Download C.5 Probability Distributions

APPENDIX C.5
C.5
Probability Distributions
A35
Probability Distributions
■ Construct frequency distributions.
■ Find probability distributions.
Frequency Distributions
As discussed in Section 11.1, a function that assigns a numerical value to each of
the outcomes in a sample space is called a random variable. For instance, if an
experiment consists of tossing three coins, then the outcomes could be assigned
the numbers 3, 2, 1, and 0, depending on the number of heads in the outcome.
Definition of Discrete Random Variable
Let S be a sample space. A random variable is a function x that assigns a
numerical value to each outcome in S. If the set of values taken on by the
random variable is finite, then the random variable is discrete.
The number of times a specific value of x occurs is the frequency of x and
is denoted by n共x兲. A table showing the frequency of all possible values of the
random variable is called a frequency distribution.
Example 1
STUDY TIP
In Example 1, note that the
sample space consists of eight
outcomes, each of which is
equally likely. The sample space
does not consist of the outcomes
“zero yes responses,” “one yes
response,”, “two yes responses,”
and “three yes responses.” You
cannot consider these events to
be outcomes because they are
not equally likely.
Finding Frequencies
Three people are asked whether they enjoy watching horror movies. A random
variable assigns the number 0, 1, 2, or 3 to each possible outcome, depending on
the number of “yes” responses.
S ⫽ 再YYY, YYN, YNY, YNN, NYY, NYN, NNY, NNN冎
3
2
2
1
2
1
1
0
Find the frequencies of 0, 1, 2, and 3.
To find the frequencies, simply count the number of occurrences of
each value of the random variable, as shown in the frequency distribution below.
SOLUTION
Random variable, x
0
1
2
3
Frequency of x, n共x兲
1
3
3
1
✓CHECKPOINT 1
You have pennies, nickels, and dimes in your pocket. You remove one coin from
your pocket at random, then another. A random variable assigns the number 0,
1, or 2 to each possible outcome, depending on the number of pennies that you
remove. Find the frequencies of 0, 1, and 2. ■
A36
APPENDIX C
Probability and Probability Distributions
A histogram is a graphical representation of a frequency distribution.
Histograms can be done in different ways. For instance, Figure C.18 shows three
different styles of histograms that can be constructed for the frequency distribution in Example 1.
n(x)
3
2
1
Example 2
x
0
1
2
Constructing a Frequency Distribution
3
Two six-sided dice are tossed. Let the random variable x be the sum of the points
on the two dice. Construct a frequency distribution for this experiment.
n(x)
3
SOLUTION
2
The sums of points on the two dice range between 2 and 12, as
follows.
1
x
x
0
1
2
1
2
3
2
3
4
5
6
7
8
9
10
11
12
3
n(x)
3
2
1
x
FIGURE C.18
n共x兲
Event
共1, 1兲
共1, 2兲, 共2, 1兲
共1, 3兲, 共2, 2兲, 共3, 1兲
共1, 4兲, 共2, 3兲, 共3, 2兲, 共4, 1兲
共1, 5兲, 共2, 4兲, 共3, 3兲, 共4, 2兲, 共5, 1兲
共1, 6兲, 共2, 5兲, 共3, 4兲, 共4, 3兲, 共5, 2兲, 共6, 1兲
共2, 6兲, 共3, 5兲, 共4, 4兲, 共5, 3兲, 共6, 2兲
共3, 6兲, 共4, 5兲, 共5, 4兲, 共6, 3兲
共4, 6兲, 共5, 5兲, 共6, 4兲
共5, 6兲, 共6, 5兲
共6, 6兲
1
2
3
4
5
6
5
4
3
2
1
The frequency distribution can be constructed as follows.
Frequency Distribution
n(x)
Frequency of x
6
5
Random variable, x
2
3
4
5
6
7
8
9
10
11
12
Frequency of x, n共x兲
1
2
3
4
5
6
5
4
3
2
1
4
3
A histogram for this frequency distribution is shown in Figure C.19.
2
1
x
0
2
4
6
8
10
Random variable
FIGURE C.19
12
✓CHECKPOINT 2
Use a graphing utility to create a histogram similar to the one shown in Figure
C.19, representing the frequencies for tossing four coins. Let the random variable be the number of heads that turn up. ■
There are often many different values of x, so it is not feasible to list each
value in a frequency distribution. In such cases, you can construct a grouped
frequency distribution, as shown in Example 3.
APPENDIX C.5
Example 3
Probability Distributions
A37
Constructing a Grouped Frequency Distribution
A city has 48 general practice physicians who treated the following numbers
of patients during a two-week period in January of 2008. Construct a grouped
frequency distribution and histogram for these data.
