chapter 9: probability and statistics

T H E M E : Sports

When the Cubs send their right-handed power hitter to the plate in theninth inning, the Mets counter with a left-handed pitcher. Why? The

manager of the Mets is simply “playing the odds.”

Since its inception, baseball has kept meticulous records. By studying the data,managers, players, announcers, and fans use the concepts of probability andchance to make predictions. Today, players and managers use computers andcalculators to record and analyze data. They know that the more effectivelythey use statistics and probability, the better they will do their jobs.

• Team Dietitians (page 391) plan meals and nutritional plans forathletes. They use their knowledge of nutrition to help team membersmaintain overall health, muscle health, and bone strength.

• Physical Therapists (page 411) determine exercises to strengthenmuscles after injuries. Through specially designed exercise programs, theyimprove mobility, relievepain, and reduce injuries.

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Probability and Statistics

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Chapter 9 Probability and Statistics

Use the tables for Questions 1–4.

1. For each player, divide the at bats by the total of home runs andround to the nearest hundredth. Compare the unit rates. Whichplayer had the best unit rate?

2. What percent of Barry Bond’s home runs were hit in August andSeptember?

3. Find the average number of at bats per game for eachplayer. Round to the nearest hundredth. Which player had the greatest average?

4. In 2001, Bonds hit 49 singles. For what percent of his at bats did he reach first base by hitting a single?

CHAPTER INVESTIGATIONBaseball has been called “America’s Pastime.” In recentyears, the game’s appeal has become international. Usingstatistics and probability, fans at home can predict what will happen when the bases are loaded in the bottom of the ninth inning.

Working TogetherGather baseball statistics and design your baseball simulation gameusing dice and percentile cards. Make a lineup and play a game.Decide whether the game’s outcome matches your predictions. Usethe Chapter Investigation icons to guide your group.

Data Activity: Home Run Greats

381

Thome: 12.30; Rodriguez: 10.95; Bonds: 6.52;Maris: 9.67; Ruth: 9.00. Bonds’s rate is best.

about 33%

About 10%

3. Thome: 3.64;Bonds: 3.11;Rodriguez: 3.85;Maris: 3.66;Ruth: 3.58;Rodriguez’saverage is thegreatest.

Home Run Greats–Then and Now

Player Year Home Games At bats Batting Runsruns played average batted in

Jim Thome 2003 47 159 578 .266 131Barry Bonds 2001 73 153 476 .328 137Alex Rodriguez 2002 57 162 624 .300 142Roger Maris 1961 61 161 590 .269 142Babe Ruth 1927 60 151 540 .356 164

Home Runs Month by Month

Player Year Apr May Jun Jul Aug Sept OctJim Thome 2003 4 8 9 6 10 10 0Barry Bonds 2001 11 17 11 6 12 12 4Alex Rodriguez 2002 9 8 7 12 12 9 0Roger Maris 1961 1 11 15 13 11 9 1Babe Ruth 1927 4 12 9 9 9 17 0

The skills on these two pages are ones you have already learned. Stretch yourmemory and complete the exercises. For additional practice on these and moreprerequisite skills, see pages 654-661.

PERCENTS, DECIMALS, AND FRACTIONS

Convert each fraction or decimal to a percent. Round to the nearest tenth if necessary.

1. �34

� 2. 0.86 3. �78

� 4. 0.93 5. �23

�

6. 0.5 7. �38

� 8. 0.42 9. �176� 10. 0.38

11. �16

� 12. 0.64576 13. �261� 14. 0.19823 15. �

24

76�

Convert each percent to a decimal. Round to the nearest thousandth if necessary.

16. 46% 17. 83% 18. 29% 19. 15% 20. 12%

21. 18.76% 22. 9.3825% 23. 78.6215% 24. 64.93% 25. 21.748%

Convert each percent to a fraction in lowest terms.

26. 80% 27. 50% 28. 68% 29. 92% 30. 75%

31. 64.2% 32. 39.8% 33. 20% 34. 19.6% 35. 51.9%

MEASURES OF CENTRAL TENDENCY

Find the mean, median, mode and range of each set of data. Round to the nearest tenth if necessary.

36. 74 75 79 76 77 37. 30 32 34 36 3874 78 72 71 39 37 35 34 33

38. 40 48 52 47 56 49 39. 17 12 13 16 22 2143 55 46 48 51 19 18 14 20 15 11

40. 88 87 81 92 86 87 89 41. 62 63 67 68 65 69 6490 93 91 85 92 94 61 65 66 60 67 65 63

REDUCING FRACTIONS

Determine the greatest common factor of the numerator and denominator.Divide both the numerator and denominator by the factor to write the fractionin lowest terms.

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Chapter 9 Probability and Statistics

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9 Are You Ready?Refresh Your Math Skills for Chapter 9

75%

50%

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mean � 16.5; median� 16.5; no mode;range � 11

mean � 48.6;median � 48; range �16; mode � 48

mean � 88.8; median� 89; mode � 87, 92;range � 13

mean � 64.6; median � 65; mode� 65; range � 9

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Chapter 9 Are You Ready?

HISTOGRAMS

Draw a histogram for each frequency chart.

54. 55.

56. 57.

58. 59.

AREA

Find the area of the shaded region of each figure. Use 3.14 for �. Round to thenearest hundredth if necessary.

60. 61. 62.

4.2 m

8

16 in.

4 in.

4 in.

7 cm

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Work with a partner.

1. Discuss: How large must a group of people be for there to be a 50% chance that two members of the group will share a birthday? Make a guess.

2. Take a survey of the students in one of your classes to look for common birthdays. What did you learn?

BUILD UNDERSTANDING

Recall that probability theory is the mathematics of chance. Probability is used to describe how likely it is that an event will occur. Probabilities are reported using fractions, decimals, and percents. The greater the probability of an event, the more likely the event is to occur.

One way to find the likelihood of an event occurring in the real world is to conduct an experiment. In an experiment, you either take a measurement or make an observation. A probability determined byobservation or measurement is called experimental probability. An outcome is theresult of each trial of an experiment.

The experimental probability of an event E is defined as follows.

P(E) �

E x a m p l e 1

RECREATION Lions fans attending a recent 3-game serieswere asked whether the team should have a mascot. Thetable shows how many fans thought it should.

According to these results, what is the probability that aLions fan wants the team to have a mascot?

SolutionUse the experimental probability formula.

P(E) �

P(fan favoring mascot) ��36

88

81

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The probability that a fan interviewed at a Lions game will favor having a team mascot is 0.75.

number of favorable observations of E��

total number of observations

number of favorable observations of E��

total number of observations

Chapter 9 Probability and Statistics384

9-1 Review Percents and ProbabilityGoals ■ Find experimental and theoretical probabilities.

Applications Sports, Card Games, Test Taking

Game AttendanceIn favor

of mascot

Friday 681 388Saturday 527 428Sunday 928 786

Results will vary. Probability theory shows that a group of 23 people has a50% chance that two members share a birthday; for a group of 40, the probability is about 90%.

The set of all possible outcomes of an experiment is called thesample space.

E x a m p l e 2

SPORTS A baseball team has 8 pitchers and 3 catchers. Themanager is choosing a pitcher-catcher combination. How many are possible?

SolutionOne way to show all possible outcomes is to use ordered pairs. Forexample, use the numbers 1–8 for pitchers and the letters A–C forthe catchers.

(1, A) (2, A) (3, A) (4, A) (5, A) (6, A) (7, A) (8, A)

(1, B) (2, B) (3, B) (4, B) (5, B) (6, B) (7, B) (8, B)

(1, C) (2, C) (3, C) (4, C) (5, C) (6, C) (7, C) (8, C)

There are 24 possible pitcher-catcher combinations.

Another way to show the sample space is to use atree diagram.

You can use probability to predict the number oftimes an event will occur.

E x a m p l e 3

SPORTS A softball player has had 24 hits in herfirst 60 times at bat. Predict her total number ofhits in 330 at bats.

SolutionFirst, use the outcomes that have already occurredto find the experimental probability of the playergetting a hit each time at bat.

P(hit) � �26

40� � 0.4

Then multiply that result by the total number oftimes at bat.

0.4(330) � 132

Based on the player’s first 60 times at bat, you can predict that she will get 132 hits in 330 at bats.

As you increase the number of trials in a probability experiment, theexperimental probability will probably get closer to the theoretical probability. Forexample, when tossing a fair coin, P(heads) � �

12

�. The more often you toss thecoin, the closer you will come to tossing an equal number of heads and tails.

Lesson 9-1 Review Percents and Probability 385

Problem SolvingTip

Probability is oftenexpressed as a percent. A percent is a ratio of anumber compared to 100.For example, 87% means

87 : 100 or �18070

� or 0.87.

A decimal can beconverted to a percent by moving the decimaltwo places to the right.So, 0.4 can also be writtenas 40%.

Pitchers Catchers Outcomes

1

2

3

4

5

6

7

8

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B

C

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C

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(1, B)

(1, C)

(2, A)

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(2, C)

(3, A)

(3, B)

(3, C)

(4, A)

(4, B)

(4, C)

(5, A)

(5, B)

(5, C)

(6, A)

(6, B)

(6, C)

(7, A)

(7, B)

(7, C)

(8, A)

(8, B)

(8, C)

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The theoretical probability of an event, P(E), is the ratio of the number offavorable outcomes to the number of possible outcomes in the sample space.

P(E) �

E x a m p l e 4

CARD GAMES A card is picked at random from a standard deckof 52 cards. Find P(face card).

SolutionThere are 52 possible outcomes. There are 12 favorable outcomes—4 jacks, 4 queens, and 4 kings.

P(face card) � �15

22� � �

133�

TRY THESE EXERCISES

1. SPORTS Of the first 1500 fans to pass through the turnstiles at the stadium,1050 had reserved seats. What is the probability that the next person throughwill have a reserved seat?

2. WRITING MATH A person flips a penny, a nickel, and a dime. Each coin canland with heads up (H) or tails up (T). Make a tree diagram to show whatdifferent outcomes are possible.

3. You roll a die 60 times. Predict the number of times you will roll an evennumber greater than 2.

4. GAMES A spinner for a game is divided into ten equal sections numbered 1through 10. What is the probability of spinning 7 or higher?

PRACTICE EXERCISES • For Extra Practice, see page 690.

A die is rolled 100 times with the following results.

What is the experimental probability of rolling each of the following results?

