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.