chapter 8: probability...374 chapter 8 probability probability words the probability of an event is...

370 Unit 4 Probability and Statistics

People often basetheir decisions aboutthe future on datathey’ve collected. Inthis unit, you willlearn how to makesuch predictions usingprobability andstatistics.

People often basetheir decisions aboutthe future on datathey’ve collected. Inthis unit, you willlearn how to makesuch predictions usingprobability andstatistics.

Probability

Statistics and Matrices

370 Unit 4 Probability and Statistics

MATH andSCIENCE

IT’S ALL IN THE GENESMirror, mirror on the wall... why do I look like my parents at all? You’ve been selectedto join a team of genetic researchers to find an answer to this very question. On thisadventure, you’ll research basic genetic lingo and learn how to use a Punnett square.Then you’ll gather information about the genetic traits of your classmates. You’ll alsomake predictions based on an analysis of your findings. So grab your lab coat andyour probability and statistics tool kits. This is one adventure you don’t want to miss.

Log on to msmath3.net/webquest to begin your WebQuest.

Unit 4 Probability and Statistics 371

http://msmath3.net/webquest

How are math and bicycles related?Bicycles come in many styles, colors, and sizes. To find how many different types of bicycles a manufacturer makes, you can use a tree diagram or the Fundamental Counting Principle.

You will solve problems about different types of bicycles in Lesson 8-2.

372 Chapter 8 Probability

Duo

mo/

CO

RB

IS

Probability

C H A P T E R


Chapter 8 Getting Started 373

Take this quiz to see whether you are ready tobegin Chapter 8. Refer to the lesson or pagenumber in parentheses if you need more review.

Vocabulary ReviewComplete each sentence.

1. The equation �165� � �

25

� is a because it contains two equivalentratios. (Lesson 4-4)

2. Percent is a ratio that compares anumber to . (Lesson 5-1)

Prerequisite SkillsWrite each fraction in simplest form.(Page 611)

3. �47

82� 4. �

36

50�

5. �29

19� 6. �

38

02�

Evaluate x(x � 1)(x � 2)(x � 3) for eachvalue of x. (Lesson 1-2)

7. x � 11 8. x � 6

9. x � 9 10. x � 7

Evaluate each expression. (Lesson 1-2)

11. �73

��

62

��

51

� 12. �122

��

111

�

13. �84

��

73

��

62

��

51

� 14. �53

��

42

��

31

�

Multiply. Write in simplest form. (Lesson 2-3)

15. �23

� � �34

� 16. �145� � �

57

�

17. �78

� � �49

� 18. �35

� � �16

�

Solve each problem. (Lessons 5-3 and 5-6)

19. Find 28% of 80. 20. Find 55% of 34.

?

?Probability Make thisFoldable to help youorganize your notes. Begin with two sheets of8�

12

�" � 11" unlined paper.

Fold in QuartersFold each sheet in quarters along the width.

Reading and Writing As you read and study the chapter,write notes and examples for each lesson on each page ofthe journal.

TapeUnfold each sheet and tape to form one long piece.

LabelLabel each page with the lesson number asshown. Refold to form a booklet.

8-58-38-2 8-48-1 8-6 8-7

Chapter 8 Getting Started 373

Aaron Haupt

Probability of Simple Events

In the game of double-six dominoes, there are 28 tiles that can bepicked. These tiles are called the . A list of all the tiles iscalled the . If all outcomes occur by chance, the outcomeshappen at .

A is a specific outcome or type of outcome. Whenpicking dominoes, one event is picking a double. is thechance that an event will happen.

Probabilitysimple event

randomsample space

outcomes

The probability that an event will happen is between 0 and 1 inclusive.A probability can be expressed as a fraction, a decimal, or a percent.


Probability

Words The probability of an event is a ratio that compares thenumber of favorable outcomes to the number of possibleoutcomes.

Symbols P(event) �

Example P(doubles) � �278� or �

14

�

number of favorable outcomes��number of possible outcomes

impossible

equally likely

certain

somewhat likelynot very likely

50%

12 or 0.5 3

4 or 0.7514 or 0.25

75% 100%

1

25%0%

0

Double

GAMES The game of double-six dominoes is played with 28 tiles. Seven of the tiles are called doubles.