5. 2 6. 6 7. a number less than 4

8. What is the theoretical probability of rolling a number less than 4?

9. CHAPTER INVESTIGATION Working with a partner, prepare to make yourown baseball simulation game. To begin, gather batting statistics on at least18 players. You may use statistics from the most recent baseball season orstatistics from prior years. For each player, you will need to know the numberAt Bats (AB), Hits (H), Doubles (2B), Triples (3B), Walks (BB) and Strike Outs(SO). This information is available in the newspaper, in sports magazines oron team websites.

number of favorable outcomes��number of possible outcomes


Reading Math

The odds in favor of anevent are expressed as theratio of the number offavorable outcomes to thenumber of unfavorableoutcomes. For example,when you roll a die, theodds of getting a 4 are1:5—because there is 1way the event can occur,and 5 ways it cannot.

0.7

20 times

�25

�

0.18 0.2 0.55

0.5

(H, H, H), (H, H, T), (H, T, H), (T, H, H), (H, T, T), (T, H, T),(T, T, H), (T, T, T)

Answers will vary.

Outcome 1 2 3 4 5 6

Frequency 15 18 22 9 16 20

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387

List all the elements of the sample space for each of the following experiments.

10. You flip a dime and a quarter.

11. You spin each of these spinners once.

Find the probability of each of the following:

12. CARD GAMES drawing a black jack from astandard deck of cards

13. rolling a die and getting a multiple of 3

14. EDUCATION guessing correctly on one true-false question on a test

15. EDUCATION guessing incorrectly on a multiple-choice question with fourchoices

16. reaching into a drawer without looking and taking out a pair of black socks,when the drawer contains 3 pairs of black socks, 2 pairs of white socks, 1 pairof red socks, and 2 pairs of blue socks

17. SPORTS A basketball player has made 96 free-throws in his last 128attempts. What is the probability he will be successful in his next attempt?How many successful free-throws do you predict this player will make in 500attempts?

18. WRITING MATH You want to predict how many students in your school areright-handed. Describe how you would do it.

19. WEATHER The weather forecaster predicts a 25% chance of rain in your areatomorrow. Describe what this forecast means.

20. TRANSPORTATION A bus breaks down while traveling between two citiesthat are 200 mi apart. What is the probability the breakdown is within 25 miof either city?

EXTENDED PRACTICE EXERCISES

Write whether each of the following probabilities can be determinedexperimentally or theoretically.

21. The probability that Player A will win when Player A plays Player B in tennis.

22. The probability that a person will win the state’s lottery.

23. The probability that a family with 4 children has all boys.

24. The probability that it will snow on January 9.

MIXED REVIEW EXERCISES

Graph the image of triangle ABC with vertices at A(�2, 1) B(4, 2),and C(1, 4), under each transformation from the original position. (Lesson 8-11)

25. 3 units up 26. reflected across the x-axis

Graph the image of parallelogram PQRS with vertices at P(�1, 1) Q(2, 3),R(2, 6), and S(�1, 4), under each transformation from the original position.(Lesson 8-11)

27. 6 units down 28. reflected across the x-axis

A

BC

1

23

Lesson 9-1 Review Percents and Probability

(H, T), (H, H), (T, H), (T, T)

(A, 1), (A, 2), (A, 3), (B, 1), (B, 2), (B, 3), (C, 1), (C, 2), (C, 3)

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�13

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0.5

0.75

0.375

0.25

0.75; 375

Sample answer: Find the ratio of the number of right-handed students in your class to the number of students in your class. Then multiply by the total number of

students in yourschool.

experimentally

theoretically

theoretically

experimentally

See additional answers.


Of past days when weatherconditions were similar to those predicted for tomorrow, it rained 25% of the time.

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Sometimes a probability problem is too difficult to solve theoretically or experimentally. One way to solve such a problem is to model it with a simulation to estimate the probability. Simulations often use random numbers; these can be readily generated and recorded by a computer. You can also find random numbers by rolling dice, flipping coins, using numbered slips of paper, or spinning a spinner.

P r o b l e m

MARKETING Each box of Batter-Up Pancake Mix contains one of 5 different classic baseball cards. Assuming that the company has evenly distributed the cards among the boxes, what is the probability that you will find all 5 cards if you buy 10 boxes of Batter-Up?

Solve the ProblemWork with a partner. Use 5 slips of paper numbered 1–5; each sliprepresents a box of cereal. Place the slips in a paper bag. Then drawone slip of paper from the bag, record its number, and place it backin the bag. Repeat the process until you have drawn and recorded 10 slips of paper. If you have drawn each of the 5 numbers at leastonce, consider the outcome of your experiment to be successful. Ifyou have not drawn every number, the outcome is unsuccessful.

Repeat the experiment 50 times, recording all results in a table.Indicate which trials are successful. Then write a ratio comparingsuccessful outcomes to the total number of outcomes. This ratio will be an estimate of the probability of getting every card in the setwhen you buy 10 boxes of cereal.

TRY THESE EXERCISES

1. COMPUTER SCIENCE A computer generates a list of random 2-digitnumbers. Zero cannot be the first digit. What is the probability that arandomly chosen number from the list contains the digit “1”?

2. MARKETING A candy company has placed 6 different prizes in its boxes.The prizes are uniformly distributed among the boxes of candy, only one perbox. Describe a simulation you could do to estimate the probability ofgetting all 6 prizes in a 12-pack of candy.

3. WRITING MATH Describe a simulation you could do to find out how manycards you would expect to have to draw from a standard deck to get twokings.

4. TEST TAKING Suppose you are going to take a 10-question true-false test onthe evolution of idiomatic phrases in Sri Lanka. You will need to guess eachtime, and you want to find out your chances of scoring 65% correct or better.Design a simulation to determine your chances. Hint: Use coin flipping.


9-2 Problem Solving Skills:Simulations

Problem SolvingStrategies

Guess and check

Look for a pattern

Solve a simplerproblem

Make a table, chartor list

Use a picture,diagram or model

Act it out

Work backwards

Eliminate possibilities

Use an equation orformula

✔

1:5 (18 of the 90 possible numbers)

Answers will vary.

Answers will vary.

Answers will vary.

Interactive Labmathmatters3.com


PRACTICE EXERCISES

5. A family wants to have 3 children. Do the following simulation to determinethe probability that, if they do have 3 children, all 3 will be the same gender.

a. Use 3 coins. Let heads � girl and tails � boy. Toss the coins and recordthe results. Repeat the coin tosses until you have recorded 50 sets of 3tosses each.

b. Count the successful outcomes—those with either 3 heads or 3 tails.

c. Write a ratio comparing successful outcomes with total outcomes. Whatis your experimental probability of having 3 children, all of whom are ofthe same gender?

6. SPORTS One baseball player always arrives at the stadium between 5:30 P.M.and 6:30 P.M. for a night game. If batting practice always starts between 6:00 P.M. and 7:00 P.M., what is the probability on any given night that thisplayer will arrive before batting practice begins? Design and do a simulationto find out.

7. PROGRAMMING A pitcher throws strikes about 60% of the time. If hethrows 80 pitches, how many might be strikes? The following computerprogramming statements can be used to simulate 80 pitches.

1 S � 0 The experiment begins with no successes.

2 FOR I � 1 TO 80 80 pitches

3 X � RND(1) Generate a random decimal.

4 IF X � .6 THEN S � S � 1 If the decimal � 0.6, increase S by 1.

5 NEXT I Simulate the next pitch.

6 PRINT S Total number of strikes.

7 END

Use the program to simulate the problem. Then describe how you wouldadjust the program for a pitcher who throws strikes 40% of the time.

8. TALK ABOUT IT Petra is designing a simulation to determine the chance ofguessing the correct answer on a multiple-choice test. Each item on the testhas three choices. Petra plans to roll a 6-sided die to simulate randomguesses. A roll of 1 or 2 will indicate choice A; a roll of 3 or 4, choice B; and aroll of 5 or 6, choice C. Will Petra’s simulation work? Explain.


Find the slope of each line using the given information. (Lesson 6-1)

9. A(�2, �1), B(5, 3) 10. C(7, 2), D(3, �2) 11. E(1, 8), F(�3, �4)

12. �3x � 2y � 9 13. 4y � 2x � 8 14. �12 � x � 4y

15. G(�3, 5), H(�3, 9) 16. I(�3, 5), J(3, �5) 17. K(2, �1), L(�8, �1)

Solve each proportion. (Lesson 7-1)

18. �5x

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52� 19. �

192� � �

2x0� 20. �

143� � �

1x6� 21. �

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Lesson 9-2 Problem Solving Skills: Simulations 389

Five-stepPlan

1 Read2 Plan3 Solve4 Answer5 Check

Answers will vary.

Answers will vary.

Change line 4 to: “IF X � .4 THEN S � S � 1”.

Yes; the six-sided die can be used as long as each of the three choices isassigned an equal number of outcomes from the die roll.

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PRACTICE LESSON 9-1A card is picked at random from a standard deck of 52 cards. Find each theoreticalprobability.

1. P(heart) 2. P( jack) 3. P(red card ) 4. P(black 2)

5. P(2 or 3) 6. P(7 of hearts) 7. P(3 � card � 8) 8. P(king of clubs)

You flip a coin four times. Find each theoretical probability.

9. P(no tails) 10. P(exactly one head) 11. P(2 tails, 2 heads) 12. P(4 tails)

13. P(� 2 heads) 14. P(0 or 1 head) 15. P(3 tails) 16. P(1, 2, 3 or 4 heads)

You roll a pair of dice and calculate the sum. Find each theoretical probability.

17. P(7) 18. P(11) 19. P(even) 20. P(1)

21. P(12) 22. P(4 or 5) 23. P(6) 24. P(� 6)

25. P(10, 11, or 12) 26. P(9) 27. P(� 11) 28. P(2)

29. A basketball player has made 48 free throws in her last 72 attempts. What isthe probability she will be successful on her next attempt? How many freethrows do you predict she will make in 600 attempts?

30. A car breaks down while traveling between two cities that are 300 mi apart.What is the probability that the breakdown is within 18 mi of either city?

PRACTICE LESSON 9-231. A computer generates a list of random 3-digit numbers. Zero cannot be the

first digit. What probability would you expect for a number in the list tocontain the digit “4”?

32. Each box of Toasted Crunchies cereal contains a single prize. There are 4different prizes uniformly distributed among all boxes of cereal at theproduction facility. Describe a simulation you could do to estimate theprobability of getting all 4 prizes if you buy 10 boxes of Toasted Crunchies?

33. Perform and document a simulation to determine the probability that afamily having 2 children will have 1 boy and 1 girl.

34. Perform and document a simulation to find the chances of scoring 50% orhigher on a 5 question multiple choice test. Each question has four choices,and you guess on each question.

35. Agnes walks her dog each night outside the grounds of Tellco Corporationbetween 7:30 and 8:00 P.M. First shift workers leave the grounds between 7:30and 8:30 P.M. each night. Describe a simulation using two spinners that youcould use to calculate the probability of Agnes seeing first-shift workersleaving the grounds during her walk.