107
105
150
109
171
153
162
193
153
171
163
107
184
167
164
150
118
124
170
149
167
138
142
162
177
195
171
100
107
192
102
127
163
174
144
134
145
193
141
147
100
187
141
191
129
153
132
177
To begin constructing a grouped frequency distribution, you must
first decide on the number of groups. There are several ways to group these data.
However, the smallest number is 100 and the largest is 195, so 10 groups of 10
each are appropriate. The first group is 100–109, the second group is 110–119,
and so on. By tallying the data in the 10 groups, you can obtain the following
grouped frequency distribution.
SOLUTION
n共x兲
Group
Event
100–109
110–119
120–129
130–139
140–149
150–159
160–169
170–179
180–189
190–199
100, 100, 102, 105, 107, 107, 107, 109
118
124, 127, 129
132, 134, 138
141, 141, 142, 144, 145, 147, 149
150, 150, 153, 153, 153
162, 162, 163, 163, 164, 167, 167
170, 171, 171, 171, 174, 177, 177
184, 187
191, 192, 193, 193, 195
Total
8
1
3
3
7
5
7
7
2
5
48
A histogram for this grouped frequency distribution is shown in Figure C.20.
✓CHECKPOINT 3
Grouped Frequency Distribution
45
63
33
30
68
27
61
38
45
35
6
34
21
33
32
60
37
71
64
54
38
43
81
52
14
n(x)
Frequency
The numbers of minutes 25
Internet subscribers spent on
the Internet in the past week are
listed below. Construct a grouped
frequency distribution and
histogram for these data.
8
7
6
5
4
3
2
1
x
100 110 120 130 140 150 160 170 180 190 200
Patients treated
■
FIGURE C.20
APPENDIX C
Probability and Probability Distributions
Probability Distributions
If each outcome in a sample space S is equally likely, then the probability of x is
given by
P共x兲 ⫽
Frequency of x
n共x兲
⫽
Number of outcomes in S n共S兲
where n共S兲 is the number of outcomes in the sample space. Note that if the range
of a discrete random variable consists of m different values 再x1, x2, x3, . . . , xm冎,
then the sum of the frequencies of the xi is equal to n共S兲. This can be written as
n共x1兲 ⫹ n共x 2兲 ⫹ n共x3兲 ⫹ . . . ⫹ n共xm 兲 ⫽ n共S兲.
The collection of probabilities corresponding to the values of the random variable
is the probability distribution of the random variable. For example, the probability distribution for the random variable representing the number of heads that
turn up when three coins are tossed and a corresponding graphical representation
are shown in Figure C.21.
Probability Distribution
P(x)
3
8
Probability
A38
Random variable, x
0
1
2
3
Probability of x, P共x兲
1
8
3
8
3
8
1
8
1
4
1
8
x
0
1
2
3
Random variable
FIGURE C.21
An important fact about the probability distribution shown is that the sum of
the probabilities of the values of the random variable is 1. So,
P共0兲 ⫹ P共1兲 ⫹ P共2兲 ⫹ P共3兲 ⫽
1 3 3 1
⫹ ⫹ ⫹ ⫽ 1.
8 8 8 8
This is true of the probability distribution for any discrete random variable.
Probability Distribution for a Discrete Random Variable
If the range of a discrete random variable consists of m different values
再x1, x 2 , x3 , . . . , x m 冎
then the sum of the probabilities of xi is equal to 1. This can be written as
P共x1兲 ⫹ P共x2 兲 ⫹ P共x3兲 ⫹ . . . ⫹ P共xm 兲 ⫽ 1.
APPENDIX C.5
Example 4
A39
Probability Distributions
Finding a Probability Distribution
Using the results of Example 2, sketch the graph of the probability distribution
for the random variable giving the sum of the points when two six-sided dice are
tossed.
As shown in Example 2, the sums of the points on the two dice range
from 2 to 12, as follows.
SOLUTION
x
n共x兲
Event
共1, 1兲
共1, 2兲, 共2, 1兲
共1, 3兲, 共2, 2兲, 共3, 1兲
共1, 4兲, 共2, 3兲, 共3, 2兲, 共4, 1兲
共1, 5兲, 共2, 4兲, 共3, 3兲, 共4, 2兲, 共5, 1兲
共1, 6兲, 共2, 5兲, 共3, 4兲, 共4, 3兲, 共5, 2兲, 共6, 1兲
共2, 6兲, 共3, 5兲, 共4, 4兲, 共5, 3兲, 共6, 2兲
共3, 6兲, 共4, 5兲, 共5, 4兲, 共6, 3兲
共4, 6兲, 共5, 5兲, 共6, 4兲
共5, 6兲, 共6, 5兲
共6, 6兲
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
5
4
3
2
1
Total
36
The number of outcomes in the sample space is 36. So, the probability of each
value of the random variable is as shown below.