1. Write the ratio that compares the number of double tiles to the total number of tiles.

2. What percent of the tiles are doubles?

3. Write a fraction in simplest form that represents the part of thetiles that are doubles.

4. Write a decimal that represents the part of the tiles that aredoubles.

5. Suppose you pick a domino without looking at the spots.Would you be more likely to pick a tile that is a double or onethat is not a double? Explain.

am I ever going to use this?

Find the probability of asimple event.

outcomesample spacerandomsimple eventprobabilitycomplementary events

percent: a ratio thatcompares a number to100 (Lesson 5-1)

Virginia SOL Standard 8.11 The studentwill analyze problem situations, including games of chance, board games, orgrading scales, and makepredictions, using knowl-edge of probability.

Lesson 8-1 Probability of Simple Events 375

Find Probabilities

A box contains 5 green pens, 3 blue pens, 8 black pens, and 4 redpens. A pen is picked at random.

What is the probability the pen is green?

There are 5 � 3 � 8 � 4 or 20 pens in the box.

P(green) � Definition of probability

� �250� or �

14

� There are 5 green pens out of 20 pens.

The probability the pen is green is �14

�. The probability can also bewritten as 0.25 or 25%.

What is the probability the pen is blue or red?

P(blue or red) � Definition of probability

� �3

2�

04

� or �270� There are 3 blue pens and 4 red pens.

The probability the pen is blue or red is �270�. The probability can also

be written as 0.35 or 35%.

What is the probability the pen is gold?

Since there are no gold pens, the probability is 0.

The spinner is used for a game.Write each probability as afraction, a decimal, or a percent.

a. P(6) b. P(odd)

c. P(5 or even) d. P(a number less than 7)

6

3

2

45

1

blue pens � red pens��total number of pens

green pens��total number of pens

Mental Math Theprobability of adefective computer is �

21,50000

� or �215�.

Since defective and nondefectivecomputers arecomplementaryevents, theprobability of a nondefectivecomputer is 1 � �

215�

or �2245�.

ProbabilityP(green) is read theprobability of green.

Suppose you roll a number cube. The events of rolling a 6 and of notrolling a 6 are . The sum of the probabilities of complementary events is 1.

complementary events

Probability of a Complementary Event

PURCHASES A computer company manufactures 2,500 computerseach day. An average of 100 of these computers are returned withdefects. What is the probability that the computer you purchasedis not defective?

2,500 � 100 or 2,400 computers were not defective.

P(not defective) � Definition of probability

� �22

,,45

00

00

� or �22

45� There are 2,400 nondefective computers.

The probability that your computer is not defective is �22

45�.

nondefective computers��total number of computers

msmath3.net/extra_examples/sol

http://msmath3.net/extra_examples/sol


1. Draw a spinner where the probability of an outcome of white is �38

�.

2. OPEN ENDED Give an example of an event with a probability of 1.

3. FIND THE ERROR Masao and Brian are finding the probability of getting a 2 when a number cube is rolled. Masao says it is �

16

�, and Brian says it is

�26

�. Who is correct? Explain.

The spinner is used for a game. Write each probability as a fraction,a decimal, and a percent.

4. P(5) 5. P(even) 6. P(greater than 5)

7. P(not 2) 8. P(an integer) 9. P(less than 7)

10. GAMES A card game has 25 red cards, 25 green cards, 25 yellow cards, 25 blue cards, and 8 wild cards. What is the probability that thefirst card dealt is a wild card?

8 5

1 4

67

32

Exercises 2 & 3

A beanbag is tossed on the square at the right. It lands at random in a small square. Write eachprobability as a fraction, a decimal, and a percent.

11. P(red) 12. P(blue)

13. P(white or yellow) 14. P(blue or red)

15. P(not green) 16. P(brown)

17. What is the probability that a month picked at random starts with J?

18. What is the probability that a day picked at random is a Saturday?

19. A number cube is tossed. Are the events of rolling a number greater than 3 and a number less than 3 complementary events? Explain.