36. Describe a simulation you could do to find out how many cards you wouldhave to draw from a standard deck to get a pair of hearts.


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Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Review and Practice Your Skills

Workplace Knowhow

PRACTICE LESSON 9-1–LESSON 9-2List all the elements of the sample space for each experiment. (Lesson 9-1)

37. Tossing a quarter and a nickel. 38. Spinning each of these two spinners:

39. Rolling a die and tossing a dime.

40. A computer randomly generates a list of 2-digitnumbers. Zero cannot be the first digit. What isthe probability that the next number generatedis a multiple of 3? (Lesson 9-2)

2 3

1 4

A B

C D

Chapter 9 Review and Practice Your Skills 391

D ietetics has applications in many different career fields. Clinical dietitiansplan meals and nutritional plans for groups such as schools and hospitals.

Community dietitians inform the public on nutritional habits to prevent diseaseand promote healthy lifestyles.

Consultant dietitians provide advice in the areas of sanitation and safetyprocedures. In the sports world, nutrition is important for maintaining musclehealth and bone strength.

As the dietitian for a baseball team, you need to determine whether the teammembers are getting enough calcium in their diets. To find out, you separate theplayers into three categories: infielders, outfielders, and pitchers. During a buffet,you chart the food selections of the players.

1. Find the average amount of calcium consumed by players in each group.

A. Infielders B. Outfielders C. Pitchers1B–300 mg RF–113 mg P1–233 mgC–220 mg CF–197 mg P2–212 mg2B–186 mg LF–262 mg P3–184 mg3B–216 mg P4–258 mgSS–102 mg

2. Your research shows that during lunch, calcium intake is �34

� of the amount

consumed during the buffet. Breakfast amounts are �35

� of the buffet amount.

How many milligrams of calcium is each player getting per day?

3. If 450 mg of calcium is recommended per player per day, which players areconsuming too little calcium?

(H, H), (T, H), (H, T), (T, T)(1, H), (4, H), (1, T), (4, T), (2, H), (5, H), (2, T), (5, T), (3, H), (6, H), (3, T), (6, T)

�13

�

{(A, 1), (B, 1), (C, 1), (D, 1), (A, 2), (B, 2),(C, 2), (D, 2), (A, 3), (B, 3), (C, 3), (D, 3),(A, 4), (B, 4), (C, 4), (D, 4)}

204.8 mg

190.67 mg221.75 mg

1B, 705 mg; C, 517mg; 2B, 437.1 mg;3B, 507.6 mg; SS,239.7 mg; RF,265.55 mg; CF,462.95 mg; LF,615.7 mg; P1,547.55 mg; P2,498.2 mg; P3, 432.4mg; P4, 606.3 mg

2B, SS, RF, P3

Career – Dietitian

mathmatters3.com/mathworks

http://mathmatters3.com/mathworks

Play this game with a partner. Use a pair of 6-sided dice.

1. Player A rolls first. If Player A rolls a 7, Player B wins the game. If not, Player B rolls.

2. If Player B rolls a 7 or an 11, Player A wins. If not, it is Player A’s turn. Continue taking turns until there is a winner.

3. Play the game several times. Do you think one player has a better chance of winning?

BUILD UNDERSTANDING

A compound event consists of two or more simple events. Compound events mayinvolve finding the probability of one event and another event occurring. Or, theprobability of one event or another event occurring. For example, when rolling adie, rolling a number that is even (event A) and greater than 2 (event B) is writtenP(A and B). Rolling a number that is even or greater than 2 is written P(A or B).

If two events cannot occur at the same time, they are called mutually exclusiveevents. For example, it is impossible to draw from a standard deck of playingcards a card that is both a heart and a club.

When two events A and B are mutually exclusive, the probability of thecompound event A or B can be found using the formula P(A or B) � P(A) � P(B).

E x a m p l e 1

SPORTS A standard deck of playing cards is used to simulate a baseball game.During the game, players draw a card at random. Spade number cards greaterthan 4 represent doubles. Home runs are represented by either a 3 or a queen.

a. Find P(spade and a number card greater than 4).

b. Find P(3 or queen).

SolutionThere are 52 possible outcomes.

a. There are 6 outcomes in which spades are greater than 4: 5, 6, 7, 8, 9, and 10 ofspades.

So, P(spade and number greater than 4) � �562� or �

236�.

The probability that the card will be a spade and a number greater than 4 is �236�.


9-3 Compound EventsGoals ■ Find probabilities of compound events.

Applications Sports, Games, Business

Answers will vary.

b. A card cannot be both a 3 and a queen at the same time, so theevents are mutually exclusive.

P(3 or queen) � P(3) � P(queen)

� �542� � �

542�

� �582� or �

123�

The probability that the card will be a 3 or a queen is �123�.

Events that can happen at the same time are not mutually exclusive.

E x a m p l e 2

GAMES You draw a card at random from a standard deck of playingcards. Find the probability that the card is a club or a jack.

SolutionThese are not mutually exclusive events, because a card can be botha club and a jack.

There are 13 clubs, so P(club) � �15

32�.

There are 4 jacks, so P(jack) � �542�.

However, 1 club is a jack. P(club and jack) � �512�.

P(club or jack) � �15

32� � �

542� � �

512�

� �15

62� � �

143�

The probability that the card is a club or a jack is �143�.

Example 2 illustrates that when two events A and B are not mutually exclusive,the probability of A or B can be found using the formula

P(A or B) � P(A) � P(B) � P(A and B)

Suppose you know the probability of event A. The set of outcomes in the samplespace, but not in A, is called the complement of the event.

P(not A) � 1 � P(A)

E x a m p l e 3

You select a marble from this jar without looking. You know �

15

� of the marbles are red and �15

� are blue. What is the probability you will select neither a red nor a blue marble?

SolutionFind the probability of selecting red or blue.

P(red or blue) � �15

� � �15

� � �25

�

Lesson 9-3 Compound Events 393

CheckUnderstanding

Classify each of thefollowing pairs of eventsas mutually exclusive ornot mutually exclusive.

1. drawing the 4 of clubs; drawing the 4 of spades

2. rolling two dice thatshow a sum of 8;rolling two dice andgetting differentnumbers

3. tossing two coins andgetting two tails;tossing two coins andgetting two heads

1. mutually exclusive2. not mutually exclusive3. mutually exclusive





Find the probability of not selecting red or blue.

P(neither red nor blue) � 1 � P(red or blue)

� 1 � �25

�

� �35

�

The probability that you will not select a red or blue marble is �35

�.

TRY THESE EXERCISES

1. A die is rolled. Find the probability of rolling a 1 or a 2.

2. Two coins are tossed. Find the probability that the coins show two heads orone tail and one head.

3. A card is drawn at random from a standard deck of playing cards. Find theprobability that it is a 7 or a black card.

4. Two dice are rolled. Find the probability that the sum of the numbers is notgreater than 5.

5. Each student in your class writes his or her full name on a piece of paper.The pieces are put in a box and one is chosen without looking. What is theprobability that your name will not be chosen?


GAMES A player rolls two 6-sided dice.

6. List the sample space for the rolls.

7. Find the probability that the sum of the numbers rolled is odd and greaterthan 5.

8. Find the probability that the sum of the numbers rolled is either 8 or 10.

9. Find P(not even).

10. Find P(neither odd nor sum of 6).

SPORTS Suppose you are on a team in the midst of a losing streak. The coachdecides to “shake up” the line-up. He chooses the batting order by putting nineplayers’ names into a hat and pulling them out one by one. The player whosename is drawn first bats first, the second bats second, and so on.

11. What is the probability you will bat second or fourth?

12. What is the probability you will bat fifth, or in the first third of the battingorder?

13. What is the probability you will bat first, or in the first third of the battingorder?

14. What is the probability you will bat in the last third of the batting order, or inan odd-numbered position?

15. BUSINESS Ms. Garrett plans to select a worker at random for a specialtraining seminar. If there are 14 workers in sales, 6 in accounting and 5 inpersonnel, what is the probability that the worker will be from either sales oraccounting?


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number of pieces of paper � 1��

number of pieces of paper


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GAMES You spin the spinner shown. Findeach probability.

16. spinning a 2 or an odd number

17. spinning an odd or an even number

18. spinning a multiple of 2 or a multiple of 3

CARD GAMES To begin a game, the dealerdraws a card at random from a standard deck of playing cards.

19. Find the probability that the card is a 2, a 5, or a face card.

20. Find the probability the card is a 7, an 8, or a red card.

A card is drawn at random from a standard deck of playing cards. For each event, estimate whether the probability is closer to 1, �

12

�, or 0.

21. P(red or face card) 22. P(2, 3, or 4)

23. P(black and face card) 24. P(black, heart, or 7)

PHOTOGRAPHY A team photo album contains photos of the players bythemselves, the coaching staff by themselves, and the players and the coachestogether. The players are in 15 of the photos and the coaches are in 12 of thephotos. In 6 photos, the players and coaches appear together.

25. How many photos are in the album?

26. If you open the album at random to one of the team photos, what is theprobability that the photo shows only coaches?


27. WRITING MATH Suppose you roll a pair of dice. Why are rolling a multipleof 6 and a multiple of 4 not mutually exclusive events?

28. A pair of dice is rolled. What is the probability that the sum of the numbers is neither 5 nor a multiple of 2?

29. SPORTS Batting average is found by dividing hits by at-bats. In 1941, TedWilliams batted over .400 when he got 185 hits in 456 official at-bats for anaverage of .406. Suppose Ted Williams had 1 more at-bat in 1941. Based onhis performance all season, what would you estimate the probability of hisnot getting a hit that time?

30. CHAPTER INVESTIGATION For each player you selected, the number ofhits (H) is equal to the sum of the singles (S), doubles (2B), triples (3B), andhome runs (HR). Since the number of singles is not usually reported as aseparate statistic, calculate the number of singles (S) for each player using the formula: H � (2B � 3B � HR) � S. You may want to use acomputer spreadsheet.


Evaluate each expression when a � 5 and b � �4. (Lesson 1-8)

31. a 2b 2 32. a 2 � b 2 33. ab 3 34. a 3b 2

35. a(a 2b 2) 36. ab � ab 2 37. a 3b 38. (a 2 � b 2)2

39. (a 2)(b 2)(a 2) 40. a 2 � b(ab)2 41. �(b 2) 42. (�b)2

Lesson 9-3 Compound Events 395

12

3

4

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8

9

10

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0

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Sample answer: 12 is a multiple of both numbers.

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400

2000

10,000

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1625

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81

16

Answers will vary.


Work with a partner. You will need a six-sided die and a coin.

1. Suppose one person rolls the die while the other tosses or flipsthe coin. What do you think the probability is of rolling a 3 andlanding the coin heads up? Record your prediction.