Random variable, x
2
3
4
5
6
7
8
9
10
11
12
Probability of x, P共x兲
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
A graph of this probability distribution is shown in Figure C.22.
Probability Distribution
P(x)
Probability
0.18
0.15
0.12
0.09
0.06
0.03
x
2
3
4
5
6
7
8
9
10 11 12
Random variable
FIGURE C.22
✓CHECKPOINT 4
Four coins are tossed. Sketch the graph of the probability distribution for the
random variable giving the number of heads that turn up. Use the results of
Checkpoint 2 and a graphing utility. ■
A40
APPENDIX C
Probability and Probability Distributions
Example 5
Finding a Probability Distribution
Amount of Claim
Probability
$0 –$1000
0.477
$1001– $2000
0.184
$2001– $3000
0.111
Amount of claim
Number of claims
$3001– $4000
0.069
$4001– $5000
0.030
$5001– $6000
0.020
$6001– $7000
0.009
$7001 and over
0.099
$0–$1000
$1001–$2000
$2001–$3000
$3001–$4000
$4001–$5000
$5001–$6000
$6001–$7000
$7001 and over
Total
405,825
156,330
94,213
58,458
25,913
17,356
7,589
84,560
850,244
A health insurance company finds that the claims it paid during the past five years
are in the following categories.
FIGURE C.23
Construct a probability distribution for this data.
Because the total number of claims paid during the five-year period
was 850,244, the probability that a claim will be in the group $0 –$1000 is
SOLUTION
Probability Distribution
P(x)
P共$0–$1000兲 ⫽
Probability
0.5
0.4
405,825
⬇ 0.477.
850,244
Similarly, the probability that a claim will be in the group $1001– $2000 is
0.3
0.2
P共$1001–$2000兲 ⫽
0.1
x
7001+
6001–7000
5001–6000
4001–5000
3001–4000
2001–3000
1001–2000
0–1000
Amount of claim (in dollars)
FIGURE C.24
156,330
⬇ 0.184.
850,244
By continuing this process, you can obtain the probability distribution shown in
Figure C.23. A graph of this probability distribution is shown in Figure C.24.
✓CHECKPOINT 5
Construct a probability distribution using the data and results of Checkpoint 3.
CONCEPT CHECK
1. What is a random variable?
2. If the set of values taken on by a random variable is finite, then the
random variable is ______.
3. A table showing the frequency of all possible values of a random variable
is a ______ ______. It can be represented graphically by a ______.
4. What is a probability distribution?
■
APPENDIX C.5
Skills Review C.5
A41
Probability Distributions
The following warm-up exercises involve skills that were covered in earlier sections. You will use
these skills in the exercise set for this section. For additional help, review Section C.1.
In Exercises 1–3, determine the sample space for the experiment.
1. Two coins are removed at random from a purse that contains pennies, nickels, dimes, and quarters.
2. Two four-sided dice are tossed and the sum of the points is recorded.
3. One six-sided die and one four-sided die are tossed and the sum of the points is recorded.
In Exercises 4– 6, find the indicated probability. Use the results of Exercises 1–3.
4. Two coins are removed at random from a purse that contains pennies, nickels, dimes, and quarters. What is the
probability that both coins are pennies?
5. Two four-sided dice are tossed and the sum of the points is recorded. What is the probability that the sum of the
points is 6?
6. One six-sided die and one four-sided die are tossed and the sum of the points is recorded. What is the probability
that the sum of the points is 6?
Exercises C.5
1. Coin Toss A coin is tossed twice. A random variable
assigns the number 0, 1, or 2 to each possible outcome,
depending on the number of tails that turn up. Find the
frequencies of 0, 1, and 2.
2. Two Coins You have a mixture of pennies, nickels,
dimes, and quarters in your pocket. You remove one coin
from your pocket at random, then another. A random
variable assigns the number 0, 1, or 2 to each possible
outcome, depending on the number of pennies that you
remove. Find the frequencies of 0, 1, and 2.
3. Dice Two four-sided dice are tossed. Let the random
variable x be the sum of the points on the two dice. Find the
frequency of each possible value of x.
4. Dice One six-sided die and one four-sided die are tossed.
Let the random variable x be the sum of the points on the
two dice. Find the frequency of each possible value of x.
5. Exam Two students answer a true-false question on an
examination. A random variable assigns the number 0, 1, or
2 to each outcome, depending on the number of answers of
true among the two students. Find the frequencies of 0, 1,
and 2.
