chapter 10- probability (in - advisory 207 -...

10A ExperimentalProbability

10-1 Probability

10-2 Experimental Probability

LAB Generate RandomNumbers

10-3 Use a Simulation

LAB Use Different Models forSimulations

10B TheoreticalProbability andCounting

10-4 Theoretical Probability

10-5 Independent andDependent Events

10-6 Making Decisions andPredictions

10-7 Odds

10-8 Counting Principles

10-9 Permutations andCombinations

518 Chapter 10

ProbabilityProbability

Letter CodeA 1000001E 1000101H 1001000I 1001001L 1001100

M 1001101O 1001111T 1010100V 1010110

Cryptographer1001001100110010011111010110

10001011001101100000110101001001000

Is this pattern of zeros and ones some kindof message or secret code? A cryptographercould find out. Cryptographers create andbreak codes by assigning number values toletters of the alphabet.

Almost all text sent over the Internet isencrypted to ensure security for the sender. Codes made up of zeros and ones, or binarycodes, are frequently used in computer applications.

Use the table to break the code above.

KEYWORD: MT7 Ch10

m807_c10_518_519 1/12/06 3:11 PM Page 518

Probability 519

VocabularyChoose the best term from the list to complete each sentence.

1. The term __?__ means “per hundred.”

2. A __?__ is a comparison of two numbers.

3. In a set of data, the __?__ is the greatest value minus theleast value.

4. A __?__ is in simplest form when its numerator and denominatorhave no common factors other than 1.

Complete these exercises to review skills you will need for this chapter.

Simplify RatiosWrite each ratio in simplest form.

5. 5:50 6. 95 to 19 7. �12000� 8. �

18902

�

Write Fractions as DecimalsWrite each fraction as a decimal.

9. �15020� 10. �10

700� 11. �

35� 12. �

29�

Write Fractions as PercentsWrite each fraction as a percent.

13. �11090� 14. �

18� 15. �

52� 16. �

23�

17. �34� 18. �2

90� 19. �1

70� 20. �

25�

Operations with Fractions Add. Write each answer in simplest form.

21. �38� � �

14� � �

16� 22. �

16� � �

23� � �

19� 23. �

18� � �

14� � �

18� � �

12� 24. �

13� � �

14� � �

25�

Multiply. Write each answer in simplest form.

25. �38� � �

15� 26. �

23� � �

67� 27. �

37� � �

12

47� 28. �

15

32� � �5

31�

29. �45� � �

141� 30. �

52� � �

34� 31. �

287� � �

49� 32. �1

15� � �

390�

fraction

percent

range

ratio

m807_c10_518_519 1/12/06 3:11 PM Page 519

Previously, you

• found the probability ofindependent events.

• constructed sample spaces forsimple or compositeexperiments.

• made inferences based onanalysis of given or collecteddata.

You will study

• finding the probabilities ofindependent and dependentevents.

• selecting and using differentmodels to simulate an event.

• using theoretical probabilitiesand experimental results tomake predictions.

You can use the skillslearned in this chapter

• to make predictions based ontheoretical and experimentalprobabilities in sciencecourses like biology.

• to learn how to create moreadvanced simulations for usein fields like computer scienceand meteorology.

Vocabulary ConnectionsTo become familiar with some of thevocabulary terms in the chapter, consider thefollowing. You may refer to the chapter, theglossary, or a dictionary if you like.

1. The word dependent means “determinedby another.” What do you think

are?

2. The prefix in- means “not.” What do yousuppose are?

3. The word simulation comes from the Latinroot simulare, which means “to represent.”What do you think a is inprobability?

simulation

independent events

dependent events

520 Chapter 10

Key Vocabulary/Vocabulariocombination combinación

dependent events sucesos dependientes

experimental probabilidad probability experimental

independent events sucesos independientes

mutually exclusive mutuamente excluyentes

outcome resultado

permutation permutación

probability probabilidad

simulation simulación

theoretical probability probabilidad teórica

Stu

dy

Gu

ide:

Pre

view

m807_c10_520_520 1/12/06 3:11 PM Page 520

Reading Strategy: Learn Math Vocabulary Mathematics has a vocabulary all its own. To learn and remember newvocabulary words, use the following study strategies.

• Try to figure out the meanings of new words based on their context.

• Use a dictionary to look up root words or prefixes.

• Relate the new word to familiar everyday words.

• Use mnemonics or memory tricks to remember the definition.

Once you know what a word means, write its definition in your own words.

Probability 521

quartile = four outlier = out

variability = variable

Term Study Notes Definition

The root word quart- Three values that Quartile means “four.” divide a data set

into fourths

Relate it to the word A value much greater Outlier out, which means or much less than the

“away from a place.” others in a data set

Relate it to the The spread, or Variability word variable, which amount of change,

is a value that can of values in a set change. of data

Try This

Term Study Notes Definition

1. Systematic sample

2. Median

3. Quartile

4. Frequency table

Complete the table below.

Read

ing

and

Writin

g M

ath

m807_c10_521_521 1/18/06 3:45 PM Page 521

522 Chapter 10 Probability

10-1 Probability

Learn to find theprobability of an event by using the definition of probability.

Vocabulary

certain

impossible

probability

event

sample space

outcome

trial

experiment

An is an activity in which results are observed. Each observation is called a

, and each result is called an . The is the set of all possible outcomes of an experiment.

Experiment Sample space

• flipping a coin • heads, tails

• rolling a number cube • 1, 2, 3, 4, 5, 6

• guessing the number of • whole numbersmarbles in a jar

An is any set of one or more outcomes. The

of an event is anumber from 0 (or 0%) to 1 (or100%) that tells you how likelythe event is to happen.

• A probability of 0 means the event is , or can neverhappen.

• A probability of 1 means the event is , or has to happen.

• The probabilities of all the outcomes in the sample space add up to 1.

Finding Probabilities of Outcomes in a Sample Space

Give the probability for each outcome.

The weather forecast shows a30% chance of snow. The probability of snow is

P(snow) � 30% � 0.3. The probabilities must add to 1, so theprobability of no snow is P(no snow) � 1 � 0.3 � 0.7, or 70%.

certain

impossible

probability

event

sample spaceoutcometrial

experiment

0.25

25%

0.5

50%

0.75

75%

1

1

100%

0

0

0%

Neverhappens

Happens abouthalf the time

Alwayshappens

14

12

34

E X A M P L E 1

Outcome Snow No snow

Probability

The probability of anevent can be writtenas P(event).

Sample space Event of rollingan odd number

Outcome ofrolling a 6

1 2 3

4 56

m807_c10_522_526 1/12/06 3:11 PM 22

Score 0 1 2 3 4 5

Probability 0.105 0.316 0.352 0.180 0.043 0.004

10-1 Probability 523


One-half of the spinner is red, so a reasonable estimate of theprobability that the spinner lands on red is P(red) � �

12�.

One-fourth of the spinner is yellow, so a reasonable estimate ofthe probability that the spinner lands on yellow is P(yellow) � �

14�.

One-fourth of the spinner is blue, so a reasonable estimate of theprobability that the spinner lands on blue is P(blue) � �

14�.

Check The probabilities of all the outcomes must add to 1.

�12� � �

14� � �

14� � 1 ✔

To find the probability of an event, add the probabilities of all theoutcomes included in the event.

Finding Probabilities of Events

A quiz contains 3 multiple-choice questions and 2 true-falsequestions. Suppose you guess randomly on every question. Thetable below gives the probability of each score.

What is the probability of guessing 4 or more correct?

The event “4 or more correct” consists of the outcomes 4 and 5.P(four or more correct) � 0.043 � 0.004

� 0.047, or 4.7%

What is the probability of guessing fewer than 3 correct?

The event “fewer than 3 correct” consists of the outcomes 0, 1,and 2.

P(fewer than 3 correct) � 0.105 � 0.316 � 0.352

� 0.773, or 77.3%

What is the probability of failing the quiz (getting 0, 1, 2, or 3correct) by guessing?

The event “failing the quiz” consists of the outcomes 0, 1, 2, and 3.

P(failing the quiz) � 0.105 � 0.316 � 0.352 � 0.18

� 0.953, or 95.3%

Outcome Red Yellow Blue

Probability

2E X A M P L E

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PROBLEM SOLVING APPLICATION

Six students are running for class president. Jin’s probability ofwinning is �

18�. Jin is half as likely to win as Monica. Petra has the

same chance to win as Monica. Lila, Juan, and Marc all have thesame chance of winning. Create a table of probabilities for thesample space.

Understand the Problem

The answer will be a table of probabilities. Each probability will be anumber from 0 to 1. The probabilities of all outcomes add to 1. Listthe important information:

• P(Jin) � �18� • P(Monica) � 2P(Jin) � 2 � �

18� � �

14�

• P(Petra) � P(Monica) � �14� • P(Lila) � P(Juan) � P(Marc)

Make a Plan

You know the probabilities add to 1, so use the strategy write anequation. Let p represent the probability for Lila, Juan, and Marc.P(Jin) � P(Monica) � P(Petra) � P(Lila) � P(Juan) � P(Marc) � 1

�18� � �

14� � �

14� � p � p � p � �

58� � 3p � 1

Solve

�58� � 3p � 1

��58� ��

58� Subtract �

58� from both sides.

�� 3p � �

38�

�13� � 3p � �

13� � �

38� Multiply both sides by �

13�.

p � �18�

Look Back

Check that the probabilities add to 1.

�18� � �

14� � �

14� � �

18� � �

18� � �

18� � 1 ✔

E X A M P L E 3

1

2

3

4

Think and Discuss

1. Give a probability for each of the following: usually, sometimes,always, never. Compare your values with the rest of your class.

2. Explain the difference between an outcome and an event.

Outcome Jin Monica Petra Lila Juan Marc

Probability �18� �

14� �

14� �

18� �

18� �

18�

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10-1 Probability 525

10-1 ExercisesExercises

1. The weather forecast calls for a 60% chance of rain. Give the probabilityfor each outcome.

A game consists of randomly selecting 4 colored ducks from a pond andcounting the number of green ducks. The table gives the probability of each outcome.

2. What is the probability of selecting at most 1 green duck?

3. What is the probability of selecting more than 1 green duck?

4. There are 4 teams in a school tournament. Team A has a 25% chance of winning. Team B has the same chance as Team D. Team C has half the chance of winning as Team B. Create a table of probabilities for thesample space.

5. Give the probability for each outcome.

Customers at Pizza Palace can order up to 5 toppings on a pizza. The tablegives the probabilities for the number of toppings ordered on a pizza.

6. What is the probability that at least 2 toppings are ordered?

7. What is the probability that fewer than 3 toppings are ordered?

8. Five students are trying out for the lead role in a school play. Kim andSasha have the same chance of being chosen. Kris has a 30% chance ofbeing chosen, and Lei and Denali are both half as likely to be chosen asKris. Create a table of probabilities for the sample space.

KEYWORD: MT7 Parent

KEYWORD: MT7 10-1

GUIDED PRACTICE

INDEPENDENT PRACTICE

Outcome Rain No rain

Probability

See Example 2

See Example 3

See Example 1

See Example 2

See Example 3

See Example 1

Number of Green Ducks 0 1 2 3 4

Probability 0.043 0.248 0.418 0.248 0.043

Number of Toppings 0 1 2 3 4 5

Probability 0.205 0.305 0.210 0.155 0.123 0.002

Outcome Red Blue Yellow Green

Probability

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19. Multiple Choice The local weather forecaster said there is a 30% chance of rain tomorrow. What is the probability that it will NOT rain tomorrow?

0.7 0.3 70 30

20. Gridded Response A sports announcer states that a runner has an 84% chance of winning a race. Give the probability, as a fraction in lowest terms, that the runner will NOT win the race.

Evaluate the powers of 10. (Lesson 4-2)

21. 10�4 22. 10�1 23. 10�5 24. 10�7

Find the slope of the line through the given points. (Lesson 7-5)

25. A(�2, 5), B(�2, 4) 26. G(4, �3), H(5, 2) 27. R(8, 4), S(10, 1) 28. J(3, 2), K(�1, 2)

DCBA

Use the table to find the probability of each event.

9. A, C, or E occurring 10. B or D occurring

11. A, B, D, or E occurring 12. A not occurring

13. Consumer A cereal company puts “prizes” in some of its boxes to attractshoppers. There is a 0.005 probability of getting two tickets to a movietheater, �

18� probability of finding a watch, 12.5% probability of getting an

action figure, and 0.2 probability of getting a sticker. What is theprobability of not getting any prize?

14. Critical Thinking You are told there are 4 possible events that mayoccur. Event A has a 25% chance of occurring, event B has a probability of�15� and events C and D have an equal likelihood of occurring. What stepswould you take in order to find the probabilities of events C and D?