2. Check your prediction by rolling the die and tossing the coinuntil you get both of these outcomes.

3. Share the results of your experiment with other groups.

BUILD UNDERSTANDING

When the outcome of one event does not affect the outcome ofanother event, the events are said to be independent. To find theprobability of both events occurring, multiply the probabilities ofeach event.

If A and B are independent events, then P(A and B) � P(A) � P(B)

To emphasize that A and B do not characterize a single event, sometimes P(A andB) is written P(A, then B).

E x a m p l e 1

A bag contains 3 white softballs, 2 yellow softballs, 3 green softballs, and 4 redsoftballs. You reach into the bag without looking and take out a ball. Youreplace it and then take out another ball at random. Find the probability thatthe first ball is red and the second ball is white.

SolutionBecause the first ball is replaced before the second is taken, the sample space of12 balls is the same for each event. The two events are independent. Multiply tofind the probability that both will occur.

P(red, then white) � P(red) � P(white)

� �

� �13

� � �14

�

� �112�

The probability of picking red, then white, is �112�.

number of white balls��total number of balls

number of red balls��total number of balls


9-4 Independent andDependent EventsGoals ■ Find the probability of dependent and independent

events.

Applications Sports, Surveys, Scheduling

When the outcome of one event is affected by the outcome ofanother, the events are said to be dependent.

E x a m p l e 2

SPORTS Six teams—the Panthers, Tigers, Lions, Bears, Cheetahs,and Elephants—are in the lottery round for this year’s draft picks.The name of each team is written on a card and placed in a box.

To determine who gets the first lottery pick, one card will be drawnat random and not replaced. Then a second card will be drawn atrandom to determine the second pick. What is the probability thatthe Bears get the first draft choice and the Lions get the seconddraft choice?

SolutionBecause the first card is not replaced, the sample space for the second drawinghas been changed. The second event is dependent on the first event.

Probability of first event

P(Bears) � � �16

�

Probability of second event

P(Lions, after Bears) � � �15

�

Multiply the probabilities.

P(Lions, after Bears) � �16

� � �15

�

� �310�

The probability of drawing the Bears first and the Lions second is �310�.

E x a m p l e 3

A bag contains 3 green marbles, 2 red marbles, 4 yellow marbles, and 1 black marble. Two are taken at random from the bag without replacement. Find P(green, then green).

SolutionP(first green marble) � � �

130�

P(second green marble) � � �130

�

�

11

� � �29

�

Multiply the probabilities.

P(green, then green) � �130� � �

29

�

� �960� � �

115�

The probability of picking green, then green, is �115�.

number of green marbles��total number of marbles

number of green marbles��total number of marbles

number of Lions cards��new total number of cards

number of Bears cards��total number of cards

Lesson 9-4 Independent and Dependent Events 397

CheckUnderstanding

Are these eventsindependent ordependent?

1. tossing a coin twice

2. picking two marblesfrom a bag withoutreplacing the first one

3. choosing a captain andthen choosing a co-captain

4. rolling three dice

1. independent2. dependent3. dependent4. independent





TRY THESE EXERCISES

A group of numbered cards contains three 3s, four 4s, and five 5s. Cards arepicked at random, one at a time, and then replaced. Find each probability.

1. P(3, then 5) 2. P(4, then odd number) 3. P(even, then not 4)

SPORTS A baseball team has 10 pitchers, 3 catchers, 5 outfielders, and 7infielders on its roster. Two players from this team will be chosen at random torepresent the league in a tour of Japan. Find each probability.

4. P(pitcher, then catcher) 5. P(outfielder, then infielder)

SURVEYS A newspaper survey asked 100 men and 100 women whether theyplanned to vote for a proposed tax increase. Twenty men and 40 women saidthey are in support of the increase. A person from the survey is chosen atrandom. Find each probability.

6. What is the probability that the person chosen is in support of the tax increase?

7. What is the probability that the person is a woman in support of the increase?

8. What is the probability that the person is a man who is against the increase?


A box contains 3 red counters, 4 yellow counters, 2 green counters, and 1 bluecounter. Counters are taken at random from the box, one at a time, and thenreplaced. Find each probability.

9. P(red, then yellow) 10. P(blue, then green) 11. P(red, then not red)

A drawer contains 2 pairs of black socks, 3 pairs of brown socks, a pair of beigesocks, and 6 pairs of white socks. One sock at a time is taken at random from thedrawer and not replaced. Find each probability.

12. P(black, then black) 13. P(white, then black) 14. P(beige, then white)

A billboard says “EAT HERE NOW.” Two letters fall off, one after the other.

15. What is the probability that both letters are vowels?

16. What is the probability that the first letter is an E, and the second letter is notan E?

17. You are given one ticket each to two soccer games at astadium with 48,000 seats. What is the probability you will sitin Section D in the first game, and then Section A in thesecond game, if Section D has 4000 seats and Section A has3000 seats?

SCHEDULING Liu and Michi plan to sign up for a drawing classnext term. Drawing is offered during the first 4 periods of the day,and students are assigned randomly to classes.

18. What is the probability that Liu and Michi will have drawingtogether?

19. What is the probability that both students will have drawingduring first period?


Maracana Stadium, Brazil

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Both spinners are spun. Find each probability.

20. P(red, red)

21. P(red, yellow)

22. P(green, not green)

A golf bag pocket contains 4 yellow golf balls, 3 white balls, 1 green ball, and 4 redballs. You pull out one ball at a time, without replacing it. Find each probability.

23. P(red, then white, then yellow) 24. P(green, then red, then white)

25. WRITING MATH Explain the difference between events that are mutuallyexclusive and those that are independent.


HISTORY Suppose that it is 1944, and the Homestead Grays of the NegroNational League are playing the Birmingham Black Barons of the NegroAmerican League in a “best two out of three” series. Then suppose the Grays are the favored team, and the probability they will win any individual game is �

34

�.

26. What is the probability that the Black Barons win a game?

27. What is the probability that the Grays win in two straight games?

28. What is the probability that the series goes for three games?

29. What is the probability that the Homestead Grays win the series?

30. DATA FILE Use the data on baseball on page 652. Suppose Davis had onemore official at-bat in the 1943 season and Wagner had one more official at-bat in the 1948 season. What is the probability that both would have gone hitless?

31. CHAPTER INVESTIGATION When a baseball player comes to bat, theplayer can get a hit—either a single, a double, a triple or a home run—or theplayer can walk, strike out, or make an out some other way. For each playeryou have chosen, find the probability expressed as a percent that each eventwill occur. For example, to find the probability that a player will walk, dividethe number of walks (BB) by the number of at bats (AB) and convert thedecimal to a percent. Use a spreadsheet or calculator.


Copy quadrilateral ABCD. Then draw its dilation image.(Lesson 8-3)

32. The center of dilation is the origin and the scale factor is 4.

33. The center of dilation is the origin and the scale factor is �34

�.

34. The center of dilation is point A and the scale factor is 2.5.

Lesson 9-4 Independent and Dependent Events 399

y

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Mutually exclusive events cannot occur at the sametime. Independent events can occur at the sametime, but neither event affects the other.



PRACTICE LESSON 9-3A card is picked at random from a standard deck of 52 cards. Find each probability.

1. P(heart and face card) 2. P(jack or queen) 3. P(red or black card)

4. P(black two) 5. P(2 or 3) 6. P(7 of hearts)

7. P(2 � card � 5) 8. P(king of clubs) 9. P(club and (ten or king))

You flip a coin four times. Find each theoretical probability.

10. P(exactly one head) 11. P(2 tails, 2 heads) 12. P(3 or 4 tails)

13. P(more than 2 heads) 14. P(0 or 1 head) 15. P(3 tails)

You roll a pair of dice. Find each theoretical probability.

16. P(sum � 7) 17. P(sum � 11) 18. P(both even)

19. P(1 is rolled) 20. P(4 or 5 is rolled) 21. P(sum � 2)

22. P(sum is odd) 23. P(sum � 6) 24. P(sum � 10, 11, or 12)

25. P(sum is even and � 7) 26. P(sum is odd and � 11) 27. P(values are equal)

28. A spinner has 20 equal sectors numbered 1–20. Are spinning a multiple of 4and multiple of 9 mutually exclusive events? Explain.

29. You spin a spinner with 8 equal sectors, numbered 1–8. What is theprobability of spinning a number that is neither odd nor greater than 6?

PRACTICE LESSON 9-4A drawer contains 7 red shirts, 5 blue shirts, and 4 white shirts. One shirt at atime is taken at random from the drawer and not replaced. Find eachprobability.

30. P(red, then blue) 31. P(red, then not white) 32. P(white, then blue)

33. P(both white) 34. P(not blue, then not red) 35. P(both not blue)

A box contains 4 red cards, 5 black cards, 10 green cards, and 2 blue cards.Cards are taken at random from the box, one at a time, and then replaced. Findeach probability.

36. P(red, then red) 37. P(red, then green, then blue)

38. P(not red, then green) 39. P(black, then black, then not green)

40. P(not green in each of three draws) 41. P(black, then not blue)

42. P(red, then red, then red, then red) 43. P(blue, then blue, then black)

44. You are given one ticket each to two hockey games in an arena with 18,000seats. What is the probability that you will sit in Section B in the first game,and then Section C in the second game, if Section B has 4500 seats andSection C has 3000 seats?


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No; 36 is a multiple of both.

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PRACTICE LESSON 9-1–LESSON 9-4List all the elements of the sample space for each of the following experiments.(Lesson 9-1)

45. tossing a quarter and a nickel 46. choosing a month of the year

47. choosing the day of the week 48. choosing a letter of the alphabet

49. Conduct and document a simulation using a six-question multiple choicetest. If each question has three choices for answers, and you guess on eachquestion, what are your chances of getting 3 or more questions correct?(Lesson 9-2)

Three coins are tossed. Find each probability. (Lesson 9-3)

50. P(at least one heads) 51. P(no tails) 52. P(0 or 1 tails)

53. P(two tails) 54. P(all three coins the same) 55. P(1 or 2 heads)

A box contains 100 cards, numbered from 1–100. Cards are taken at random from the box, one at a time, and not replaced. Find each probability. (Lesson 9-4)

56. P(even number, then odd number) 57. P(multiple of 3, then 50)

58. P(45, then 45) 59. P(99, then 100)

60. P(number � 40, then number � 80) 61. P(prime number, then prime number)

Mid-Chapter Quiz1. How many outcomes are there for an outfit chosen from three pairs of pants

and five shirts? (Lesson 9-1)

2. A basketball player has made 60 out of 125 attempts. How many shots is helikely to make in 500 attempts? (Lesson 9-2)

3. A family has five children. What is P(three boys and two girls)? (Lesson 9-2)

A card is picked at random from a standard deck of 52 cards. (Lesson 9-3)

4. Find P(heart and less than 5) 5. Find P(heart or less than 5)

6. Find P(7 or king) 7. Find P(neither 5 nor diamond)

8. Find P(neither 3 nor red)

A bag contains 7 green marbles, 3 red marbles, and 5 blue marbles. A marble ispicked and replaced. Then another marble is picked. (Lesson 9-4)

9. Find P(green, then red) 10. Find P(two red marbles)

From the same bag, a marble is picked and not replaced. Then another marbleis picked. (Lesson 9-4)

11. Find P(green, then red) 12. Find P(two red marbles)


{HH, HT, TH, TT}

{Sun., Mon., Tues., Wed., Thurs., Fri., Sat.} {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z}

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Answers will vary.