6. Exam Five students answer a true-false question on an
examination. A random variable assigns the number 0, 1, 2,
3, 4, or 5 to each outcome, depending on the number of
answers of true among the five students. Find the frequencies of 0, 1, 2, 3, 4, and 5.
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
In Exercises 7–10, use the following data
26
23
37
35
5
36
18
9
19
28
18
39
34
32
23
15
17
4
8
21
33
35
6
13
19
10
14
20
16
39
12
32
32
2
16
26
12
20
21
25
7. Construct a frequency distribution using intervals of width
5, starting with the interval 0 < x ≤ 5.
8. Construct a frequency distribution using intervals of width
8, starting with the interval 0 < x ≤ 8.
9. Construct a frequency distribution using intervals of width
7, starting with the interval 1 < x ≤ 8.
10. Construct a frequency distribution using intervals of width
9, starting with the interval 1 < x ≤ 10.
11. Numbers of Employees Twenty medium-sized companies (between 50 and 100 employees) are located in the
same city. The numbers of employees of the companies are
as follows.
54
73
94
70
78
82
100
79
87
91
67
80
100
78
97
72
75
60
92
86
(a) Construct a frequency distribution using the following
ranges: 50–59, 60–69, 70–79, 80–89, 90–100.
(b) Construct a histogram for the frequency distribution.
A42
APPENDIX C
Probability and Probability Distributions
12. Retirement Contributions The employees of a company contribute 7% of their biweekly salary to a companysponsored retirement plan. The contributed amounts (in
dollars) for the company’s 35 employees are as follows.
100
126
140
152
120
200
135
124
104
136
130
98
172
126
148
136
114
127
155
112
161
117
143
92
116
156
168
157
194
146
209
133
124
115
96
(a) Construct a frequency distribution using groups of 20,
with the first group 90 < x ≤ 110.
(b) Construct a histogram for the frequency distribution.
13. Basketball Scores A basketball team’s scores during
the playing season are as follows.
54
60
76
68
52
54
50
62
56
65
56
62
61
78
49
72
74
60
58
53
58
55
58
70
46
57
64
69
71
45
17. Use a graphing utility to construct a probability distribution
for the percentages of profit. Use intervals of width 0.5,
starting with the interval 14.0 < x ≤ 14.5.
18. Use a graphing utility to construct a probability distribution
for the percentages of profit. Use intervals of width 1.0,
starting with the interval 14.0 < x ≤ 15.0.
19. Use a graphing utility or spreadsheet software program to
construct a probability distribution for the following frequency distribution.
Random variable, x
1
2
3
4
5
6
Frequency of x, n共x兲
2
4
6
6
4
2
20. Use a graphing utility or spreadsheet software program
to construct a probability distribution for the following
frequency distribution.
(a) Construct a frequency distribution using groups of five,
with the first group 45 ≤ x < 50.
Random variable, x
2
3
4
5
6
7
(b) Construct a histogram for the frequency distribution.
Frequency of x, n共x兲
2
4
6
8
10
12
Random variable, x
8
9
10
11
12
Frequency of x, n共x兲
10
8
6
4
2
14. Qualifying Times
event are as follows.
1.14
1.30
1.15
1.26
1.54
1.24
1.13
1.07
1.22
1.19
1.15
2.16
The qualifying times for a sports
2.00
1.34
1.47
1.41
2.06
2.01
1.20
1.42
In Exercises 21–24, use a graphing utility to construct a
probability distribution for the experiment.
(a) Construct a frequency distribution using intervals of
width 0.2, starting with the interval 1.00 ≤ x < 1.20.
(b) Construct a histogram for the frequency distribution.
21. Three cards are drawn (without replacement) from a standard deck of 52 playing cards, and the number of face cards
is counted.
15. Dice Two four-sided dice are tossed. Sketch the graph of
the probability distribution for the random variable
giving the sum of the points on the two dice.
22. Three cards are drawn (without replacement) from a standard deck of 52 playing cards, and the number of kings is
counted.
16. Dice Three four-sided dice are tossed. Sketch the graph
of the probability distribution for the random variable
giving the sum of the points on the three dice.
23. A baseball player with a batting average of 0.330 comes
to bat three times in a game, and the number of hits is
counted.
Profit In Exercises 17 and 18, use the following data,
which show the percentages of profit for a medical
supply company for the years 1993 through 2008.
Year
Profit
Year
Profit
1993
1994
1995
1996
1997
1998
1999
2000
15.9%
14.9%
14.5%
14.6%
17.1%
15.8%
16.0%
15.5%
2001
2002
2003
2004
2005
2006
2007
2008
15.6%
14.9%
15.6%
18.4%
17.7%
17.0%
17.8%
17.1%
24. Three dice are tossed and the number of 5’s is counted.