15. Give an example of an event that has 0 probability of occurring.

16. What’s the Error? Two people are playing a game. One of them says,“Either I will win or you will. The sample space contains two outcomes, sowe each have a probability of one-half.” What is the error?

17. Write About It Suppose an event has a probability of p. What can yousay about the value of p? What is the probability that the event will notoccur? Explain.

18. Challenge List all possible events in the sample space with outcomes A,B, and C.

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page 800.

Outcome A B C D E

Probability 0.306 0 0.216 0.115 0.363

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10-2 Experimental Probability 527

Despite the rising price of gasoline, sportsutility vehicles (SUV’s) remain popular. Thepublic perception of the safety of SUV’s varieswidely. The accident rate of SUV’s is about thesame as with other vehicles, but SUV’s tend tohave a higher rate among accidents involvingfatalities. Insurance companies estimate theprobability of accidents by studying accidentrates for different types of vehicles.

In , the likelihood of an event is estimatedby repeating an experiment many times and observing the number oftimes the event happens. That number is divided by the total numberof trials. The more the experiment is repeated, the more accurate theestimate is likely to be.

probability �

Estimating the Probability of an Event

After 1000 spins of the spinner, the following information wasrecorded. Estimate the probability of the spinner landing on red.

probability � � �1206070� � 0.267

The probability of landing on red is about 0.267, or 26.7%.

A marble is randomly drawn out of a bag and then replaced. Thetable shows the results after 100 draws. Estimate the probabilityof drawing a yellow marble.

probability � � �12010� � 0.21

The probability of drawing a yellow marble is about 0.21, or 21%.

number of yellow marbles drawn��total number of draws

number of spins that landed on red��total number of spins

number of times the event occurs��total number of trials

experimental probability


Learn to estimateprobability usingexperimental methods.

Vocabulary

experimental probability

E X A M P L E 1

Outcome Blue Red Yellow

Spins 448 267 285

Outcome Green Red Yellow Blue White

Draws 12 35 21 18 14

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A researcher has been observing the types of vehicles passingthrough an intersection. Of the last 50 cars, 29 were sedans,9 were trucks, and 12 were SUV’s. Estimate the probability thatthe next vehicle through the intersection will be an SUV.

probability � � �15

20� � 0.24 � 24%

The probability that the next vehicle through the intersection willbe an SUV is about 0.24, or 24%.

Safety Application

Use the table to compare the probability of being involved in afatal traffic crash in an SUV with being in a fatal traffic crash in a mid-size car.

probability �

probability of SUV � �751,75293� � 0.0023

probability of mid-size car � �20406,4433� � 0.0023

In 2004, an SUV was just as likely to be involved in a fatal traffic crashas a mid-size car.

number of fatal crashes��total number of crashes

number of SUV’s��total number of vehicles

Think and Discuss

1. Compare the probability in Example 1A of the spinner landing onred to what you think the probability should be.

2. Give a possible number of marbles of each color in the bag inExample 1B. Explain your reasoning.

Outcome Sedan Truck SUV

Observations 29 9 12

2E X A M P L E

Traffic Crashes in Ohio, 2004

Number of Total Number of Vehicle Class Fatal Crashes Crashes

Sub-compact cars 23 7, 962

Compact cars 266 110,598

Mid-size cars 464 200,433

Full-size cars 161 76,570

Minivan 97 45,043

SUV’s 172 75,593Source: Ohio Department of Public Safety

m807_c10_527_530 1/12/06 3:12 PM Page 528

10-2 Experimental Probability 529

10-2 ExercisesExercisesKEYWORD: MT7 Parent

KEYWORD: MT7 10-2

1. A game spinner was spun 500 times. It was found that A was spun 170 times, B was spun 244 times, and C was spun 86 times. Estimate the probability that the spinner will land on A.

2. A coin was randomly drawn from a bag and then replaced. After 300draws, it was found that 45 pennies, 76 nickels, 92 dimes, and 87 quartershad been drawn. Estimate the probability of drawing a quarter.

3. Use the table to compare the probability that astudent walks to school to the probability thata student bikes to school.

4. Use the table to compare the probability that astudent takes the bus to school to theprobability that a student rides in a car toschool.

5. A researcher polled 260 students at a university and found that 83 of themowned a laptop computer. Estimate the probability that a randomlyselected college student owns a laptop computer.

6. Keisha made 12 out of her last 58 shots on goal. Estimate the probabilitythat she will make her next shot on goal.

7. Stefan polled 113 students about the number ofsiblings they have. Use the table to compare theprobability that a student has one sibling to theprobability that a student has two siblings.

8. Use the table to compare the probability that astudent has no siblings to the probability that astudent has three siblings.

Use the table for Exercises 7–11.Estimate the probability of each event.

9. A batter hits a single.

10. A batter hits a double.

11. A batter hits a triple.

12. A batter hits a home run.

13. A batter makes an out.

GUIDED PRACTICE


See Example 2

See Example 1

See Example 1

See Example 2

Mode of Number ofTransportation Students

Bus 265

Car 313

Walk 105

Bike 87

Number of Number ofSiblings Students

0 14

1 45

2 27

3 15

4� 12

Single

Result Number

20

Double 12

Home run 8

Walk 10

Out 28

Total 80

Triple 2


See page 800.

m807_c10_527_530 1/12/06 3:12 PM Page 529


18. Multiple Choice A spinner was spun 220 times. The outcome was red 58 times. Estimate the probability of the spinner landing on red.

about 0.126 about 0.225 about 0.264 about 0.32

19. Short Response A researcher observed students buying lunch in a cafeteria. Of the last 50 students, 22 bought an apple, 17 bought a banana, and 11 bought apear. If 150 more students buy lunch, estimate the number of students who willbuy a banana. Explain.

Evaluate each expression for the given value of the variable. (Lesson 2-3)

20. 45.6 � x for x � �11.1 21. 17.9 � b for b � 22.3 22. r � (�4.9) for r � 31.8

A spinner is divided into 8 equal sections. There are 3 red sections, 4 blue,and 1 green. Give the probability of each outcome. (Lesson 10-1)

23. red 24. blue 25. green

DCBA

The strength of an earthquake is measured on the Richter scale. A major earthquake measures between 7 and 7.9 on the Richter scale, and a great earthquake measures 8 or higher. The table shows the number of major and great earthquakes per year worldwide from 1985 to 2004.

14. Estimate the probability that there willbe more than 15 major earthquakesnext year.

15. Estimate the probability that there will be fewer than 12 major earthquakes next year.

16. Estimate the probability that there willbe no great earthquakes next year.

17. Challenge Estimate the probability that there will be more than one major earthquake in the next month.

Earth Science

Number of Earthquakes Worldwide

Year Major Great Year Major Great

1985 13 1 1995 22 3

1986 5 1 1996 14 1

1987 11 0 1997 16 0

1988 8 0 1998 11 1

1989 6 1 1999 18 0

1990 12 0 2000 14 1

1991 11 0 2001 15 1

1992 23 0 2002 13 0

1993 15 1 2003 14 1

1994 13 2 2004 14 2KEYWORD: MT7 Quake

m807_c10_527_530 1/12/06 3:12 PM Page 530

Generate Random Numbers

A spreadsheet can be used to generate random decimal numbers that are greater than or equal to 0 but less than 1. By using formulas, you can shift these numbers into a useful range.

Use a spreadsheet to generate five random decimal numbers that are between 0 and 1. Then convert these numbers to integers from 1 to 10.

a. Type �RAND() into cell A1 and press ENTER. A random decimal number appears.

b. Click to highlight cell A1. Go to the Edit menu and Copy thecontents of A1. Then click and drag to highlight cells A2 throughA5. Go to the Edit menu and use Paste to fill cells A2 through A5.

Notice that the random number in cell A1 changed when you filled the other cells.

RAND() gives a decimal number greater than or equal to 0, but lessthan 1. To generate random integers from 1 to 10, you need to do thefollowing:

• Multiply RAND() by 10 (to give a number greater than or equal to 0 but less than 10).

• Use the INT function to drop the decimal part of the result (to give an integer from 0 to 9).

• Add 1 (to give an integer from 1 to 10).

c. Change the formula in A1 to �INT(10*RAND()) � 1 and press ENTER. Repeat the process in part b to fill cells A2 through A5.

The formula �INT(10*RAND()) � 1 generates random integers from 1 to 10.

1. Explain how INT(10*RAND()) � 1 generates random integers from 1 to 10.

1. Use a spreadsheet to simulate 3 spins of a spinner with 4 equal regions.

Activity

Think and Discuss

10-3 Technology Lab 531

Use with Lesson 10-3

10-3

KEYWORD: MT7 Lab10

1

Try This

m807_c10_531_531 1/12/06 3:12 PM Page 531


Learn to use asimulation to estimateprobability.

Vocabulary

random numbers

simulation

In football, many factors are used toevaluate how good a quarterback is. Oneimportant factor is the quarterback’sability to complete passes.

If a quarterback has a completionpercentage of 64%, he completes about64 of every 100 passes he throws. Whatis the probability that he will completeat least 6 of 10 passes thrown? Asimulation can help you estimate thisprobability.

A is a model of a realsituation. In a set of ,each number has the same probabilityof occurring, and no pattern can beused to predict the next number.Random numbers can be used tosimulate random events in real situations. The table is a set of 280 random digits.

PROBLEM SOLVING APPLICATION

A quarterback has a completion percentage of 64%. Estimate theprobability that he will complete at least 6 of his next 10 passes.

1. Understand the Problem

The answer will be the probability that he will complete at least 6 ofhis next 10 passes. It must be a number between 0 and 1. List theimportant information:

• The probability that the quarterback will complete a pass is 0.64.

87244 11632 85815 61766 19579 28186 18533 42633

74681 65633 54238 32848 87649 85976 13355 46498

53736 21616 86318 77291 24794 31119 48193 44869

86585 27919 65264 93557 94425 13325 16635 28584

18394 73266 67899 38783 94228 23426 76679 41256

39917 16373 59733 18588 22545 61378 33563 65161

96916 46278 78210 13906 82794 01136 60848 98713

random numberssimulation

10-3 Use a SimulationProblem Solving Strategy

1E X A M P L E

1

During the 2004 season, Indianapolis’s PeytonManning had a completion percentage of 67.6%.

m807_c10_532_535 1/12/06 3:19 PM Page 532

10-3 Use a Simulation 533

1. Make a Plan

Use a simulation to model the situation. Use digits from the table,grouped in pairs. The numbers 01–64 represent completed passes,and the numbers 65–00 represent incomplete passes. Each group of20 digits represents one trial. You can start anywhere on the table.

1. Solve

The first 20 digits in the table are shown below.

87244 11632 85815 61766

The digits can be grouped in ten pairs, as shown below.

87 85 81 66This represents 6 of 10 completed passes.

If you continue using the table, the next nine trials are as follows.

92 81 86 7 completed passes


87 98 76 98 6 completed passes

73 86 87 72 91 5 completed passes

79 69 8 completed passes



66 67 89 93 87 83 4 completed passes

94 82 76 67 94 5 completed passes

Out of the 10 trials, 7 represented 6 or more completed passes. Based on this simulation, the probability of completing at least 6 of 10 passes is about 0.70, or 70%.

1. Look Back

A completion percentage of 64% means the quarterback completesabout 64 of every 100 passes. This ratio is equivalent to 6.4 out of 10passes, so he should make at least 6 passes most of the time. Theanswer is reasonable.

561226342232473918

5263162533514257354926195258

48341948191143243116166253

6454351359644828235433561633263453185719

175632164124

Think and Discuss

1. Explain why a random number generator on a computer orcalculator is useful for estimating probability by simulation.

2. Tell how you could use a simulation to estimate the probabilitythat a quarterback who has a completion percentage of 50% willmake at least 7 of 10 passes.

Calculators andcomputers cangenerate sets ofapproximatelyrandom numbers. A formula is used to generate thenumbers, so they arenot truly random, butthey work for mostsimulations.

2

3

4

m807_c10_532_535 1/12/06 3:19 PM Page 533



Use the table of random numbers to simulate each situation. Use at least 10 trials for each simulation.

49064 12830 66783 14965 81537 24935 69675 32681

42893 42668 70963 58827 17354 42190 36165 29827

21705 89446 38703 21274 90049 19036 37971 05322

52737 40117 54132 11152 02985 82873 28197 89796

1. Liza makes free throws at a rate of 81%. If she takes 8 free throws during agame, estimate the probability that she will make at least 6 free throws.

2. During the summer, a city has a 15% chance of a day with a temperatureover 90°F. Estimate the probability that at least 3 days have a temperatureover 90°F during the last week in August.

3. Customers at a carnival game win about 25% of the time. Estimate theprobability that no more than 1 of the next 6 customers will win the game.

4. Marcelo completes a sale with approximately 32% of the customers hemeets. If he has 6 customer appointments tomorrow, estimate theprobability that he will complete at least 3 sales.