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Work with a partner to answer the questions.

You have just applied for your first set of license plates. Each license plate has 3different letters followed by 3 different digits. The letters and digits are assignedrandomly by the Department of Motor Vehicles.

1. How many arrangements of 3 different letters are possible?

2. How many arrangements of 3 different digits are possible? Remember, toinclude 0 as a digit.

3. Suppose you had hoped that your plate would read “ACE 123.” What is theprobability that you will receive this plate? How do you know?

BUILD UNDERSTANDING

Thus far, you have used either a tree diagram or a set of ordered pairs to find thenumber of outcomes in a sample space. But sometimes the sample space is toolarge to use either of these methods.

Another way to find the total number of outcomes is to use the fundamentalcounting principle. This principle can be used to calculate the number of ways twoor more events can happen in succession. It states that, if an event A can occur in m ways and an event B can occur in n ways, then events A and B can happen inm � n ways.

E x a m p l e 1

TRAVEL Suppose the Carthage College Women’s Basketball team will travel on a road trip to Grand Rapids, Peoria, and Battle Creek. They can go from Kenosha to Grand Rapids by car, train, or bus, then from Grand Rapids to Peoria by bus, train, or plane, and from there to Battle Creek by car, bus, train, or plane. Finally, from Battle Creek, they can either take the bus or the train back to Kenosha. How many different routes are possible on this road trip?

SolutionUse the fundamental counting principle.

The 3 possible routes for the first leg of the trip are car, train, or bus.Then they have 3 possible routes for the second leg of the trip, 4 for the third leg of the trip, and 2 for the return trip to Kenosha.

3 � 3 � 4 � 2 � 72

Seventy-two different routes are possible.


9-5 Permutations and CombinationsGoals ■ Find the number of permutations and combinations

of a set.

Applications Travel, Sports, Office Work

15,600

720

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1600� � �

7120� � �

11,2312,000�

An arrangement of items in a particular order is called a permutation. For the fourletters M, A, T, H, there are 24 different 4-letter permutations.

MATH MAHT MHAT MHTA MTHA MTAHAMTH AMHT AHTM AHMT ATHM ATMHTAMH TAHM TMAH TMHA THAM THMAHAMT HATM HMAT HMTA HTAM HTMA

You can use the fundamental counting principle to find the number ofpermutations of any group of items.

For each arrangement of M, A, T, H, there are 4 choices for the first letter, 3choices for the second, 2 choices for the third, and 1 choice for the fourth.

4 � 3 � 2 � 1 � 24.

4 � 3 � 2 � 1 can be written in factorial notation as 4!

In general, the number of permutations of n different items is written n! and readas n factorial.

E x a m p l e 2

In how many different ways can you arrange your math,science, social studies, language arts, and literature anthologybooks in a row on a shelf?

SolutionThere are five books. Find the number of permutations of five items.

number of permutations of five items � 5!

� 5 � 4 � 3 � 2 � 1 or 120

There are 120 different ways to line up five books on a shelf.

Sometimes you need only part of a set of items, such as selectingtwo of nine players. The number of permutations of n differentitems, taken r items at a time, with no repetitions, is written nPr .Use the formula below to find the number of permutations whenonly part of a set is used.

nPr � �(n

n�

!r)!

�

E x a m p l e 3

SPORTS Eight teams enter a tournament. How many different arrangements offirst-, second-, and third-place winners are possible?

SolutionUse the formula: nPr � �

(nn�

!r)!

�

8P3 � �(8 �

8!3)!

�

� � 8 � 7 � 6 � 336

There are 336 ways for teams to finish first, second, and third.

8 � 7 � 6 � 5 � 4 � 3 � 2 � 1��

5 � 4 � 3 � 2 � 1

Lesson 9-5 Permutations and Combinations 403

Cancel common factors tosimplify the computation.

CheckUnderstanding

What is wrong with thenotation 4P5?

A group of 4items cannot betaken 5 at atime.



For each situation described in the preceding Examples, the orderof the items in consideration is important. A set of items in noparticular order is called a combination. The number ofcombinations of n different items, taken r items at a time, where 0 � r � n, is written nCr. You can use the formula below to find thenumber of combinations of a set of items.

nCr � �(n �

n!r)!r!�

E x a m p l e 4

How many different four-person ensembles can be chosen from apool of 10 musicians?

SolutionThere are 10 people from which to pick, 4 at a time. So, n � 10 andr � 4. Use the formula:

nCr � �(n �

n!r)!r!�

10C4 � �(10 �

10!4)!4!�

�

� �520440�

� 210

There can be 210 different four-person ensembles.

TRY THESE EXERCISES

1. SPORTS A manager is choosing her infield from among 4 third-base players,3 shortstops, 2 second-base players, and 5 first-base players. How manydifferent ways can an infield be chosen?

Tell whether each question involves a permutation or a combination. Then solve.

2. In how many different ways can you arrange the letters a, c, e, g, i, k, and 1?

3. How many different selections of three tapes can be made by a consumerchoosing from among a collection of six tapes?

4. In how many different ways can a starting lineup of 5 players be selectedfrom a group of 12 basketball players?

5. In how many different ways can 4 winners be chosen from 15 contestants?


6. OFFICE WORK A secretary has to create ten new customer files. In howmany different orders can he do this?

7. HIRING Six applicants apply for two jobs. How many different outcomes arepossible?

10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1��(6 � 5 � 4 � 3 � 2 � 1)(4 � 3 � 2 � 1)


Cancel common factors to simplify thecomputation.

Math: Who,Where, When?

Although several 16th-and 17th-centurymathematicians, notablyPascal and Fermat,investigated probability,Jacques Bernoulli isconsidered by some to bethe founder of probabilitytheory. His book, ArsConjectandi, published in1713, is the first bookdevoted entirely to thesubject of probability. Thisbook contains a theory ofcombinations, essentiallythe same as weunderstand it today, aswell as the firstappearance, with today’smeaning, of the wordpermutation.

120

permutation; 5040

combination; 20

combination; 792

permutation; 32,760

10! or 3,628,800

30



8. Ralph is a tour guide. In how many ways can he choose 3museums to visit from the 8 museums in a city?

9. In how many ways can a disk jockey select 5 of the 20 top hits?

10. ERROR ALERT On a test, students must choose 3 out of 5 essayquestions to answer. Dale calculates that there are 60 ways to dothis. What has Dale done wrong?

11. SPORTS Bill, Phil, and Jill are among 12 players competing for 3spots on a table-tennis team. Every player has an equal chance ofmaking the team. Find the probability that all three will make theteam.

12. What is n, if nP3 � 120?

13. Suppose that license plates contain three different letters. What is the probability that Meg’s plates will spell her name?

14. SPORTS How many ways can a batting order be made for 9players if you know that one player has already been designatedto bat first and another to bat fourth?

15. WRITING MATH Find the number of permutations of the letters in the wordshutout. Explain how you did it.

16. Two students out of 8 will be chosen to speak at a school assembly. Howmany different outcomes are possible?

DATA FILE For Exercises 17–18, use the information about the All-AmericanGirls Professional Baseball League on page 652.

17. Suppose you could interview all eight women who were batting championsof the All-American Girls Professional Baseball League to discover whatplaying in this league was like. How many orders for these interviews wouldbe possible?

18. If you were to interview only four of the eight women, what is the probabilitythat you would first interview a player from either Fort Wayne or Rockford?


19. Compare the values of 8C 5 and 8C3. What do you notice?

20. Find the values of 7C 4 and 7C 3. What do you notice?

21. What can you say about the sum of the number of items taken at one timefor each combination shown in Exercises 19 and 20?

22. CRITICAL THINKING Use what you have discovered to quickly find 67C 64.


Add. (Lesson 8-5)

23. � � � � � 24. � � � � �

Find the measure of each angle. (Lesson 3-2)

25. �BZC 26. �CZD 27. �AZC

�23

�41

37

31

02

6�5

34

1�2

40

23

Lesson 9-5 Permutations and Combinations 405

Technology Note

Some calculators have a

factorial key, marked .

To find 7!, enter 7, then

. On graphing

calculators, you canchoose the factorialfunction from a displayedmathematical menu. Thismenu may also includeoperations for findingpermutations andcombinations.

B

A Z

C

D

(4x � 9)°

(3x � 6)°

56

15,504

10. Dale has foundthe number ofpermutations insteadof combinations. Inthis situation, orderdoesn’t matter. Thereare only 10combinations.

�2120�

6

�15,

1600�

7! or 5040

28

8! or 40,320

�114�

They are the same.

Both have values of 35.

They equal the total number of items.

47,905

� �74

31 � �1

4�4

392

1260; Sample answer: Divide 7! by 2! � 2! � 1! � 1! � 1!

51° 39°141°



Work in groups of 3–4 students.

Find out your classmates’ favorite music performer. Make a list of ten popularmusic groups or artists. Using a rating scale of 1–10, survey 25 students. Make a graph to display your findings. Compare findings and graphs with those ofclassmates.

BUILD UNDERSTANDING

Data can be presented in many ways. Graphs are usefulbecause they can help identify characteristics of data. Recallthat a histogram shows frequencies of intervals of data. Astem-and-leaf plot shows all data ordered as in a frequencytable, but also visually, as in a bar graph. It shows how dataare clustered.

A scatter plot is another type of visual display used to explorethe relationship between two sets of data, represented byunconnected points on a grid.

E x a m p l e 1

MANUFACTURING The scatter plot shows the relationshipbetween years of experience and hourly pay at one factory.

a. Why are the scales different?

b. What does each • represent?

c. Find the hourly pay of an employee with 8 years of experience.

d. Describe the relationship between experience and pay?

Solutiona. There are two different sets of data—hourly pay and years

of experience.

b. Each • shows the hourly pay given the years of experience.

c. $10.50

d. Hourly pay usually increases with years of work experience.

A pattern may emerge that shows a relationship between the two setsof data. If data clusters around a line of best fit, or trend line, from thebottom left upward to the top right of the graph, this shows a positivecorrelation between the sets of data. If the line slopes downward fromleft to right, it indicates a negative correlation between the data.


