42. Multiple Choice What is the probability that the
spinner shown does not land on green?
1
3
F. 1
2
G. 2
3
H. 5
6
I. Lesson 13.5
Disjoint Events
661