Use the table of random numbers to simulate each situation. Use at least 10 trials for each simulation.

63415 12776 31960 42974 36444 23826 46320 48308

41591 43536 64118 53147 23544 61352 12954 57628

26446 12734 22435 42612 24834 21961 12526 22832

16522 33043 21997 15738 25788 33205 55699 33357

53040 39923 29591 64384 58166 39164 54474 38970

5. Veronica gets a hit 32% of the time she bats. Estimate the probability thatshe will get at least 5 hits in her next 10 at bats.

6. At a local fast-food restaurant, about 83% of the customers order theirfood to go. Estimate the probability that 6 of the next 7 customers willorder their food to go.

7. A local radio station is having a contest. Each time you call in, yourchances of winning are 6%. If you call in 10 times, estimate theprobability that you will win more than once.

8. Kyle works at a juice bar. He knows about 45% of the customers by name.Estimate the probability that he will know the names of at least 7 of thenext 9 customers.

KEYWORD: MT7 Parent

KEYWORD: MT7 10-3

GUIDED PRACTICE


See Example 1

See Example 1

m807_c10_532_535 1/12/06 3:19 PM Page 534

10-3 Use a Simulation 535

Use the table of random numbers for Exercises 9 and 10. Use at least 10 trialsto simulate each situation.

19067 26149 88557 80696 88246 56652 73023 56838

98048 26387 65953 94163 66233 57325 65618 76782

32958 47253 24960 32052 16921 54925 44766 33115

89164 06342 98577 44523 72304 38221 33506 63923

9. Entertainment A radio station plays a song from the 1980s about 60%of the time. What is the probability that at least 4 of the next 5 songs arefrom the 1980s?

10. Life Science About 7% of the deer population in an area was captured,tagged, and released. If 8 deer were randomly captured later, estimate theprobability that at least 1 deer would have a tag.

11. Write About It A silk screener checks the prints on T-shirts. If 2% of theT-shirts were defective, how could you estimate the probability that nomore than 1 T-shirt in each case of 144 would be defective?

12. Challenge A box of tulip bulbs has bulbs in 5 colors. The probabilitiesfor each color are given below. Vijay likes red and yellow best. Rosa likesblue and red best. If each chooses 6 bulbs randomly, estimate theprobability that they will each get at least one of their favorites.


See page 800.

Life Science

The capture-release-recapturemethod usesratios to estimatethe size of wildpopulations.

Color Red Yellow Blue Orange Pink

Probability 0.3 0.1 0.3 0.2 0.1

13. Multiple Choice Diego hit a home run 25% of his times at bat. He wantsto estimate his probability of hitting at least 3 home runs during his next10 at bats. Using a random number table, he performs 10 trials. Of the 10trials, 2 represented 3 or more home runs. What is the probability thatDiego will hit 3 home runs during his next 10 at bats?

2% 20% 25% 30%

14. Short Response In a certain area, 11% of the wolf population is taggedand released. Describe how to use a random number table to determine theprobability that at least 2 out of the next 10 wolves caught will be tagged.

Find the appropriate factor for each conversion. (Lesson 5-3)

15. quarts to pints 16. inches to yards 17. millimeters to meters

A spinner was spun 400 times. It landed on red 192 times, blue 144 times,and green 64 times. Estimate the probability of each event. (Lesson 10-2)

18. spinner landing on red 19. spinner landing on green 20. spinner landing on blue

DCBA

m807_c10_532_535 1/12/06 3:19 PM Page 535

Use Different Models forSimulations

Use with Lesson 10-3

10-3


Activity 1

Activity 2

Try This

NumberRolled Frequency

1 (prize)

2 (prize)

3 (prize)

4 (prize)

5 (prize)

6 (no prize)

Heads TailsTrial (new) (not new)

1

2

3

4

5

Think and Discuss

You can use a simulation to model an experiment that would be difficult to perform.

A cereal company discovered that 1 out of 6 boxes did not contain aprize. Suppose you buy 10 boxes of the cereal. What is the probabilitythat you will buy a cereal box without a prize?

Use a number cube to simulate buying a box of cereal. Let 6 represent a box without a prize and 1–5 represent a box with a prize.

a. Copy the table. Then roll the number cube 10 times to represent buying 10 boxes of cereal. Tally your results.

b. Find the experimental probability of buying a box without aprize.

1. What other methods could you use to simulate this situation? Which methods are best? Explain.

1. Roll the number cube 100 times. What is the experimentalprobability of buying a box without a prize? How does thisprobability compare with your earlier result?

Each Thursday, a radio station randomly plays new releases 50% of the time. What is the probability that 6 of the next 10 songs will be new releases on any given Thursday?

You can use a coin to simulate playing a new release. Let headsrepresent a new release and tails represent a song that is not a new release.

a. Copy the table. For each trial, toss the coin 10 times to represent playing 10 songs. Tally your results. Complete 5 trials.

b. In how many trials did heads appear 6 or more times?

c. Find the experimental probability that 6 of the next 10 songs on any given Thursday will be new releases.

KEYWORD: MT7 Lab10

m807_c10_536_537 1/12/06 3:12 PM Page 536

1. Why is tossing a coin a good way to simulate this situation?

2. What other methods could you use to simulate this situation? Which methods are best? Explain.

1. Toss the coin 100 times. What is the experimental probability that 6of the next 10 songs are new releases? How does this probabilitycompare with your earlier result?

Belinda makes 80% of her free throws. What is the probability thatshe will make 8 out of her next 10 free throws?

1. You can use an area model to simulate Belinda’s shooting a freethrow. Color 80 squares on a sheet of 10-by-10 grid paper torepresent free throws made. The remaining 20 blank squareswill represent free throws missed.

a. For each trial, flip a dime 10 times onto the grid paper. Do not count a result if the dime does not land completelywithin the grid. Flip the dime again. If the dime landscompletely within the shaded area, count it as a free throwmade. If the dime lands in the unshaded area or partly in boththe shaded and unshaded areas, count it as a free throw missed.Tally your results. Complete 10 trials.

b. Find the experimental probability of Belinda’s making 8 out ofthe next 10 free throws.

1. What other methods could you use to simulate this situation?

2. Why are 80 squares filled in and 20 squares left blank? Does it matterwhich 80 squares are filled in? Explain.

Select and conduct a simulation to find the experimental probability.Explain which method you chose and why.

1. Raul works for a pet groomer. He knows about 70% of the pets fromprevious visits. Estimate the probability that he will know at least 6of the next 8 pets that arrive.

2. At a local restaurant, about 50% of the customers order dessert.Estimate the probability that 4 out of the next 10 customers willorder dessert.

10-3 Hands-On Lab 537

Think and Discuss

Think and Discuss

Try This

Try This

Activity 3

m807_c10_536_537 1/12/06 3:12 PM Page 537

Quiz for Lessons 10-1 Through 10-3

10-1 Probability


1. P(C) 2. P(not B) 3. P(A or D) 4. P(A, B, or C)

5. There are 4 students in a race. Jennifer has a 30% chance of winning.Anjelica has the same chance as Jennifer. Debra and Yolanda haveequal chances. Create a table of probabilities for the sample space.


A colored chip is randomly drawn from a box and then replaced.The table shows the results after 400 draws.

6. Estimate the probability of drawing a red chip.

7. Estimate the probability of drawing a green chip.

8. Use the table to compare the probability of drawing a blue chip to the probability of drawing a yellow chip.

10-3 Use a Simulation

Use the table of random numbers to simulate each situation. Use atleast 10 trials for each simulation.

93840 03363 31168 57602 19464 52245 98744 6104068395 76832 56386 45060 57512 38816 51623 2325216805 92120 74443 49176 49898 62042 65847 1538085178 78842 16598 28335 84837 76406 53436 45043

9. At a local school, 58% of the tenth-grade students play a musicalinstrument. Estimate the probability that at least 6 out of 8 randomlyselected tenth-grade students play a musical instrument.

10. Kayla has a package of 100 multicolored beads that contains 15 purplebeads. If she randomly selects 8 beads to make a friendship bracelet,estimate the probability that she will get more than 1 purple bead.

Rea

dy

to G

o O

n?


Outcome A B C D

Probability 0.3 0.1 0.4 0.2

Outcome Red Green Blue Yellow

Draws 76 172 84 68

m807_c10_538_538 1/12/06 3:12 PM Page 538

A point in the circumscribed triangle ischosen randomly. What is the probabilitythat the point is in the circle?

A chef observed the number of peopleordering each antipasto from the evening’sspecials. Estimate the probability that thenext customer will order gnocchi al veneta.

Evelina and Ilario play chess 3 times a week.They have had 6 stalemates in the last 10weeks. Estimate the probability that Evelinaand Ilario will have a stalemate the next timethey play chess.

A pula has a coat of arms on the obverse anda running zebra on the reverse. If a pula istossed 150 times and lands with the coat ofarms facing up 70 times, estimate theprobability of its landing with the zebrafacing up.

4

3

2

1

Copy each problem, and circle any words that you do not understand.Look up each word and write its definition, or use context clues toreplace the word with a similar word that is easier to understand.

Understand the Problem• Understand the words in the problem

Words that you don’t understand can make a simple problem seemdifficult. Before you try to solve a problem, you will need to knowthe meaning of the words in it.

If a problem gives a name of a person, place, or thing that isdifficult to understand, such as Eulalia, you can use another nameor a pronoun in its place. You could replace Eulalia with she.

Read the problems so that you can hear yourself saying the words.

Focus on Problem Solving 539

30 cm

40 cm

20 cm

Saltimbocca Gnocchi Al Galleto AllaAntipasto Alla Romana Veneta Griglia

NumberOrdered

16 21 13

m807_c10_539_539 1/12/06 3:12 PM Page 539



Learn to estimateprobability usingtheoretical methods.

Vocabulary

disjoint events

mutually exclusive

fair

equally likely

theoretical probability

In the game of Monopoly®, you can get out of jail if you roll doubles, butif you roll doubles three times in a row, you have to go to jail. Your turn is decided by the probability that both dice will be the same number.

is used to estimate probabilities by making certain assumptions about an experiment. Suppose a sample space has 5 outcomes that are , that is, they all have the same probability, x. The probabilities must add to 1.

x � x � x � x � x � 1

5x � 1

x � �15�

A coin, die, or other object is called if all outcomes are equally likely.

Calculating Theoretical Probability

An experiment consists of rolling a fair number cube. Find theprobability of each event.

P(5)

The number cube is fair, so all 6 outcomes in the sample spaceare equally likely: 1, 2, 3, 4, 5, and 6.

P(5) � � �16�

P(even number)

There are 3 possible even numbers: 2, 4, and 6.

P(even number) � � �36� � �

12�

number of possible even numbers��6

number of outcomes for 5��6

fair

equally likely

Theoretical probability

THEORETICAL PROBABILITY FOR EQUALLY LIKELY OUTCOMES

Suppose there are n equally likely outcomes in the sample space of anexperiment.

• The probability of each outcome is �n1

�.

• The probability of an event is .number of outcomes in the event��n

E X A M P L E 1

m807_c10_540_544 1/12/06 3:12 PM Page 540

10-4 Theoretical Probability 541

Suppose you roll two fair number cubes. Are all outcomes equally likely? It depends on how you consider the outcomes. You could look at the number on each number cube or at the total shown on the number cubes.

If you look at the total, all outcomes are not equally likely. Forexample, there is only one way to get a total of 2, 1 � 1, but a total of5 can be 1 � 4, 2 � 3, 3 � 2, or 4 � 1.

Calculating Probability for Two Fair Number Cubes

An experiment consists of rolling two fair number cubes. Find the probability of each event.

P(total shown � 1)

First find the sample space that has all outcomes equally likely.

There are 36 possible outcomes in the sample space. Then findthe number of outcomes in the event “total shown � 1.” There isno way to get a total of 1, so P(total shown � 1) � �3

06� � 0.

P(at least one 6)

There are 11 outcomes in the event rolling “at least one 6”:

P(at least one 6) � �13

16�

Altering Probability

A bag contains 5 blue chips and 8 green chips. How many yellowchips should be added so that the probability of drawing a greenchip is �

25�?

Adding chips to the bag will increase the number of possibleoutcomes. Let x equal the number of yellow chips.

�138� x� � �

25� Set up a proportion.

2(13 � x) � 8(5) Find the cross products.

26 � 2x � 40 Multiply.

� 26 � 26 Subtract 26 from both sides.��

�22x� � �

124� Divide both sides by 2.

x � 7

Seven yellow chips should be added to the bag.

2E X A M P L E

3E X A M P L E

You can write theoutcome of a red 3and a blue 6 as theordered pair (3, 6).

m807_c10_540_544 1/12/06 3:12 PM Page 541


Two events are , or , if they cannotboth occur in the same trial of an experiment. For example, rolling a 5and an even number on a number cube are mutually exclusive eventsbecause they cannot both happen at the same time. Suppose A and Bare two mutually exclusive events.