9-6 Scatter Plots and Box-and-Whisker PlotsGoals ■ Interpret and make scatter plots and box-and-

whisker plots.

Applications Manufacturing, Sales, Sports

2 4 6 8 10 12 14 16

12

10

8

6

4

2

0

Ho

url

y Pa

y in

Do

llars

Factory Wages

Years of Experience

Answers will vary.

E x a m p l e 2

SALES Use the scatter plot at the right for thesequestions.

a. What can you say about the correlation between theage of a car and its resale value?

b. Predict the resale value of an 8-year-old car with anoriginal cost of $15,000.

Solutiona. The trend line slopes downward from upper left to

lower right, so there is a negative correlation between a car’s age and its resalevalue.

b. Extend the pattern. A reasonable assumption would be for the resale value tobe about 30% of the original cost for an 8-year-old car. So, a car that cost$15,000 originally might sell for about $4500 after 8 years.

Another way to display data is with a box-and-whisker plot, also known as box plot.This plot shows how data are dispersed around a median, but does not showeach specific item in the data. By examining a box-and-whisker plot, you can tellif data are clustered closely together or spread far apart.

A box-and-whisker plot shows both the median and the extremes of a set of data.It also shows the median of the lower half of the data, called the lower quartile,and the median of the upper half of the data, called the upper quartile. Bothquartiles include the median if the data contain an odd number of items.

E x a m p l e 3

SPORTS Joe DiMaggio played center field for the New York Yankees for 13 years.During each year of his career, he hit the following number of home runs: 29, 46,32, 30, 31, 30, 21, 25, 20, 39, 14, 32, and 12. Make a box-and-whisker plot for thisdata.

SolutionWrite the data in numerical order. Find the least and greatest values, the median,the lower quartile, and the upper quartile.

least value median greatest value� � �

12 14 20 21 25 29 30 30 31 32 32 39 46� �

lower quartile upper quartile

Use points to mark the values below a number line. Draw a boxthat starts and stops at the lower and upper quartiles, and a verticalline at the point for the median. Then draw whiskers, or linesegments, from each end of the box to the least and greatest values.Finally, give your graph a title.

Lesson 9-6 Scatter Plots and Box-and-Whisker Plots 407

0 5 10 15 20 5025 30 35 40 45

DiMaggio’s Home Runs

12 21 30 32 46

CheckUnderstanding

In a box-and-whisker plot,what percent of a set ofdata is represented by thebox? By the whisker tothe right of the box?

1 2 3 4 5 6 7 8

100

80

60

40

20

0

Car

Val

ue

as %

of

Stic

ker

Pric

e

Car Resale Values

Age of Car9 10

50%; 25%





Box-and-whisker plots can be used to compare sets of data.

E x a m p l e 4

Use the box-and-whisker plots below to answer questions about the math testscores of two different classes.

a. Which class had the higher median score?

b. What was the lower quartile in Mr. Pascal’s class?

c. Which class had its scores grouped more closely around itsmedian?

d. For which class were the lowest scores clustered more closely?

e. Which class, as a whole, scored better on the test?

Solutiona. Ms. Cotter’s

b. about 7.2

c. Mr. Pascal’s; the range of the middle 50% of the scores is about 1.8. The rangefor the middle 50% in Ms. Cotter’s class is about 3.5.

d. Ms. Cotter’s; the range for the lowest quarter is about 1. In Mr. Pascal’s class, it is about 1.2.

e. Ms. Cotter’s

TRY THESE EXERCISES

FITNESS Use the scatter plot at theright for Exercises 1–3.

1. What is the weight of the studentwho is 67 in. tall?

2. Does the scatter plot show apositive or negative correlation?

3. Give an estimate of the height of astudent who weighs 145 lb.

4. SPORTS Make a box-and-whiskerplot for the following data.

TOP PRICES OF TICKETS TOSPORTING EVENTS (IN DOLLARS)

45, 55, 40, 60, 15, 25, 35, 30, 10, 40

5. Use the box-and-whisker plot you made in Exercise 4. Are thedata clustered more closely above or below the median?


Test Scores From Two Classes

5 6 7 8 9 10 11 12 13 14

Ms. Cotter’s Class

Mr. Pascal’s Class

120 140 160 180

75

70

65

60

Hei

gh

t in

Inch

es

Heights and Weights

Weight in Pounds

Reading Math

The abbreviations Q1, Q2,and Q3 are sometimesused for the lowerquartile, the median, andthe upper quartile. Theinterquartile range is thedifference between thevalues of the upper andlower quartiles.

Technology Note

Use a graphing calculatorto make a box-and-whisker plot.

1. Enter the data as a listusing the STAT feature.

2. Select the STATPLOTmenu and choosePlot1. Under Type,select the box-and-whisker plot diagram.

3. Adjust the windowdimensions if necessaryand press GRAPH.

4. Use the TRACE featureto find the median andupper and lowerquartiles.

about 130 lbs

positive

above

67–68 in.

Check students’ plots;look for median; 37.5;upper quartile: 45; lowerquartile: 25; extremes:10 and 60


SPORTS This table shows how many points a basketball player scored during his career. Use this information for Exercises 6–8.

6. Make a scatter plot.

7. What is the range of this player’s scoring average?

8. Does your scatter plot show a positive correlation, a negative correlation, or no correlation?

SPORTS These box-and-whisker plots show batting averages for 3 baseball teams.

9. Which team has the highest median batting average?

10. Which team has the smallest range of batting averages?

11. WRITING MATH Why is the right whisker for the Meatballs longer than theleft whisker?

12. CHAPTER INVESTIGATION Create a 10-by-10 table for each player.Number both the columns and rows from 1 to 10. The table represents all thepossible outcomes for a player at bat. Using the percents you calculated, fillin the cells of the table with appropriate abbreviations. For example, in 1998Gary Sheffield hit a home run in 5% of his at bats. To create a table forSheffield, you would write HR in any 5 cells.


Choose the graph you think works best to display the data described.

13. MANUFACTURING To show the relationshipbetween the percent of polyester in an article ofclothing and the price of the article of clothing

14. To show that the test scores in your classclustered around the middle-most score

15. WRITING MATH Is it possible for the mean ofa set of data to fall outside the box part of abox-and-whisker plot? Explain.


Multiply. (Lesson 8-5)

16. 6 � � � 17. 4 � � � 18. 7 � � ��50

�36

74

81

97

32

6�4

Lesson 9-6 Scatter Plots and Box-and-Whisker Plots 409

200 210 220 230 240 250 260 270 280 300290

Player Batting Averages

Artichokes

Onions

Meatballs

Check students’ drawings.

10 points

No correction

Artichokes

Onions

There is a greater range of batting averages in the upper25% of its players than in the lower 25%.

Answers will vary. Sample answers are given.

scatterplot

box-and-whisker-plot

yes, when there are data values far from the median

� �1812

36�24 � �32

43628

� ��350

�2142

4928

Answers will vary.


Age Scoring Average23 1824 17.525 22.526 2427 21.528 2629 23.530 22.531 27.532 20.5


PRACTICE LESSON 9-5Evaluate.

1. 5P3 2. 8P7 3. 4P1 4. 5P5

5. 7P4 6. 9P0 7. 14P7 8. 9P6

9. 5C3 10. 9C6 11. 4C0 12. 6C1

13. 8C7 14. 5C4 15. 6C2 16. 12C4

17. In how many ways can the positions of president, vice president, andsecretary be chosen from a club containing 20 members?

18. In how many ways can a committee of three people be chosen from a clubcontaining 20 members?

19. In how many ways can a volleyball coach choose 6 starters from a team of 14 players?

20. In how many ways can a disc jockey play 3 of the top ten hits?

21. In how many ways can first-place and runner-up winners be chosen from 15 entrants in a contest?

22. In how many ways can the numbers 1, 2, 3, 4, and 5 be arranged in a 5-digitpassword?

23. What is n, if nP2 � 72?

24. Find the values of 10C4 and 10C6. What do you notice?

PRACTICE LESSON 9-6This table shows the appraised value of a house over time.

25. Make a scatter plot of the data.

26. What is the range of appraised values?

27. Does your scatter plot show a positive correlation, a negative correlation, orno correlation?

28. Draw a box-and-whisker plot that has the following attributes: (Lesson 9-6)a. range of 87 d. upper quartile of 162b. median value of 135 e. range of middle 50% of data of 38c. low value of 107

29. Draw a box-and-whisker plot that has the same value for its maximum valueand its upper quartile. Define a collection of data points that would yield thistype of plot. (Lesson 9-6)

Age (years) 0 3 6 9 12 15 18Value (thousands) 140 148 160 162 185 178 194


60

840

10

8

40,320

1

84

5

4

17,297,280

1

15

120

60,480

6

495

6840

1140

3003

720

210

120

9

210, 210; They are equal.


$54,000

positive


Answers will vary.


Workplace Knowhow

PRACTICE LESSON 9-1–LESSON 9-6A spinner with 8 equal sectors labeled A through H is spun. Find each probability.(Lesson 9-1)

30. P(spinning E) 31. P(spinning vowel) 32. P(spinning H)

33. P(spinning a letter before F) 34. P(spinning B or G) 35. P(spinning M)

36. Describe a simulation you could do to find out how many cards you wouldexpect to have to draw from a standard deck to get three clubs. (Lesson 9-2)

For each situation, tell whether order does or does not matter. (Lesson 9-5)

37. You are selecting three-number combinations for school lockers.

38. You are selecting five books to check out from the library.

39. You are choosing the 9 starters on a baseball team.

40. You are choosing a 5-member committee from your leadership board.


Physical therapists work with people who have been injured. They improvemobility, relieve pain and prevent or limit permanent physical disabilities.

To relieve pain and treat injuries, physical therapists use massages, electricalstimulation, hot and cold packs and traction. The sports world depends onphysical therapists to help athletes who are injured during practices, exercisesessions or games. Some injuries require surgery, but many can be treated by restfollowed by proper exercise. Physical therapists often travel with teams.

1. As a physical therapist, you have treated 236 injuries during the past year. Ofthe total number, 108 injuries were caused by an improper warm-up. Whatpercent of the injuries resulted from an improper warm-up?

2. A team of 45 players suffered 15 ankle injuries during the season. If the squadis increased to 52 players, how many ankle injuries would you expect to seeduring a season?

3. A physical therapist employs 4 kinds of massage, 3 kinds of baths, 1 type ofelectrical stimulation, and 8 exercise programs. If each injury is treated with allfour types of therapies, how many combinations of therapies does thistherapist offer?

4. You are examining the player files for 30 players who have been injured duringthe season. For this sport, 1 out of 3 injuries are to the knee and 1 out of 2 ofthese cases require surgery. What is the probability that the first file you selectwill belong to a player requiring knee surgery?