• P(both A and B will occur) � 0• P(either A or B will occur) � P(A) � P(B)

Finding the Probability of Mutually Exclusive Events

Suppose you are playing a game of Monopoly and havejust rolled doubles two times in a row. If you rolldoubles again, you will go to jail. You will alsogo to jail if you roll a total of 3 because youare 3 spaces away from the “Go to Jail”square. What is the probabilitythat you will go to jail?

It is impossible to roll doublesand a total of 3 at the same time,so the events are mutually exclusive.Add the probabilities to find theprobability of going to jail on the next roll.

The event “doubles” consists of six outcomes—(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6)—soP(doubles) � �3

66�.

The event “total � 3” consists of two outcomes, (1, 2)and (2, 1), so P(total of 3) � �3

26�.

P(going to jail) � P(doubles) � P(total � 3)

� �366� � �3

26�

� �386�

The probability of going to jail is �386� � �

29�, or about 22.2%.

disjoint eventsmutually exclusive

Think and Discuss

1. Describe a sample space for tossing two coins that has alloutcomes equally likely.

2. Give an example of an experiment in which it would not bereasonable to assume that all outcomes are equally likely.

E X A M P L E 4

m807_c10_540_544 1/12/06 3:13 PM Page 542

10-4 Theoretical Probability 543


An experiment consists of rolling a fair number cube. Find the probability of each event.

1. P(odd number) 2. P(2 or 4)

An experiment consists of rolling two fair number cubes. Find theprobability of each event.

3. P(total shown � 10) 4. P(rolling two 2’s)

5. P(rolling two odd numbers) 6. P(total shown � 8)

7. What color should you shade the blank region so that theprobability of the spinner landing on that color is �

12�?

8. Suppose you are playing a game in which two fair dice arerolled. To make the first move, you need to roll doubles or asum of 3 or 11. What is the probability that you will be able tomake the first move?

An experiment consists of rolling a fair number cube. Find the probability of each event.

9. P(9) 10. P(not 6)

11. P(� 5) 12. P(� 3)


13. P(total shown � 3) 14. P(at least one even number)

15. P(total shown � 0) 16. P(total shown � 9)

17. A bag contains 20 pennies, 25 nickels, and 15 quarters. How many dimesshould be added so that the probability of drawing a quarter is �

16�?

18. Suppose you are playing a game in which two fair dice are rolled. You need 9 to land on the finish by an exact count or 3 to land on a “roll again”space. What is the probability of landing on the finish or rolling again?

Three fair coins are tossed: a penny, a dime, and a quarter. Find the samplespace with all outcomes equally likely. Then find each probability.

19. P(TTH) 20. P(THH) 21. P(dime heads)

22. P(exactly 2 tails) 23. P(0 tails) 24. P(at most 1 tail)

KEYWORD: MT7 Parent

KEYWORD: MT7 10-4

GUIDED PRACTICE


See Example 1

See Example 1

See Example 2

See Example 3

See Example 4

See Example 2

See Example 4

See Example 3


See page 800.

m807_c10_540_544 1/12/06 3:13 PM Page 543


28. Multiple Choice A bag has 3 red marbles and 6 blue marbles in it. What is the probability of drawing a red marble?

1 �23� �

13� �

12�

29. Gridded Response On a fair number cube, what is the probability, written as a fraction, of rolling a 2 or higher?

Determine whether each ordered pair is a solution of y � 3x � 2. (Lesson 3-1)

30. (3, 11) 31. (0, �2) 32. (�1, �5) 33. (�4, 10)

34. Wallace completed 27 of his last 38 passes. Estimate the probability thathe will complete his next pass. (Lesson 10-2)

DCBA

Life Science

What color are your eyes? Can you roll your tongue? These traits are determined by the genes you inherited from your parents. A Punnett square shows all possible gene combinationsfor two parents whose genes are known.

To make a Punnett square, draw a two-by-two grid. Write the genes of one parent above the top row and the other parent along the side. Then fill in the grid as shown.

25. In the Punnett square above, one parent has the genecombination Bb, which represents one gene for brown eyesand one gene for blue eyes. The other parent has the genecombination bb, which represents two genes for blue eyes. If all outcomes in the Punnett square are equally likely, what is the probability of a child with the gene combination bb?

26. Make a Punnett square for two parents who both have the gene combination Bb.

a. If all outcomes in the Punnett square are equally likely, what is theprobability of a child with the gene combination BB?

b. The gene combinations BB and Bb will result in brown eyes, and thegene combination bb will result in blue eyes. What is the probabilitythat the couple will have a child with brown eyes?

27. Challenge The combinations Tt and TT represent the ability to roll your tongue, while tt means you cannot roll your tongue. Draw aPunnett square that results in a probability of �

12� that the child can roll

his or her tongue. Explain whether the parents can roll their tongues.

B b

b Bb bb

b Bb bb

m807_c10_540_544 1/12/06 3:13 PM Page 544

10-5 Independent and Dependent Events 545

Skydivers carry two independentparachutes. One parachute is theprimary parachute, and the other isfor emergencies.

A compound is made up oftwo or more separate events. To findthe probability of a compoundevent, you need to know if the events are independent or dependent.

Events are if the occurrence of one event doesnot affect the probability of the other. Events are if the occurrence of one does affect the probability of the other.

Classifying Events as Independent or Dependent

Determine if the events are dependent or independent.

a coin landing heads on one toss and tails on another toss

The result of one toss does not affect the result of the other, so the events are independent.

drawing a 6 and then a 7 from a deck of cards

Once one card is drawn, the sample space changes. The events are dependent.

Finding the Probability of Independent Events

An experiment consists of spinning the spinner 3 times.

What is the probability of spinning a 2 all 3 times?

The result of each spin does not affect the results of the otherspins, so the spin results are independent.

For each spin, P(2) � �15�.

P(2, 2, 2) � �15� � �

15� � �

15� � �1

125� � 0.008 Multiply.

dependent eventsindependent events

event

10-5

Learn to find theprobabilities ofindependent anddependent events.

Vocabulary

dependent events

independent events

compound event

E X A M P L E 1

FINDING THE PROBABILITY OF INDEPENDENT EVENTS

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

5

4

3

21E X A M P L E 2

Independent and Dependent Events

m807_c10_545_549 1/12/06 3:13 PM Page 545


An experiment consists of spinning thespinner 3 times. For each spin, all outcomesare equally likely.

What is the probability of spinning an evennumber all 3 times?

For each spin, P(even) � �25�.

P(even, even, even) � �25� � �

25� � �

25� � �1

825� � 0.064 Multiply.

What is the probability of spinning a 2 at least once?

Think: P(at least one 2) � P(not 2, not 2, not 2) � 1.

For each spin, P(not 2) � �45�.

P(not 2, not 2, not 2) � �45� � �

45� � �

45� � �1

6245� � 0.512 Multiply.

Subtract from 1 to find the probability of spinning at least one 2.

1 � 0.512 � 0.488

To calculate the probability of two dependent events occurring, dothe following:

1. Calculate the probability of the first event.

2. Calculate the probability that the second event would occur if the firstevent had already occurred.

3. Multiply the probabilities.

Suppose you draw 2 marbles without replacement from a bag that contains 3 purple and 3 orange marbles. On the first draw,

P(purple) � �36� � �

12�.

The sample space for the second draw depends on the first draw.

If the first draw was purple, then the probability of the second draw being purple is

P(purple) � �25�.

So the probability of drawing two purple marbles is

P(purple, purple) � �12� � �

25� � �

15�.

5

4

3

21

FINDING THE PROBABILITY OF DEPENDENT EVENTS

If A and B are dependent events, then P(A and B) � P(A) � P(B after A).

Before first draw

After first draw

Outcome of first draw Purple Orange

Sample space for 2 purple 3 purplesecond draw 3 orange 2 orange

m807_c10_545_549 1/12/06 3:13 PM Page 546


Finding the Probability of Dependent Events

A jar contains 16 quarters and 10 nickels.

If 2 coins are chosen at random, what is the probability of getting2 quarters?

Because the first coin is not replaced, the sample space isdifferent for the second coin, so the events are dependent. Findthe probability that the first coin chosen is a quarter.

P(quarter) � �12

66� � �1

83�

If the first coin chosen is a quarter, then there would be 15 quarters and a total of 25 coins left in the jar. Find theprobability that the second coin chosen is a quarter.

P(quarter) � �12

55� � �

35�

�183� � �

35� � �

26

45� Multiply.

The probability of getting two quarters is �26

45�.

If 2 coins are chosen at random, what is the probability of getting2 coins that are the same?

There are two possibilities: 2 quarters or 2 nickels. The probabilityof 2 quarters was calculated in Example 3A. Now find theprobability of getting 2 nickels.

P(nickel) � �12

06� � �1

53� Find the probability that the second coin

chosen is a nickel.

If the first coin chosen is a nickel, there are now only 9 nickels and 25 total coins in the jar.

P(nickel) � �295� Find the probability that the second coin

chosen is a nickel.

�153� � �2

95� � �6

95� Multiply.

The events of 2 quarters and 2 nickels are mutually exclusive, soyou can add their probabilities.

�26

45� � �6

95� � �

36

35� P(quarters) � P(nickels)

The probability of getting 2 coins the same is �36

35�.

Think and Discuss

1. Give an example of a pair of independent events and a pair ofdependent events.

2. Tell how you could make the events in Example 1B independentevents.

E X A M P L E 3

Two mutuallyexclusive eventscannot both happenat the same time.

m807_c10_545_549 1/12/06 3:13 PM Page 547




1. drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles

2. drawing a heart from a deck of cards and a coin landing on tails

An experiment consists of spinning each spinner once.

3. Find the probability that the first spinner landson yellow and the second spinner lands on 8.

A sock drawer contains 10 white socks, 6 blacksocks, and 8 blue socks.

4. If 2 socks are chosen at random, what is the probability of getting a pair ofwhite socks?

5. If 3 socks are chosen at random, what is the probability of getting first ablack sock, then a white sock, and then a blue sock?


6. drawing the name Roberto from a hat without replacing it and thendrawing the name Paulo from the hat

7. rolling 2 fair number cubes and getting both a 1 and a 6

An experiment consists of tossing 2 fair coins, a penny and a nickel.

8. Find the probability of heads on the penny and tails on the nickel.

9. Find the probability that both coins will land the same way.

A box contains 4 berry, 3 cinnamon, 4 apple, and 5 carob granola bars.

10. If Dawn randomly selects 2 bars, what is the probability that they willboth be cinnamon?

11. If two bars are selected randomly, what is the probability that they will bethe same kind?

A box contains 6 red marbles, 4 blue marbles, and 8 yellow marbles.

12. Find P(yellow then red) if a marble is selected, and then a second marbleis selected without replacing the first marble.

13. Find P(yellow then red) if a marble is selected, and replaced, and then asecond marble is selected.

KEYWORD: MT7 Parent

KEYWORD: MT7 10-5

GUIDED PRACTICE

PRACTICE AND PROBLEM SOLVING


Extra PracticeSee page 801.

See Example 2

See Example 2

See Example 3

See Example 3

See Example 1

See Example 1

12

345

6

78

m807_c10_545_549 1/12/06 3:13 PM Page 548


14. You roll a fair number cube twice. What is the probability of rolling two 3’sif the first roll is a 5? Explain.

15. School On a quiz, there are 5 true-false questions. A student guesses on all 5 questions. What is the probability that the student gets all 5questions right?

16. Games The table shows the Scrabble® tilesavailable at the start of a game. There are 100 tiles:42 vowels, 56 consonants, and 2 blanks. To beginplay, each player draws a tile. The player with the tileclosest to the beginning of the alphabet goes first. A blank tile beats any letter.

a. If you draw first, what is the probability that youwill select an A?

b. If you draw first and do not replace the tile, whatis the probability that you will select an E andyour opponent will select an I?

c. If you draw first and do not replace the tile, whatis the probability that you will select an E andyour opponent will win the first turn?

17. Write a Problem Write a problem about the probability of an event in aboard game, and then solve it.

18. Write About It In an experiment, two cards are drawn from a deck.How is the probability different if the first card is replaced before thesecond card is drawn than if the first card is not replaced?

19. Challenge Suppose you deal yourself 7 cards from a standard 52-carddeck. What is the probability that you will deal all red cards?

20. Multiple Choice If A and B are independent events such that P(A) � 0.14and P(B) � 0.28, what is the probability that both A and B will occur?

0.0392 0.0784 0.24 0.42

21. Gridded Response A bag contains 8 red marbles and 2 blue marbles.What is the probability, written as a fraction, of choosing a red marble anda blue marble from the bag at the same time?

Find the first and third quartiles for each data set. (Lesson 9-4)

22. 19, 24, 13, 18, 21, 8, 11 23. 56, 71, 84, 66, 52, 11, 80

24. An experiment consists of rolling two fair number cubes. Find theprobability of rolling a total of 14. (Lesson 10-4)

DCBA

Games

This giant Scrabblegame was held on the 50thanniversary ofScrabble. Each tilewas 100 times thesize of a standardtile, and the boardwas nearly 100 ftby 100 ft.

Scrabble Letter Distribution

A-9 B-2 C-2

D-4 E-12 F-2

G-3 H-2 I-9

J-1 K-1 L-4

M-2 N-6 O-8

P-2 Q-1 R-6

S-4 T-6 U-4

V-2 W-2 X-1

Y-2 Z-1 blank-2

m807_c10_545_549 1/12/06 3:13 PM Page 549


10-6 Making Decisions and Predictions

Learn to use probabilityto make decisions andpredictions.

Aliza works for a store thatsells socks. She conducted asurvey to learn about colorpreferences. She recordedthe colors of the last 100pairs of socks sold. Aliza canuse the results of her surveyto decide how many pairs ofsocks of each color to orderfrom the maker.

Probability can be used to make decisions or predictions. Use theprobability of an event’s occurring to set up a proportion to find thenumber of times an event is likely to occur.

Using Probability to Make Decisions and Predictions

The table shows the colors of the last 100 pairs of socks sold.Aliza plans to place an order for 1200 pairs of socks. How manyblue pairs of socks should she order?

� �12000�, or �

15�

�15� � �12

n00� Set up a proportion.

1 � 1200 � 5n Find the cross products.

�12

500� � �

55n� Divide both sides by n.

240 � n

Aliza should order 240 blue pairs of socks.

number of blue pairs of socks sold��total number of pairs of socks sold

E X A M P L E 1

Pairs of Socks Sold

Color Number

Black 9

Blue 20

Gold 6

Green 22

Purple 25

Red 18

Find the probability of selling a blue pair of socks.

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10-6 Making Decisions and Predictions 551

At a carnival, a spinner is used todetermine a player’s prize. If thespinner lands on red, the player gets astuffed animal. Suppose the spinner isspun 160 times. What is the bestprediction of the number of stuffedanimals that will be given away?

� �18�

�18� � �1


1 � 160 � 8n Find the cross products.

�1

860� � �

88n� Divide both sides by 8.

20 � n

Approximately 20 stuffed animals will be given away.

Probability is often used to determine whether a game is fair. A gameinvolving chance is fair if each player is equally likely to win.

Deciding Whether a Game Is Fair

In a game, two players each roll two fair dice and add the twonumbers. Player A wins with a sum of 6 or less. Otherwise playerB wins. Decide whether the game is fair.List all possible outcomes.

1 � 1 � 2 2 � 1 � 3 3 � 1 � 4 4 � 1 � 5 5 � 1 � 6 6 � 1 � 7

1 � 2 � 3 2 � 2 � 4 3 � 2 � 5 4 � 2 � 6 5 � 2 � 7 6 � 2 � 8

1 � 3 � 4 2 � 3 � 5 3 � 3 � 6 4 � 3 � 7 5 � 3 � 8 6 � 3 � 9

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

1 � 5 � 6 2 � 5 � 7 3 � 5 � 8 4 � 5 � 9 5 � 5 � 10 6 � 5 � 11

1 � 6 � 7 2 � 6 � 8 3 � 6 � 9 4 � 6 � 10 5 � 6 � 11 6 � 6 � 12

Find the theoretical probability of each player’s winning.

P(player A winning) � �13

56�

P(player B winning) � �23

16�

Since �13

56� � �

23

16�, the game is not fair.

number of possible red outcomes��total possible outcomes

Think and Discuss

1. Give an example of a game that is fair. Explain how you know.

Find the theoreticalprobability of spinning red.

2E X A M P L E

There are 15 combinations with a sumof 6 or less

There are 21 combinations with a sumgreater than 6.

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A store sells cases to hold CDs. The table shows thecapacities of the last 200 cases sold. The store is going toorder 1500 more CD cases. Use probability to decide howmany of each type of case to order.

1. 24-CD case 2. 96-CD case

3. Players use a spinner to movearound a game board. Suppose thespinner is spun 40 times. Predicthow many times the spinner willland on “Get a Clue!”

Decide whether each game is fair.

4. Roll two fair number cubes labeled 1–6. Add the two numbers. Player Awins if the sum is odd. Player B wins if the sum is even.

5. Toss three fair coins. Player A wins if exactly 2 heads land up. OtherwisePlayer B wins.

6. In her last ten 10K runs, Celia had the following times in minutes: 50:30,50:37, 48:29, 50:46, 51:12, 49:19, 49:50, 51:19, 53:39, and 53:54. Based onthese results, what is the best prediction of the number of times Celia willrun faster than 50 minutes in her next 30 runs?

7. Football games begin with a coin toss to decide who kicks off and whoreceives. The Cougars won the coin toss in their first 2 games. Predict howmany coin tosses the Cougars will win in their next 10 games.

Decide whether each game is fair.

8. Roll two fair number cubes labeled 1–6. Add the two numbers. Player Awins if the sum is a multiple of 3. Otherwise Player B wins.

9. A spinner is divided evenly into 8 sections. There are 4 blue sections, 2 red, 1 green, and 1 yellow. Player A wins if the spinner lands on blue.Otherwise Player B wins.

A fair number cube is labeled 1–6. Predict the number of outcomes for thegiven number of rolls.

10. outcome: 3 11. outcome: even numbernumber of rolls: 36 number of rolls: 50

12. outcome: not 2 13. outcome: greater than 6number of rolls: 72 number of rolls: 100

KEYWORD: MT7 Parent

KEYWORD: MT7 10-6

GUIDED PRACTICE




See Example 2

See Example 1

See Example 1

See Example 2

CD Cases Sold

Capacity Number

24 CDs 70

32 CDs 13

64 CDs 24

96 CDs 52

160 CDs 41

1

2

34

Get aClue!

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10-6 Making Decisions and Predictions 553

14. School Before a school election, a sample of voters gave Karim 28 votes,Marisol 41, and Richard 11. Based on these results, predict the number ofvotes for each candidate if 1600 students vote.

15. Critical Thinking Jack suggested the following game to Charlie: “Let’sroll two dice. We’ll subtract the smaller number from the larger. If thedifference is 0, 1, or 2, I get a point. If the difference is 3, 4, or 5, you get apoint.” Charlie thought the game sounded fair. Decide whether Charliewas correct. If he was not, describe a way to make the game fair.

16. Estimation An ice-skating rink inspects 23 pairs of skates and finds 2pairs to be defective. Estimate the probability that a pair of skates chosenat random will be defective. The rink has 121 pairs of ice skates. Estimatethe number of pairs that are likely to be defective.

17. School There are 540 students in Marla’s school. In her classroom, thereare 2 left-handed students and 18 right-handed students. Predict thenumber of left-handed students in the whole school.

18. Write a Problem Use sports statistics from the newspaper or Internet towrite a prediction problem using probability.

19. Write About It If you make a prediction based on experimentalprobability, how accurate will your prediction be?

20. Challenge A bag contains 10 number tiles labeled 1–10. Which 2 number tiles would you remove from the bag to increase the chances of the following events: drawing an even tile, drawing a multiple of 3, and drawing a number less than 5? Explain.

21. Multiple Choice In a survey of 500 potential voters, Susan Wilson waspicked by 182 people, Anthony Altimuro by 96, Laura Carson by 128, andPaul Johannson by 94. In the actual election, which is the best estimate ofthe percent of votes Anthony Altimuro can expect to receive?

19% 24% 48% 96%

22. Short Answer A game consists of spinning the spinner twice andadding the results. Player A wins if the sum is 4. Otherwise Player Bwins. Decide whether the game is fair.

Find the area of each figure with the given dimensions. (Lesson 8-2)

23. triangle: b � 26, h � 16 24. trapezoid: b1 � 14, b2 � 18, h � 9

25. triangle: b � 10m, h � 8 26. trapezoid: b1 � 6.2, b2 � 11, h � 5.4

27. A company manufactures a toy cube that is 4 in. on each edge. If thelength of each edge is doubled, what will be the effect on the volume ofthe cube? (Lesson 8-5)

DCBA

21

3

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10-7 Odds

Learn to convertbetween probabilities and odds.

Vocabulary

odds against

odds in favor

Schools often sell raffle tickets as a way to raise money. Familyand friends buy the tickets andhave a chance to win a prize. The odds of winning depend on the number of tickets sold and the number of prizes raffled.

The of an event is the ratio of favorable outcomes tounfavorable outcomes. The an event is the ratio of unfavorable outcomes to favorable outcomes.

Finding Odds

Jordan Middle School sold 552 raffle tickets for the chance to be ateacher for the day. Minnie bought 6 raffle tickets.

What are the odds in favor of Minnie’s winning the raffle?

The number of favorable outcomes is 6, and the number ofunfavorable outcomes is 552 � 6 � 546. Minnie’s odds in favor of winning the raffle are 6 to 546, or 1 to 91.

What are the odds against Minnie’s winning the raffle?

The odds in favor of Minnie’s winning are 1 to 91, so the oddsagainst her winning are 91 to 1.

Probability and odds are related. The odds in favor of rolling a two ona fair number cube are 1:5. There is 1 way to get a two and 5 ways notto get a two. The sum of the numbers in the ratio is the denominatorof the probability, �

16�.

odds against

odds in favor

odds in favor

odds against

a � number of favorable outcomesb � number of unfavorable outcomesa � b � total number of outcomes

E X A M P L E 1

CONVERTING ODDS TO PROBABILITIES

If the odds in favor of an event are a:b, then the probability of theevent’s occurring is �a �

ab�.

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10-7 Odds 555

Converting Odds to Probabilities

If the odds in favor of winning movie passes are 1:10, what is theprobability of winning movie passes?

P(movie passes) � �1 �1

10� � �111� On average, there is 1 win for

every 10 losses, so someonewins 1 out of every 11 times.

If the odds against winning a flat-screen television are 39,999:1,what is the probability of winning a flat-screen television?

If the odds against winning the television are 39,999:1, then theodds in favor of winning the television are 1:39,999.

P(television) = �1� 319,999� � �40,

1000� � 0.000025

Suppose that the probability of an event is �13�. This means that, on

average, it will happen in 1 out of every 3 trials, and it will not happenin 2 out of every 3 trials. The odds in favor of the event are 1:2, andthe odds against the event are 2:1.

Converting Probabilities to Odds

The probability of winning a CD player is �715�. What are the odds

in favor of winning a CD player?

On average, 1 out of every 75 people wins, and the other 74people lose. The odds in favor of winning the CD player are 1:(75 � 1), or 1:74.

The probability of winning an electric scooter is �1251,000�. What are

the odds against winning a scooter?

On average, 1 out of every 125,000 people wins, and the other124,999 people lose. The odds against winning the scooter are(125,000 � 1):1, or 124,999:1.

Think and Discuss

1. Explain the difference between probability and odds.

2. Compare the odds in favor of an event with the odds against it.

2E X A M P L E

CONVERTING PROBABILITIES TO ODDS

If the probability of an event is �mn�, then the odds in favor of the event

are m:(n � m) and the odds against the event are (n � m):m.

3E X A M P L E

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Monroe Middle School is holding a raffle for a new bicycle. Jonathan bought7 of the 945 tickets that were sold.

1. What are the odds in favor of Jonathan’s winning the raffle?

2. What are the odds against Jonathan’s winning the raffle?

3. If the odds in favor of winning a trip for two to Hawaii are 1:141,999, whatis the probability of winning the trip?

4. If the odds against winning a digital camera are 25,999:1, what is theprobability of winning the camera?

5. The probability of winning a CD is �810�. What are the odds in favor of

winning the CD?

6. The probability of winning a vacation is �22,1750�. What are the odds against

winning the vacation?

A teachers’ convention is giving away a new computer as a door prize. Eachof the 2240 attendees is given 5 tickets for chances to win the computer.

7. What are the odds in favor of winning the computer?

8. What are the odds against winning the computer?

9. If the odds in favor of winning a new portable music player are 1:8999,what is the probability of winning the player?

10. If the odds against being randomly selected for a committee are 19:1,what is the probability of being selected?

11. The probability of winning a gift certificate is �6120�. What are the odds in

favor of winning the gift certificate?

12. The probability of winning a portable DVD player is �12,1000�. What are the

odds against winning the player?

You roll two fair number cubes. Find the odds in favor of and against each event.

13. rolling two 1’s 14. rolling a total of 6

15. rolling a total of 4 16. rolling doubles

17. rolling an odd and an even number 18. rolling a 5 and a 3

19. The probability of choosing a black card from a standard deck is 50%.What are the odds in favor of choosing a black card?