�18

�

does

does not

does

does not

�14

�

�14

��58

�

�18

�

0

Answers will vary.

about 46%

96

�16

�

17 or 18 ankle injuries

Career – Physical Therapist

mathmatters3.com/mathworks

http://mathmatters3.com/mathworks

Work in groups of 4–5 students.

1. Collect a set of data about students in class, such as heights, arm lengths,head circumference, lengths of thumbs and so on.

2. Study the data. Look for new ways to describe the data. Instead of focusing oncentral tendencies, study how the data are spread out, or dispersed.

3. Consider these questions: How much do individual values in your data differfrom the greatest value? The least value? The mean, median or mode of thevalues?

4. Share your results with your classmates.

BUILD UNDERSTANDING

Statistics that show how data is spread out are called measures of dispersion. Forexample, you know that the range of a set of data is the difference between thelargest and smallest item.

Variance is another measure of dispersion. The variance of a set of numbers is themean of the squared differences between each number in the set and the meanof all numbers in the set. For the set of numbers x1, x2, . . . xn, with a mean of m,use this formula.

v �

E x a m p l e 1

SPORTS During a basketball tournament, the five starters for the Bulldogs madethe following number of 3-pointers: Bowen, 3; White, 4; Fillmore, 5; Graham 6;and Bonilla, 7. Find the variance for the set of numbers.

Solution1. Divide the sum of scores by 5 to find the mean, m.

(m � 5)

2. Find the difference between each number and the mean. Then find the square of each difference.

3. Find the mean of all the squares in Step 2.

4 � 1 � 0 � 1 � 4 � 10 10 � 5 � 2

The variance is 2.

The standard deviation, s, of a set of numbers is the square root of the variance.

(x1 � m)2 � (x2 � m)2 � . . . � (xn � m)2

��n


9-7 Standard DeviationGoals ■ Find the variance, standard deviation and z-scores for

a set of data.■ Use standard deviation to interpret data.

Applications Sports, Test-Taking, Education

Answers will vary.

number x � m (x � m)2

3 3 � 5 (�2)2 � 44 4 � 5 (�1)2 � 15 5 � 5 02 � 06 6 � 5 12 � 17 7 � 5 22 � 4

E x a m p l e 2

Find the standard deviation for the set of numbers in Example 1.

SolutionFind the square root of the variance.

�2� � 1.4 The standard deviation is 1.4

E x a m p l e 3

Molly took two tests. On which did she scorebetter, compared with others in her class?

Solution1. Compare both of her scores with the mean. She scored 20 points higher than

the mean on both tests.

2. Use the standard deviation.

In Test A, Molly’s score was �280�, or 2.5 standard deviations above the mean score.

In Test B, it was �21

00�, or 2 standard deviations above the mean score.

Relative to her classmates, Molly scored better on Test A.

The number of standard deviations between a score and the mean score isindicated by a z-score. Molly’s z-score was 2.5 on Test A. A score below the meanwould have a negative z-score.

TRY THESE EXERCISES

Compute the variance and standard deviation for each set of data.

1. 4, 4, 4, 4, 4 2. 7, 3, 5, 9, 11 3. 2.7, 4.7, 6.7, 8.7, 10.7


Compute the variance and standard deviation for each set of data.

4. 6, 6, 6, 6 5. 1.5, 2.5, 3.5, 4.5, 5.5

6. 4.2, 9.2, 14.2, 19.2, 24.2 7. 8.9, 4, 9.4, 26.5, 14.9

8. Raymond took two tests. On the first test, his score was 45, while the meanscore was 55 and the standard deviation was 5. On the second test, his scorewas 55, while the mean score was 65 and the standard deviation was 10. Onwhich test did Raymond score better, relative to the scores of his classmates?

9. On a science test taken by 28 students, the mean score was 82.5. Thestandard deviation for the scores was 5.3. What was the sum of all the scores?

10. WRITING MATH What can you say about the relationship between thestandard deviation of a set of scores and how spread out the scores are?

Lesson 9-7 Standard Deviation 413

Test A Test BMolly’s score 85 80Mean score 65 60Standard deviation 8 10

0; 0

0; 0 2; �2�

� 60; 2�15�50; 5�2�

second test

2310

Generally, the higher the standard deviation, the more spread out the scores are.

8; 2�2� 8; 2�2�





A visual display that shows the relative frequency of data is called a frequency distribution. A histogram is often used for this purpose.

11. Find the mean, median, mode, variance, and standard deviation to thenearest whole number for the data presented below.

TEST TAKING Find out how well your classmates would scoreon a test on which they had to guess the answer to everyquestion. Work in a small group of 3–4 students. Make up a 20-question multiple-choice test using the topic of obscure and unimportant sports data. Use an almanac, a sportsencyclopedia, a book of records, or any other source to find facts unfamiliar to anyone in your class. Ask everybody to take the test. Then grade the test as a class. Record and analyzethe results.

12. Find the range, mean, median and mode for the scores.

13. Find the standard deviation.

14. Make a visual display of the scores.

Have the class retake the test. Then analyze the results.

15. Find the range, mean, median and mode for the new scores.

16. Find the standard deviation.

17. Make a visual display of the scores.

Compare both sets of results.

18. On which test did the class perform better? Explain.

19. On which test did you perform better relative to your classmates? What wasyour z-score on that test?

20. CHAPTER INVESTIGATION Play the baseball simulation game. Draw abaseball diamond and use coins for markers. To play, two people choosenine baseball players each. Put the players’ tables in batting order.

Use 10-sided polyhedral dice or a deck of standard playing cards with thekings, queens and jacks removed. Either roll two dice or draw two cards fromthe deck. The first die or card indicates the row on the player’s table. Thesecond die or card indicates the column. Find the cell at the intersection ofthe row and column to see what happens in the game.

If the baseball player gets a hit, place a marker in the appropriate place onthe baseball diamond. Keep score as you would in a real baseball game. Playnine innings. Did your team do as well as you expected?


10 20 30 40 50 60Scores

Freq

uen

cy7

6

5

4

3

2

1

mean: 38; median: 40; mode: 30; variance: 256; standard deviation: 16

For Exercises 12–19, answers will vary.

Check students’ work.


If you were to use a smoothcurved line to connect themidpoints of the histogram onthe previous page, you wouldform a frequency distributionknown as a bell curve.

The normal curve is the best knownfrequency distribution. In a normalcurve, the mean, median, andmode are the same.

Normal curves are determinedby the mean and the standarddeviation. In every normalcurve, about 68% of the dataare within one standarddeviation unit of the mean.About 95% of the data arewithin two standard deviationunits of the mean. Finally,about 99.7% of the data arewithin three standarddeviation units.

21. Suppose you drew two normal distributions on the same set of axes.Compare the appearances of these two curves if one has a greater mean thanthe other, but their variances are the same?

22. Suppose two normal curves drawn on the same set of axes have differentvariances but equal means. Compare the curves.

23. Which of these bell curves do youthink might show the distributionof scores if your class were totake a third-grade spelling test?

24. As items in a set of data are dispersed more and more widely from the mean,what happens to the standard deviation?


Find each product. If not possible, write NP. (Lesson 8-6)

25. [3 2 4] � � � 26. � � � � � 27. � � � � �Solve. (Lesson 3-1)

28. On a number line, the coordinate of point F is �4. The length of F�G�is 13.Give 2 possible coordinates of point G.

29. On a number line, the coordinate of point Q is �18. The length of Q�R�is 76.Give 2 possible coordinates of point R.

30. Point S is between points R and T. The length of R�S�is twice the length of S�T�,and RT � 57. Find RS and ST.

6�4

�58

02

�41

53

24

38

05

21

�16

�5

Lesson 9-7 Standard Deviation 415

�3 �2 �1 0mean68%95%

99.7%

�1 �2 �3

The one with the higher mean isfarther to the right.

The one with the greater varianceis not as high as the one with the smaller variance.

the right one

It increases.

[�11] � �422

643

� �4614

�57�7

�82

9, �17

58, �94

RS � 38; ST � 19




Chapter 9 ReviewVOCABULARY

Choose the word from the list that best completes each statement.

1. The set of all possible outcomes of an experiment is the ___?__.

2. A set of items in no particular order is called a(n) ___?__.

3. If events cannot occur at the same time, they are ___?__.

4. An upward sloping trend line on a scatter plot suggests a(n)___?__ correlation between the data.

5. The whiskers of a box-and-whisker plot show ___?__ of the data.

6. The ___?__ is a measure of dispersion that compares a number to the mean of a set of data.

7. If one event affects another event, they are ___?__.

8. The ___?__ uses multiplication to find the number of outcomes.

9. A(n) ___?__ is an arrangement of items in a particular order.

10. ___?__ is the number of favorable events divided by the totalnumber of outcomes.

LESSON 9-1 Review Percents and Probability, p. 384

� Divide the number of favorable observations by the number of totalobservations to find the experimental probability of an event. To find thetheoretical probability of an event, divide the number of possible favorableoutcomes by the number of possible outcomes.

A set of 30 cards is numbered 1, 2, 3, …, 30. Suppose you choose one cardwithout looking. Find the probability of each event.

11. P(12) 12. P(odd) 13. P(integer)

14. P(less than 1) 15. P(greater than 18) 16. P(ends in 0)

17. Find the probability of drawing a red 7 from a standard deck of playing cards.

LESSON 9-2 Problem Solving Skills: Simulations, p. 388

� One way to find a probability is to model the situation using a simulation.Simulations rely on random numbers.

18. A computer generates a list of random 2-digit numbers. What probabilitywould you expect for a number in the list to contain the digit 2? �

19. A fast food restaurant is putting 3 different toys in their children’s meals. Ifthe toys are placed in the meals at random, create a simulation to determinethe experimental probability that a child will have all 3 toys after buying 5 meals.

20. Rodolfo must wear a tie when he works at the mall on Friday, Saturday, andSunday. Each day, he picks one of his 6 ties at random. Create a simulation to findthe experimental probability that he wears a different tie each day of the weekend.

a. combination

b. dependent

c. extremes

d. fundamentalcounting principle

e. independent

f. mutually exclusive

g. permutation

h. positive

i. probability

j. sample space

k. simulation

l. standard deviation

Chapter 9 Review 417

LESSON 9-3 Compound Events, p. 392

� A compound event consists of two or more simple events. If A and B are mutuallyexclusive events, they cannot occur at the same time, and P(A or B) � P(A) �P(B). If A and B are not mutually exclusive events, they can occur at the sametime, and P(A or B) � P(A) � P(B) � P(A and B).

21. Two 1–6 spinners are spun. Find the probability that the sum of the numbersspun is 9 or less than 2.

22. A card is drawn at random from a standard deck. Find the probability that itis a black card or an 8.

23. There are 3 science books, 4 math books, and 2 history books on a shelf. If abook is randomly selected, what is the probability of selecting a science bookor a history book?