KEYWORD: MT7 Parent

KEYWORD: MT7 10-7

GUIDED PRACTICE




See Example 2

See Example 2

See Example 3

See Example 3

See Example 1

See Example 1

m807_c10_554_557 1/18/06 3:46 PM Page 556

10-7 Odds 557

20. Earth Science A newspaper reports that there is a 70% probability of anearthquake of magnitude 6.7 or greater striking the San Francisco BayArea by 2030. What are the odds in favor of the earthquake’s happening?

21. Ruben and Manuel play dominoes twice a week. Over the last 12 weeks, Ruben has won 16 times. Estimate the odds in favor of Manuel’s winning the next match.

22. Business To promote sales, a cereal companyis putting game pieces inside 2,000,000 of its cereal boxes. Of these pieces, 50 win a DVD player, and 10 win a trip to New York City.

a. What are the odds in favor of winning a DVD player?

b. What is the probability of winning a prize in the contest?

c. What are the odds against winning a prize in the contest?

23. Critical Thinking Suppose you are in two contests that are independentof each other. You are given the odds of winning one at 1:4 and the odds ofwinning the other at 3:20. How would you find the odds of winning both?

24. What’s the Error? A company receives 6 applications for one job. All ofthe candidates are equally likely to be selected for the job. One of thecandidates figures that the odds in favor of her being selected are 1:6.What error has the candidate made?

25. Write About It A computer randomly selects a digit from 0 to 9.Describe how to determine the odds that the number selected will begreater than 6.

26. Challenge A spinner has three outcomes, regions A, B, or C. Region A istwice as large as regions B or C. Regions B and C have the same size. Findthe odds in favor of the spinner landing on A.

27. Multiple Choice The probability of winning a raffle is �12100�. What are the

odds in favor of winning the raffle?

1:1200 1:1199 1199:1 1200:1

28. Gridded Response The odds of winning a bicycle is 1:149. What is theprobability, written as a fraction, of winning a bicycle?

Find the interest and the total amount to the nearest cent. (Lesson 6-7)

29. $300 at 5% per year for 2 years 30. $750 at 4.5% per year for 4 years

31. Toss two fair coins. Player A wins if the coins land with two heads or twotails facing up. Otherwise, Player B wins. Decide whether the game is fair.

DCBA

m807_c10_554_557 1/12/06 3:14 PM Page 557



Learn to find thenumber of possibleoutcomes in anexperiment.

Vocabulary

Addition CountingPrinciple

tree diagram

FundamentalCounting Principle The demand for new telephone numbers is exploding as people are

using extra phone lines, cellular phones, pagers, computer modems,and fax machines. To meet the demand, state regulators are addingnew area codes.

Phone numbers have ten digits beginning with the three-digit areacode. This results in over a billion possible phone numbers!

Using the Fundamental Counting Principle

A telephone company is assigned a new area code and can issuenew 7-digit phone numbers. All phone numbers are equally likely.

Find the number of possible 7-digit phone numbers.

Use the Fundamental Counting Principle.

first second third fourth fifth sixth seventhdigit digit digit digit digit digit digit

10 10 10 10 10 10 10choices choices choices choices choices choices choices

10 � 10 � 10 � 10 � 10 � 10 � 10 � 10,000,000

The number of possible 7-digit phone numbers is 10,000,000.

Find the probability of being assigned the phone number 555-1234.

P(555-1234) �

� �10,0010,000�

� 0.0000001

1��number of possible phone numbers

???????

THE FUNDAMENTAL COUNTING PRINCIPLEIf there are m ways to choose a first item and n ways to choose asecond item after the first item has been chosen, then there arem � n ways to choose all the items.

E X A M P L E 1

© S

cott

Ad

ams/

Dis

t. b

y U

nit

edFe

atu

re S

ynd

icat

e, In

c.

m807_c10_558_562 1/12/06 3:14 PM Page 558

10-8 Counting Principles 559

A telephone company is assigned a new area code and can issuenew 7-digit phone numbers. All phone numbers are equally likely.

Find the probability of a phone number that does not contain an 8.

First use the Fundamental Counting Principle to find the numberof phone numbers that do not contain an 8.

9 � 9 � 9 � 9 � 9 � 9 � 9 � 4,782,969 possible phone numbers withoutan 8

There are 9 choices for any digit except 8.

P(no 8) � �140,7,0

8020,9,0

6090� � 0.478

The Fundamental Counting Principle tells you only the number ofoutcomes in some experiments, not what the outcomes are. A

is a way to show all of the possible outcomes.

Using a Tree Diagram

You pack 2 pairs of pants, 3 shirts, and 2 sweaters for yourvacation. Describe all of the outfits you can make if each outfitconsists of a pair of pants, a shirt, and a sweater. You can find all of the possible outcomes by making a tree diagram.There should be 2 � 3 � 2 � 12 different outfits.

Each “branch” of the tree diagram represents a different outfit. Theoutfit shown in the circled branch could be written as (black, red, gray). The other outfits are as follows:(black, red, tan), (black, green, gray), (black, green, tan), (black, yellow, gray), (black, yellow, tan), (blue, red, gray), (blue, red, tan), (blue, green, gray), (blue, green, tan), (blue, yellow, gray), (blue, yellow, tan).

tree diagram

2E X A M P L E

THE ADDITION COUNTING PRINCIPLE

If one group contains m objects and a second group contains nobjects, and the groups have no objects in common, then there are m � n total objects to choose from.

m807_c10_558_562 1/12/06 3:14 PM Page 559


Using the Addition Counting Principle

How many items can you choose from Bergen’s Deli menu?

None of the lists contains identical items, so use the AdditionCounting Principle.

10

There are 10 items to choose from.

�3�3�4�T

Soups�Salads�Sandwiches�Total Choices

Bergen’s Deli Menu

Sandwiches Salads Soups

Turkey Cobb Salad TomatoHam Taco Salad Chicken Noodle

Roast Beef Grilled Chicken Salad Split PeaRueben

Employee identification codes at a company contain 2 letters followed by 3 digits. All codes are equally likely.

1. Find the number of possible identification codes.

2. Find the probability of being assigned the ID AB123.

3. Find the probability that an ID code does not contain the number 5.

4. The soup choices at a restaurant are clam chowder, baked potato, andsplit pea. The sandwich choices are egg salad, roast beef, and pastrami.Describe all of the different soup and sandwich options available.

5. Fahti checked out 3 mysteries, 3 historical fiction books, and 2 biographiesfrom the library. How many choices of books does she have to read?

GUIDED PRACTICE

E X A M P L E 3

10-8 ExercisesExercisesKEYWORD: MT7 10-8

KEYWORD: MT7 Parent

See Example 1

See Example 2

See Example 3

Think and Discuss

1. Suppose in Example 2 you could pack one more item. Whichwould you bring, another shirt or another pair of pants? Explain.

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10-8 Counting Principles 561

INDEPENDENT PRACTICELicense plates in a certain state contain 3 letters followed by 4 digits. Assumethat all combinations are equally likely.

6. Find the number of possible license plates.

7. Find the probability of not being assigned a plate containing C or D.

8. Find the probability of receiving a plate containing no vowels (A, E, I, O, U).

9. A clothing catalog offers a shirt in red, blue, yellow, or green, with a choiceof petite or regular, and in small, medium, or large sizes. Describe all ofthe different shirts that are available.

10. There are 3 ways to travel from Los Angeles to San Francisco (car, train, orplane) and 2 ways to travel from San Francisco to Honolulu (plane orboat). Describe all the ways a person can travel from Los Angeles toHonolulu with a stopover in San Francisco.

11. A company makes cell phone face plates. It offers 6 solid colors, 6 prints,and 6 transparent colors. How many different face plates does thecompany offer?

See Example 1

See Example 2

See Example 3

Find the number of possible outcomes.

12. dogs: terrier, retriever, hound, poodletoys: bone, ball

13. sausage: Polish, bratwurst, chicken applecondiment: ketchup, mustard, relish

14. car: sedan, coupe, minivancolor: red, blue, white, black

15. destinations: Paris, London, Romemonths: May, June, July, August

16. An airline confirmation code is 6 letters that can repeat. How manyconfirmation codes are possible?

17. A personal code for an online account must be 6 characters, either lettersor numbers, which can repeat. How many codes are possible?

18. A car model is sold in 6 colors, with or without air conditioning, with orwithout a moon roof, and with either automatic or standard transmission.In how many different ways can this car model be sold?

19. Sarah needs to register for one course in each of the six subject areas. Theschool offers 5 math courses, 4 foreign language courses, 3 sciencecourses, 3 English courses, 5 social studies courses, and 6 elective courses.In how many ways can she register?

20. A computer password consists of 4 letters. The password is case sensitive,which means upper-case and lower-case letters are different characters.What is the probability of randomly being assigned the password YarN?


See page 801.

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21. Technology Tim is buying a newcomputer from an online store.His options are shown at right. Hecan choose a color, one softwarepackage, and one hardwareoption.

a. How many computer choicesare available?

b. Tim decides he wants a redcomputer. Describe all of thechoices available to him.

22. Write About It Describe whenyou would want to use theFundamental Counting Principleinstead of a tree diagram.Describe when a tree diagram would be more useful than theFundamental Counting Principle.

23. Challenge A password can have letters, numerals, or 32 other specialsymbols in each of its 6-character spaces. There are two restrictions. Thepassword cannot begin with a special symbol or 0, and it cannot end witha vowel (A, E, I, O, U ). Find the total number of passwords.

24. Multiple Choice Lynnwood High School requires all staff members to have a 6-character computer password that contains 2 letters and 4 numbers. Find the number of possible passwords.

2,600,000 6,760,000 17,576,000 45,697,600

25. Gridded Response A password contains 3 letters from the alphabet and2 digits (0–9). Find the probability, written as a decimal, of NOT having apassword with a B or D.

Evaluate each expression. (Lesson 4-5)

26. �121� � �25� 27. (4 � 3)2 28. 29. �52 � 1�22�

Use the table of random numbers to answer the following question. Use atleast 10 trials to simulate the situation. (Lesson 10-3)

30. At a local restaurant, 45% of the customers order spaghetti. Estimate theprobability that 6 of the next 10 customers will order spaghetti.

�441��

DCBA

82 78 3 56 86 14 96 96 46 23 62 28 75 61 64 6 304 12 62 54 98 30 94 11 46 22 24 55 89 92 41 79 15

58 69 73 19 73 95 45 26 39 37 91 57 90 19 7 38 44

Technology

In the process ofspin-coating aCD-ROM, the discis rotated at highspeeds. Thisprocess is usedto apply layers asthin as �

18� of a

micron, which is640 times thinnerthan a humanhair.

Red Blue Purple Green

Computer SelectionsComputer SelectionsColors

Software packagesBusinessGraphicsWord processing

HardwareCD ROM

DVD

DVD/CD RW

Checkout

Back Forward Stop Refresh Home

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10-9 Permutations and Combinations 563

Most MP3 players have a shuffle feature thatallows you to play songs in a random order. Youcan use factorials to find out how many songorders are possible.

The of a number is the product of allthe whole numbers from the number down to 1.The factorial of 0 is defined to be 1.

Evaluating Expressions Containing Factorials

Evaluate each expression.

8!

8 � 7 � 6 � 5 � 4 � 3 � 2 � 1 � 40,320

�74

!!�

�7 � 64� 5

� 3� 4

� 2� 3

� 1� 2 � 1� Write out each factorial and simplify.

7 � 6 � 5 � 210 Multiply remaining factors.

�(11

1�4!

4)!� Subtract within parentheses.

�174!!

�

14 � 13 � 12 � 11 � 10 � 9 � 8 � 17,297,280

A is an arrangement of things in a certain order.

If no letter can be used more than once, there are 6 permutations ofthe first 3 letters of the alphabet: ABC, ACB, BAC, BCA, CAB, and CBA.

first letter second letter third letter

3 choices � 2 choices � 1 choice

The product can be written as a factorial.

3 � 2 � 1 � 3! � 6

???

permutation

14 � 13 � 12 � 11 � 10 � 9 � 8 � 7 � 6 � 5 � 4 � 3 � 2 � 1��7 � 6 � 5 � 4 � 3 � 2 � 1

factorial

10-9

Learn to findpermutations andcombinations.

Vocabulary

combination

permutation

factorial

Permutations andCombinations

E X A M P L E 1

Read 8! as “eightfactorial.”

m807_c10_563_567 1/18/06 3:46 PM Page 563


If no letter can be used more than once, there are 60 permutations ofthe first 5 letters of the alphabet, when taken 3 at a time: ABC, ABD,ABE, ACD, ACE, ADB, ADC, ADE, and so on.

first letter second letter third letter

5 choices � 4 choices � 3 choices � 60 permutations

Notice that the product can be written as a quotient of factorials.

60 � 5 � 4 � 3 � �5 � 4

2� 3

� 1� 2 � 1� � �

52

!!�

Finding Permutations

There are 7 swimmers in a race.

Find the number of orders in which all 7 swimmers can finish.

The number of swimmers is 7.

7P7 � �(7 �7!

7)!� � �70

!!� ��7

� 6 � 5 � 41

� 3 � 2 � 1�� 5040

All 7 swimmers are taken at a time.

There are 5040 permutations. This means there are 5040 orders inwhich 7 swimmers can finish.

Find the number of ways the 7 swimmers can finish first, second,and third.

The number of swimmers is 7.

7P3 � �(7 �7!

3)!� � �74

!!� ��7

� 64� 5

� 3� 4

� 2� 3

� 1� 2 � 1�� 210

The top 3 places are taken at a time.

There are 210 permutations. This means that the 7 swimmers canfinish in first, second, and third in 210 ways.

A is a selection of things in any order.

If no letter can be used more than once, there is only 1 combinationof the first 3 letters of the alphabet. ABC, ACB, BAC, BCA, CAB, andCBA are considered to be the same combination of A, B, and Cbecause the order does not matter.

combination

???

PERMUTATIONS

The number of permutations of n things taken r at a time is

nPr � �(nn�

!r)!�.

2E X A M P L E

By definition, 0! � 1.

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If no letter is used more than once, there are 10 combinations of thefirst 5 letters of the alphabet, when taken 3 at a time. To see this, lookat the list of permutations below.

In the list of 60 permutations, each combination is repeated 6 times.The number of combinations is �

660� � 10.

Finding Combinations

A gourmet pizza restaurant offers 10 topping choices.

Find the number of 3-topping pizzas that can be ordered.

10 possible toppings

10C3 � �3!(1010

�!

3)!� � �31!07!!� � � 120

3 toppings chosen at a time

There are 120 combinations. This means that there are 120different 3-topping pizzas that can be ordered.

Find the number of 6-topping pizzas that can be ordered.

10 possible toppings

10C6 � �6!(1010

�!

6)!� � �61!04!!� � � 210

6 toppings chosen at a time

There are 210 combinations. This means that there are 210different 6-topping pizzas.

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

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

Think and Discuss

1. Explain the difference between a combination and a permutation.

2. Give an example of an experiment where order is important andone where order is not important.

3E X A M P L E

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

ACB ADB AEB ADC AEC AED BDC BEC BED CED

BAC BAD BAE CAD CAE DAE CBD CBE DBE DCE

BCA BDA BEA CDA CEA DEA CDB CEB DEB DEC

CAB DAB EAB DAC EAC EAD DCB EBC EBD ECD

CBA DBA EBA DCA ECA EDA DBC ECB EDB EDC

COMBINATIONS

The number of combinations of n things taken r at a time is

nCr � �nrP!r� � �r!(n

n�!

r)!�.

These 6permutations are all the samecombination.

m807_c10_563_567 1/12/06 3:14 PM Page 565




1. 6! 2. �73

!!� 3. �(7 �

9!3)!� 4. �(4 �

5!1)!�

There are 11 runners in a race.

5. In how many possible orders can all 11 runners finish the race?

6. How many ways can the 11 runners finish first, second, and third?

A group of 8 people are forming several committees.

7. Find the number of different 3-person committees that can be formed.

8. Find the number of different 6-person committees that can be formed.


9. 4! 10. �82

!!� 11. �(3 �

4!2)!� 12. �(8 �

9!5)!�

Ann has 7 books she wants to put on her bookshelf.

13. How many possible arrangements of books are there?

14. Suppose Ann has room on the shelf for only 4 of the 7 books. In howmany ways can she arrange the books now?

If Dena joins a CD club, she gets 8 free CDs.

15. If Dena can select from a list of 32 CDs, how many groups of 8 different CDs are possible?

16. If Dena can select from a list of 48 CDs, how many groups of 8 different CDs are possible?


17. �(8 �8!

3)!� 18. �6!(1111

�!

6)!� 19. 10P10 20. 8C3

21. 15C15 22. 10C7 23. �11

20

!!� 24. 8P4

Simplify each expression.

25. nCn 26. �(nn�!

1)!� 27. nC0 28. nCn � 1

29. nP0 30. nPn 31. nC1 32. nP1

33. Sports How many ways can a coach choose the first, second, third, andfourth runners in a relay race from a team of 10 runners?

KEYWORD: MT7 Parent

KEYWORD: MT7 10-9

GUIDED PRACTICE




See Example 2

See Example 2

See Example 3

See Example 3

See Example 1

See Example 1

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34. Cooking Cole is making a fruit salad. He can choose from the followingfruits: oranges, apples, pears, peaches, grapes, strawberries, cantaloupe,and honeydew melon. If he wants to have 4 different fruits, how manypossible fruit salads can he make?

35. Art An artist is making a painting of three squares, one inside the other.He has 12 different colors to choose from. How many different paintingscould he make if the squares are all different colors?

36. Sports At a track meet, there are 5 athletes competing in the decathlon.

a. Find the number of orders in which all 5 athletes can finish.

b. Find the number of orders in which the 5 athletes can finish in first,second, and third places.

37. Life Science There are 11 different species of birds in a forest. In howmany ways can researchers capture, tag, and release birds of 6 differentspecies?

38. What’s the Question? There are 12 different items available at a buffet.Customers can choose up to 4 of these items. If the answer is 495, what isthe question?

39. Write About It Explain how you could use combinations andpermutations to find the probability of an event.

40. Challenge How many ways can a local chapter of the MathematicalAssociation of America schedule 4 speakers for 4 different meetings inone day if all of the speakers are available on any of 3 dates?

Art

Josef Albers usedthe simple designof nested squaresto investigatecolor relation-ships. He did notmix colors, butinstead createdhundreds ofvariations usingpaint straightfrom the tube.

41. Multiple Choice In how many ways can 8 students form a single-fileline if each student’s place in line must be considered?

40,320 5040 8 1

42. Short Response A group of 15 people are forming committees. Find thenumber of different 4-person committees that can be formed. Then findthe number of different 5-person committees that can be formed. Showyour work.

43. Draw the front, top, and side views of the figure at right. (Lesson 8-4)

Describe the number of different combinations that can be made using one item from each category. (Lesson 10-8)

44. 3 shirts 45. 4 kinds of bread 4 pairs of shorts 5 kinds of meat7 pairs of socks 3 kinds of chips

DCBA

Front Side

A

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Rea

dy

to G

o O

n?


Quiz for Lessons 10-4 Through 10-9


An experiment consists of rolling two fair number cubes. Find the probability of each event.

1. P(total shown � 7) 2. P(two 5’s) 3. P(two even numbers)

10-5 Independent and Dependent Events

4. An experiment consists of tossing 2 fair coins, a penny and a nickel. Find the probability of tails on the penny and heads on the nickel.

5. A jar contains 5 red marbles, 2 blue marbles, 4 yellow marbles, and 4 green marbles. If two marbles are chosen at random, what is the probability that they will be the same color?

10-6 Making Decisions and Predictions

6. Players use the spinner shown to move around a game board.Suppose the spinner is spun 50 times. Predict how many times it will land on “Lose your turn.”

7. A spinner is divided evenly into 6 sections. There are 3 blue sections, 2 red, and 1 white. Player A wins if the spinner lands on blue. Otherwise Player B wins. Decide whether the game is fair.

10-7 Odds

8. If the odds in favor of winning a trip for two to New York City are 1:259,999, what is the probability of winning the trip?


Family identification codes at a preschool contain 3 letters followed by 3 digits. All codes are equally likely.

9. Find the probability of being assigned the ID BCD352.

10. A catalog company offers backpacks in 5 solid colors, 4 prints, and 4 cartoon characters. How many choices of backpacks are there?

10-9 Permutations and Combinations


11. 7! 12. 5! 13. �62

!!� 14. �(6 �

8!3)!�

15. There are 10 cross-country skiers in a race. In how many possible orders can all 10 skiers finish the race?

23

1 Loseyourturn.

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Perplexing Polygons In Mrs. Mac’s class, eachstudent is given a set of polygon cards to cut outalong the dotted lines. Each student places the cardsin his or her own brown paper bag.

Mu

lti-Step Test Prep

Multi-Step Test Prep 569

1. Juan draws a polygon from his bag at the same time that Monica draws a polygon from her bag. They do this experiment 50 times and replace the polygons each time before drawing again. How many times would Juan be expected to draw a shaded polygon at the same time that Monica draws a triangle?

2. Kyle draws twice from his bag of polygons. After the first draw, he does not put the polygon back in the bag. Predict the number of times he might draw a square and then a triangle if he conducts this experiment 24 times.

3. Eight students place all of their polygons into a hat. What is the probability of drawing a hexagon?

4. The students are asked to remove 16 polygons from the hat. How can they do this so that the probability of drawing a hexagon remains the same?

5. Eight of the students place all of their polygons into a single bag. Describe how to remove polygons from this bag so that the probability of drawing a hexagon is �

15�.

m807_c10_569_569 1/12/06 3:14 PM Page 569

Use a set of Scrabble™ tiles, or make a similar set of lettered cards. Draw 2 vowelsand 3 consonants, and place them face up in the center of the table. Each player tries to write as many permutations as possible in 60 seconds. Score 1 point per permutation, with a bonus point for eachpermutation that forms an English word.

A complete copy of the rules is available online.

The Paper ChaseStephen’s desk has 8 drawers. When he receives a paper, he usually chooses a drawer at random to put it in. However, 2 out of 10 times he forgets toput the paper away, and it gets lost.

The probability that a paper will get lost is �120�, or �

15�.

• What is the probability that a paper will get put into a drawer?

• If all drawers are equally likely to be chosen,what is the probability that a paper will get putin drawer 3?

When Stephen needs a document, he looks first in drawer 1 and then checks each drawer in orderuntil the paper is found or until he has looked inall the drawers.

If Stephen checked drawer 1 and didn’t find the paper he was looking for, what is the probability that the paper will be found in one of the remaining 7 drawers? If Stephen checked drawers 1, 2, and 3, and didn’t find the paper he was looking for, what is the probability that the paper will be found in one of the remaining 5 drawers? If Stephen checked drawers 1–7 and didn’t find the paper he was looking for, what is the probability that the paper will be found in the last drawer?

Try to write a formula for the probability of finding a paper.

3

2

1

PermutationsPermutations


KEYWORD: MT7 Games

m807_c10_570_570 1/18/06 3:48 PM Page 570

It’s in the Bag! 571

A

B

Probability Post-Up

PROJECT

Materials • 7 large sticky

notes• glue• markers

Fold sticky notes into an accordion booklet. Thenuse the booklet to record notes about probability.

DirectionsMake a chain of seven overlapping sticky notesby placing the sticky portion of one note on thebottom portion of the previous note. Figure A

Glue the notes together to make sure they stayattached.

Accordion-fold the sticky notes. The foldsshould occur at the bottom edge of each notein the chain. Figure B

Write the name and number of the chapter onthe first sticky note.

Taking Note of the MathUse the sticky-note booklet to record keyinformation from the chapter. Be sure to include definitions, examples of probability experiments, and anything else that will help you review the material in the chapter.

4

3

2

1

m807_c10_571_571 1/12/06 3:15 PM Page 571


Complete the sentences below with vocabulary words from thelist above. Words may be used more than once.

1. The ___?___ of an event tells you how likely the event is to happen. • A probability of 0 means it is ___?___ for the event to occur. • A probability of 1 means it is ___?___ that the event will occur.

2. The set of all possible outcomes of an experiment is called the ___?___.

3. A(n) ___?___ is an arrangement where order is important. A(n) ___?___ is an arrangement where order is not important.

Vocabulary

Probability (pp. 522–526)10-1

E X A M P L E EXERCISES

■ Of the garbage collected in a city, it isexpected that about �

15� of the garbage will

be recycled.