24. In a drama club, 7 of the 20 girls are seniors, and 4 of the 14 boys are seniors.What is the probability of randomly selecting a boy or a senior to representthe drama club at an arts symposium?

LESSON 9-4 Independent and Dependent Events, p. 396

� Two events are independent if the outcome of one does not affect the outcomeof the other. If A and B are independent events, P(A, then B) � P(A) � P(B).

� Two events are dependent if the outcome of one affects the outcome of theother. If A and B are dependent events, P(A, then B) � P(A) � P(B, after A).

25. A die is rolled two times. Find P(3, then even number).

26. A box contains 4 red marbles, 3 green marbles, 1 white marble, 2 yellowmarbles, and 3 blue marbles. Two marbles are chosen at random and notreplaced. Find P(red, then yellow).

27. A coin is tossed three times. What is the probability that all three times thecoin shows heads?

28. Reiko has 3 quarters, 5 dimes, and 2 nickels in her pocket. She picks twocoins at random without replacement. What is the probability that shechooses a quarter followed by a dime?

LESSON 9-5 Permutations and Combinations, p. 402

� The fundamental counting principle states that if an event A can occur in m waysand an event B can occur in n ways, then events A and B can occur in m � n ways.

� A permutation is a set of items arranged in a particular order. You can arrange a set of n items in n! ways. To find the number of permutations of a set of n items taken r at a time, use the formula at the right.

� A set of items without consideration of order is called a combination. To findthe number of combinations of a set of n items taken r at a time, use theformula at the right.

29. In how many different ways can 6 pies be awarded first- through third-placeprizes?

30. How many groups of 3 students can be chosen from a class of 20 students?

31. How many 3-digit whole numbers can you write using the digits 1, 3, 5, 7,and 9 if no digit can be used twice?

nPr � �(n

n�

!r)!

�

nCr � �(n �

n!r) r!�


32. An ice cream store has 31 flavors of ice cream. Lamel wants to buy threepints of ice cream. If each pint of ice cream is a different flavor, how manydifferent purchases can he make?

LESSON 9-6 Scatter Plots and Box-and-Whisker Plots, p. 406

� A scatter plot displays data as unconnected points. The trend line indicateswhether the items being compared have a positive correlation, a negativecorrelation, or no correlation.

� A box-and-whisker plot shows extremes of data and how data are distributed.

33. The scatter plot shows the number of calories in different fruitscompared to the number of milligrams of calcium they offer.Does there appear to be a positive, a negative, or no relationshipbetween calories and calcium in fruit?

34. Name a situation where a scatter plot would have a negativecorrelation. Sketch how it might look.

35. Some students at Johnson High rated the performance of theirbasketball team from 0 to 100, with 100 as the highest. These arethe ratings: 67, 71, 58, 53, 65, 73, 64, 50, 52, 74, 48, 47, 53, 82, 63,59, 67, 85, 45, 43, and 56. Make a box-and-whisker plot of thisdata.

36. Make a box-and-whisker plot for the following set of test scores.

77, 80, 75, 73, 77, 81, 62, 87, 99, 85, 82, 81, 77, 72, 78, 83, 86, 79, 80, 78

What does the plot tell you about the scores?

LESSON 9-7 Standard Deviation, p. 412

� The variance of a set of numbers is a measure of how the data are dispersed. Tofind the variance of a set of numbers x1, x2, . . . xn, with a mean of m, use thefollowing formula.

� The standard deviation, s, of a set of numbers is the square root of the variance.

37. Find the variance and standard deviation for the set of numbers 4, 7, 10, 13, and 16.

38. Compute the variance and standard deviation for 3, 5, 10, 7, 5.

39. Find the variance and standard deviation for the set of numbers 5, 2, 6, 8,and 19.

40. Kendra scored 95, 90 and 95 on three tests. The class mean and standarddeviation for the first test were 75 and 10; for the second, 75 and 5; and forthe third, 80 and 6. On which test did Kendra do best relative to herclassmates?

CHAPTER INVESTIGATION

EXTENSION Present your game to your class. After listening to everyone’spresentation, compare your game to those of your classmates. List theadvantages and disadvantages to your game. Make improvements to your game based on your list of disadvantages.

(x1 � m)2 � (x2 � m)2 � � � � � (xn � m)2

��n

Calories and Calcium

60

50

40

30

20

10

0100 200 300

Calories

Cal

ciu

m (

mg

)

y

x

Chapter 9 Assessment1. A bowler got 32 strikes in her first 80 frames. What is the experimental

probability that she will not get a strike in her next frame?

2. Find the probability of drawing a red king or a 5 from a standard deck ofplaying cards.

3. A pair of dice are rolled twice. Find P(sum is an even number, then sum is anodd number).

4. There are 3 red T-shirts, 2 white T-shirts, 2 green T-shirts, and 1 blue T-shirt in a drawer. You reach in without looking and take out two shirts. Find P(red, green).

5. How many different two-flavor ice cream cones can be chosen from a menuof 15 flavors?

6. Forty students are in the running for the science prize. In how many wayscan a winner, a runner-up, and an alternate be chosen?

Use the scatter plot for Exercises 7and 8.

7. Does the scatter plot show anegative or a positive correlation?

8. Estimate the number of stolenbases Davis will have when he is 32 years old.

Groups of students and of community leaders rated the performance of theschool superintendent on a scale from 0 to 100. The box-and-whisker plotsbelow show the results.

9. Which group gave the superintendent a higher median score?

10. For which group were the scores more widely spread?

11. Find the variance and standard deviation for the set of numbers: �10, �5, 0, 5, 10.

12. Patty scored 75 on a test in which the mean score in her class was 60 and thestandard deviation was 10. Jamaal took the same test in his class. His scorewas 70, the class mean score was 50, and the standard deviation was 10. Whoscored better, relative to his or her classmates?

Superintendent's Performance Rating

CommunityLeaders

Students

0 10 20 30 40 50 60 70 80 90 100

Chapter 9 Assessment 419mathmatters3.com/chapter_assessment

25 26 27 28 29 30 31 32 33 34

50

40

30

20

10

0

Player’s Age

Sto

len

Bas

esDavis’ Stolen Bases

http://mathmatters3.com/chapter_assessment


Standardized Test Practice6. What is the y-intercept for the line with

equation 3y � x � 6? (Lesson 6-1)

�13

� 2

3 6

7. A 12-m flagpole casts a 9-m shadow. At the same time, the building next to it casts a 27-m shadow. How tall is the building? (Lesson 7-7)

20.25 m 36 m

40 m 84 m

8. Find the value of y. (Lesson 8-6)

� � � � � � � �14 18

22 46

9. In the spinner, what colorshould the blank portionof the spinner be so thatthe probability of landingon this color is �

38

�?(Lesson 9-1)

blue green

red yellow

10. The weather forecaster says there is a 35%chance of rain. What is the probability that itwill not rain? (Lesson 9-3)

50% 65%

70% 75%

11. Use the box-and-whisker plot to determinethe mean of the data. (Lesson 9-6)

0 10

25 45DC

BA

DC

BA

DC

BA

DC

BA

y14

�57�7

�82

6�4

�58

02

�41

53

DC

BA

DC

BA

Part 1 Multiple Choice

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

1. The first super computer, the Cray-1, wasinstalled in 1976. It was able to perform 160million different operations in a second. Whichexpression represents this number in scientificnotation? (Lesson 1-8)

1.6 � 106 1.6 � 108

160 � 106 160 � 108

2. For which value of x is the y value in theequation 3x � 2y � 6 the greatest?(Lesson 2-5)

x � �2 x � 0

x � 2 x � 4

3. If AB�� CD��, what is m�ECD?(Lesson 3-4)

38°

56°

124°

153°

4. In the figure below, C�D� is a median of trapezoidRSTV. What is the length of C�D�? (Lesson 4-9)

60 cm 65 cm 70 cm

cannot be determined

5. What is the approximate area of the shadedregion? (Lesson 5-2)

14 ft2

34 ft2

50 ft2

114 ft2D

C

B

A

D

CBA

R S

TV

C D

46 cm

84 cm

D

C

B

A

DC

BA

DC

BA

(3x � 10)�

A B

C

E

D

(4x � 28)�

green blue

yellow

blue

redblue

yellow

0 10 20 30 40 50

4 ft

Chapter 9 Standardized Test Practice 421mathmatters3.com/standardized_test

Preparing for Standardized TestsFor test-taking strategies and morepractice, see pages 709-724.

Part 2 Short Response/Grid In

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

12. Mrs. Hayashi made a tablecloth for herkitchen. She bought 4�

34

� yd of material. Sheused 3�

18

� yd of material to make the tablecloth.How much material was not used to make thetablecloth? (Lesson 1-4)

13. The Huang family had weekly grocery bills of$105, $115, $120, and $98 last month. Whatwas their mean weekly grocery bill lastmonth? (Lesson 2-7)

14. Find the value of x. (Lesson 3-2)

15. Change 0.08 kg to grams. (Lesson 5-1)

16. For a cleaning solution, bleach is mixed withwater in the ratio of 1:8. How much bleachshould be added to 12 qt of water to make the proper solution? (Lesson 7-1)

17. Suppose the segment shown below istranslated 3 units to the left. What are thecoordinates of the endpoints of the resultingsegment? (Lesson 8-1)

(2x � 3)°(3x � 8)°

18. Only two of the five school newspaper editorscan represent the school at the state awardsbanquet. How many different combinationsof two editors can be selected to go to thebanquet? (Lesson 9-5)

Part 3 Extended Response

Record your answers on a sheet of paper. Showyour work.

19. Tiffany has a bag of 10 yellow marbles, 10 redmarbles, and 10 green marbles. Tiffany pickstwo marbles at random and gives them to hersister. (Lesson 9-4)

a. What is the probability of choosing 2yellow marbles?

b. Of the marbles left, what is the probabilityof choosing a green marble next?

c. Of the marbles left, what color has a

probability of �13

� of being picked?

Explain.

20. Kenneth is recording the times it takes him torun various distances. The results are shown.(Lesson 9-6)

a. Make a scatter plot of the data.

b. How many minutes do you think it will take Kenneth to run 4 mi? Explain.

Distance (mi) 2 3 5 7 9

Time (mi) 13 20 35 53 72

O

y

x

Test-Taking TipQuestion 19Extended response questions often involve several parts.When one part of the question involves the answer to aprevious part of the question, make sure you check youranswer to the first part before moving on. Also, remember to show all of your work. You may be able to get partial creditfor your answers, even if they are not entirely correct.

http://mathmatters3.com/standardized_test

chapter 9: probability and statistics

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