P(recycled) � �15� � 0.2 � 20%

P(not recycled) �1 � �15� � �

45� � 0.8 � 80%


4. About 85% of the people attending aband’s CD signing have already heardthe CD.

Stu

dy

Gu

ide:

Rev

iew

. . . . . . . . . . . . . 559

. . . . . . . . . . . . . . . 522

. . . . . . . . . . 564

. . . . . 545

. . . . . . . . 542

. . . . . . . . . 540

. . . . . . . . . . . . . . . . 522

. . . . . . . . . . . 522

. . . . . . . . . . . 527

. . . . . . . . . . . . . 563

. . . . . . . . . . . . . . . . . 540

. . . . . . . . . . . . . 558

. . . . . . . . . . . 522

. . . 545

. . . . 542

. . . . . . . . . . 554

. . . . . . . . . 554

. . . . . . . . . . . . . 522

. . . . . . . . . 563

. . . . . . . . . . . 522

. . . . . 532

. . . . . . . . . 522

. . . . . . . . . . . 532

. 540

. . . . . . . . . 559

. . . . . . . . . . . . . . . . . 522trial

tree diagram

theoretical probability

simulation

sample space

random numbers

probability

permutation

outcome

odds in favor

odds against

mutually exclusive

independent events

impossible

PrincipleFundamental Counting

fair

factorial

probabilityexperimental

experiment

event

equally likely

disjoint events

dependent events

combination

certain

PrincipleAddition Counting

Outcome Recycled Not Recycled

ProbabilityOutcome Heard Not Heard

Probability

m807_c10_572_574 1/12/06 3:15 PM Page 572

Stud

y Gu

ide: R

eview

Experimental Probability (pp. 527–530)10-2


■ The table shows the results of spinning aspinner 72 times. Estimate the probabilityof the spinner landing on red.

probability � �27

82� � �1

78� � 0.389 � 38.9%

5. The table shows the result of rolling anumber cube 80 times. Estimate theprobability of rolling a 4.

Outcome White Red Blue Black

Spins 18 28 12 14

Outcome 1 2 3 4 5 6

Rolls 13 15 10 12 5 25

Use a Simulation (pp. 532–535)10-3


■ At a local school, 75% of the studentsstudy a foreign language. If 5 students arechosen randomly, estimate the probabilitythat at least 4 study a foreign language.Use the random number table to make asimulation with at least 4 trials.

92 8778 88

The probability is about �34�, or 75%.

08570 99275 27378 75236 16732

93973 78658 80242 53191 86579

6. On an assembly line, 20% of the itemsare rejected. Estimate the probabilitythat at least 3 of the next 6 items arerejected. Use the random number table to make a simulation with at least4 trials.4202657339297316

3652372775095708

Theoretical Probability (pp. 540–544)10-4


■ A fair number cube is rolled once. Find theprobability of getting a 4.

P(4) � �16�

7. A marble is drawn at random from abox that contains 8 red, 15 blue, and 7white marbles. What is the probabilityof getting a red marble?

Independent and Dependent Events (pp. 545–549)10-5


■ Two marbles are drawn from a jarcontaining 5 blue marbles and 4 green.What is P(blue, green) if the first marbleis not replaced?

8. A fair number cube is rolled four times.What is the probability of getting a 6 allfour times?

9. Two cards are drawn at random from adeck that has 26 red and 26 black cards.What is the probability that the firstcard is red and the second card is black?

P(blue) P(green) P(blue, green)

Not replaced �59� �

48� �

2702� � 0.28

Study Guide: Review 573

m807_c10_572_574 1/12/06 3:15 PM Page 573


Stu

dy

Gu

ide:

Rev

iew

Making Decisions and Predictions (pp. 550–553)10-6


■ A director needs to order 600 T-shirts.Last summer she gave out 210 blue and150 red T-shirts. Approximately howmany red T-shirts should she order?

�13

56

00� � �1

52� Find the probability of red.

�152� � �6


12n � 3000 Solve for n.

n � 250

She should order 250 red T-shirts.

10. The speeds of each of 10 laps by aNASCAR racer were measured. Theapproximate speeds in miles per hourwere 188.2, 188.8, 191.2, 191.4, 189.1,187.6, 186.3, 191.1, 190.3, and 189.5. Ifthe driver goes 50 more laps, what is thebest prediction of the number of lapsthat will be at a speed greater than 190miles per hour?

Odds (pp. 554–557)10-7


■ A digit from 1 to 9 is selected at random.What are the odds in favor of selectingan even number?

favorable 4:5 unfavorable

11. A letter is selected at random from thealphabet. What are the odds in favor ofgetting a letter in the word RANDOM?

Counting Principles (pp. 558–562)10-8


■ A code contains 4 letters. How manypossible codes are there?

26 � 26 � 26 � 26 � 456,976 codes

ID codes contain 1 letter followed by 5digits. All codes are equally likely.

12. Find the number of possible ID codes.

13. Find the probability that a code doesnot contain the digit 0.

Permutations and Combinations (pp. 563–567)10-9


■ Blaire has 5 plants to arrange on a shelfthat will hold 3 plants. How many waysare there to arrange the plants if theorder is important? if the order is notimportant?

important: 5P3 � �(5 �5!

3)!� � �52

!!� � 60 ways

not important: 5C3 � �3! (55�!

3)!�� 10 ways

14. Five children are arranged in a row ofswings. How many differentarrangements are possible?

15. A school’s mock trial team has 10 members. A team of 6 students willbe chosen to represent the school at acompetition. How many different teamsare possible?

m807_c10_572_574 1/12/06 3:15 PM 4

Chapter 10 Test 575


1. P(D) 2. P(not A)

3. P(B or C)

4. There are 4 cyclists in a race. Kyle has a 50% chance of winning.Lance has the same chance as Miguel. Eddie has a �

15� chance of

winning. Create a table of probabilities for the sample space.

A coin is randomly drawn from a box and then replaced. The tableshows the results.

5. Estimate the probability of eachoutcome.

6. Estimate P(penny or nickel).

7. Estimate P(not dime).

8. In Eastwood neighborhood, 37% of the families have a cat. Eachblock has 16 families, 8 on each side. Estimate the probability that 3 or more families on one side of a given block have a cat. Use therandom number table to make a simulation with at least 10 trials.

97120 08320 17871 21826 74838 37240 36810 20423

12562 45677 88983 94930 31599 76585 61429 05379

34628 46304 66531 96270 21309 31567 30762 47240

30883 71946 25948 97988 26267 21350 59356 43952


9. P(total shown � 3) 10. P(rolling two 6’s) 11. P(total � 2)

12. A jar contains 6 red tiles, 2 blue, 3 yellow, and 5 green. If two tilesare chosen at random, what is the probability that they both will begreen?

13. A spinner is divided evenly into 9 sections. They are numbered 1 to9. Player A wins if the spinner lands on odd. Otherwise Player Bwins. Decide whether the game is fair.

14. The probability of winning a new widescreen TV is �1,0010,000�. What are

the odds against winning the TV?

15. A code contains 4 letters and 2 numbers. How many possible codesare there?

16. There are 8 swimmers in a race. In how many possible orders canall 8 swimmers finish the race?

Ch

apter Test

Outcome A B C D

Probability 0.2 0.2 0.1 0.5

Outcome Penny Nickel Dime Quarter

Probability 26 35 19 20

m807_c10_575_575 1/12/06 3:15 PM Page 575

Stan

dar

diz

ed T

est

Prep

KEYWORD: MT7 TestPrep

1. In a box containing marbles, 78 areblue, 24 are orange, and the rest aregreen. If the probability of selecting agreen marble is �

25�, how many green

marbles are in the box?

30 102

68 150

2. In the chart below, the amountrepresented by each shaded square istwice that represented by eachunshaded square. What is the ratio ofgold to silver?

�1292� �

2129�

�1139� �

1191�

3. For which set of data are the mean,median, and mode all the same?

3, 1, 3, 3, 5 2, 1, 1, 1, 5

1, 1, 2, 5, 6 10, 1, 3, 5, 1

4. What is the value of (�2 � 4)3 � 30?

�215 217

�8 219

5. About what percent of 75 is 55?

25% 75%

66% 135%

6. Which does NOT describe ��

255�

�?

real integer

rational median

7. The figure formed by the vertices (�2, 5), (2, 5), (4, �1), and (0, �1) canbe best described by which type ofquadrilateral?

square parallelogram

rectangle trapezoid

8. How many vertices are in the prismbelow?

7 10

8 12

9. The triangular reflecting pool has anarea of 350 ft2. If the height of thetriangle is 25 ft, what is the length ofthe hypotenuse to the nearest tenth?

28.7 ft 38.9 ft

37.5 ft 42.3 ft

10. If the probability of selecting a redmarble is �1

14�, what are the odds in favor

of selecting a red marble?

�1134� �1

23� �1

13� �1

15�JHGF

DB

CA

JG

HF

DB

CA

JG

HF

DB

CA

JG

HF

DB

CA

JG

HF

DB

CA

Cumulative Assessment, Chapters 1–10Multiple Choice


Precious Metals CompanySupply of Gold and Silver

Silver

Gold

25 ft x

m807_c10_576_577 1/12/06 3:15 PM Page 576

Cumulative Assessment Chapters 1–10 577

11. If the diameter of the dartboard is 10 in., what is the area of the 50-pointportion, to the nearest tenth of asquare inch?

3.1 in2 9.4 in2

6.3 in2 25.1 in2

Gridded Response

Use the following graph for items 12 and 13.

12. Find the x-coordinate of the orderedpair whose y-coordinate is 1.

13. Determine the value of y when x � 6.

14. What is the probability of rolling aneven number on a number cube andtossing a heads on a coin?

15. Teresa has to create a password thatcontains 1 digit and 2 letters. Find thenumber of possible passwords.

16. What is the value of x for the equation 7 � �

23�x � 3?

Short Response

17. A dart thrown at the square boardshown lands in a random spot. What isthe probability that it lands in the bluesquare? Show your work.

18. The pilot of a hot-air balloon is tryingto land in a 2 km square field. There isa large tree in each corner. The ropeswill tangle in a tree if the balloon landswithin �

17� km of the tree’s trunk. What is

the probability the balloon will landwithout getting caught in a tree?Express your answer to the nearesttenth of a percent. Show your work.

Extended Response

19. Students are choosing a new mascotand color. The mascot choices are abear, a lion, a jaguar, or a tiger. Thecolor choices are red, orange, or blue.

a. How many different combinationsdo the students have to choosefrom? Show your work.

b. If a second school color is added,either gold or silver, how manydifferent combinations do thestudents have to choose from? Show your work.

c. How would adding a choice fromamong n names change the numberof combinations to choose from?

DB

CA

Stand

ardized

Test Prep

2 in.

25 points 50 points

100 points 2 in.

Draw a picture to help you see if youranswer is reasonable.

O 4

4

321

321�1�1

�2

�2

�3�4

�3�4

x

y

2 km

2 km

km17

km17

m807_c10_576_577 1/12/06 3:15 PM Page 577

Clemson Tigers FootballFor the past two decades, Clemson University’s football team has beenone of the sport’s greatest success stories. Since 1985, the ClemsonTigers have appeared in 16 bowl games, placing them among thenation’s most consistently winning teams during that period.

Choose one or more strategies to solve each problem.

1. The Clemson Tigers’ team colors are orange, purple, and white. Theplayers’ jerseys and pants are available in all three colors. How manydifferent uniforms can the team make by choosing a color for thejersey and a color for the pants?

2. The entrance to the team’s locker room has an enormous photo ofthe university’s stadium. The perimeter of the photo is 78 feet. Thelength is 21 feet greater than the width. What are the length andwidth of the photo?

For 3, use the graph.

3. During the 1998 season, the Tigers scored 104 fewerpoints than during the 1999 season. During the 1999 season, they scored 43 fewer points than the median of the seasons shown in the graph. How many points did the Tigers score duringthe 1998 season?

S O U T H C A R O L I N A


Clemson Tigers: Total Points Per Season

2000

2001

2002

2003

2004

1000 200 300 400Total Points Scored

Seas

on

500

Myrtle Beach

Clemson

m807_c10_578_579 1/12/06 3:15 PM Page 578

The South Carolina Hall of FameWhat do President Andrew Jackson, jazz musician Dizzy Gillespie, and athlete Lucile Godbold have in common? All of them were born in South Carolina, and all have been inducted into the South Carolina Hall of Fame. Located in Myrtle Beach, the hall honors citizens of the state who have made lasting contributions in a wide range of fields.

Choose one or more strategies to solve each problem.

1. Each year, 10 living nominees and 10 deceased nominees are selected for induction into the hall. The judges pick one from each group. How many different pairs of possible inductees are there?

2. The hall includes five inductees from the field of medicine. Their portraits are to be lined up next to each other. In how many different ways can the portraits be arranged?

For 3 and 4, use the graph.

3. The graph shows the total number of inductees in the SouthCarolina Hall of Fame. Assume that new inductees continue to be added to the hall at the same rate. Predict the totalnumber of inductees in 2020.

4. In what year will there be more than 100 inductees forthe first time?

South CarolinaHall of Fame Inductees

80

70

60

50

40

30

20

10

0

Year

Tota

l num

ber

of in

duct

ees

1990 1995 2000 2005

Problem SolvingStrategies

Draw a DiagramMake a ModelGuess and TestWork BackwardFind a PatternMake a TableSolve a Simpler ProblemUse Logical ReasoningAct It OutMake an Organized List

m807_c10_578_579 1/12/06 3:15 PM Page 579

chapter 10- probability (in - advisory 207 -...

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