20. A coin is tossed twice and shows heads both times. What is the probability that the coin will show a tail on the next toss? Explain.

21. WEATHER A weather reporter says that there is a 40% chance of rain. What is the probability of no rain?

22. WRITE A PROBLEM Write a real-life problem with a probability of �16

�.

23. RESEARCH Use the Internet or other resource to find the probability thata person from your state picked at random will be from your city orcommunity.

Extra Practice See pages 635, 655.

For Exercises

11–20, 24–25

21

See Examples

1–3

4

Lesson 8-1 Probability of Simple Events 377James Balog/Getty Images

HISTORY For Exercises 24–26, use the table at the right and the information below.The U.S. Census Bureau divides the United States into four regions: Northeast, Midwest, South, and West.

24. Suppose a person living in the United States in 1890 was picked at random. What is the probability that the person lived in the West? Write as a decimal to the nearest thousandth.

25. Suppose a person living in the United States in 2000 was picked at random. What is the probability that the person lived in the West? Write as a decimal to the nearest thousandth.

26. How has the population of the West changed?

27. CRITICAL THINKING A box contains 5 red, 6 blue, 3 green, and 2 yellow crayons. How many red crayons must be added to the box so that the probability of randomly picking a red crayon is �

23

�?

EXTENDING THE LESSON The odds of an event occurring is a ratio thatcompares the number of favorable outcomes to the number of unfavorableoutcomes. Suppose a number cube is rolled.

Find the odds of each outcome.

28. a 6 29. not a 6 30. an even number

Source: U.S. Census Bureau

Region 1890 2000

Northeast 17,407 53,594

Midwest 22,410 64,393

South 20,028 100,237

West 3,134 63,198

U.S. Population (thousands)

For Exercises 31 and 32, the following cards are put into a box.

31. MULTIPLE CHOICE Emma picks a card at random. The number on thecard will most likely be

a number greater than 6. a number less than 6.

an even number. an odd number.

32. MULTIPLE CHOICE What is the probability of not getting an 8?

25% 30% 50% 75%

Analyze each measurement. Give the precision, significant digits if appropriate, greatest possible error, and relative error to two significant digits. (Lesson 7-9)

33. 8 cm 34. 0.36 kg 35. 4.83 m 36. 410 cm

37. GEOMETRY Find the surface area of a cone with radius of 5 inches andslant height of 12 inches. (Lesson 7-8)

IHGF

DC

BA

48985762

BASIC SKILL Multiply.

38. 5 � 6 � 2 39. 5 � 5 � 8 40. 12 � 5 � 3 41. 7 � 8 � 2

msmath3.net/self_check_quiz


SOL Practice

/sol

http://msmath3.net/self_check_quiz/sol

Laura Sifferlin

What You’ll LearnSolve problems bymaking an organizedlist.

8-2a Problem-Solving StrategyA Preview of Lesson 8-2


1. Explain why the list of possible bouquets was divided into four-color, three-color, two-color, and one-color bouquets.

2. Explain why a red and white bouquet is the same as a white and red bouquet.

3. Write a problem that can be solved by making an organized list. Include the organized list you would use to solve the problem.

We have all the orders for the Valentine’sDay bouquets. Each student could chooseany combination of red, pink, white, oryellow carnations for their bouquets.

How many different bouquetsdo you think there are?

Make an Organized List

Explore We want to know how many different bouquets can be made from fourdifferent colors of carnations.

Plan Let’s make an organized list.

Four-color bouquets: red, pink, white, yellow

Three-color bouquets: red, pink, white red, pink, yellowred, white, yellow pink, white, yellow

Two-color bouquets: red, pink red, white

Solvered, yellow pink, whitepink, yellow white, yellow

One-color bouquets: red pinkwhite yellow

There is 1 four-color bouquet, 4 three-color bouquets, 6 two-color bouquets,and 4 one-color bouquets. There are 1 � 4 � 6 � 4 or 15 bouquets.

Examine Check the list. Make sure that every color combination is listed and that nocolor combination is listed more than once.

Lesson 8-2a Problem-Solving Strategy: Make an Organized List 379

Solve. Make an organized list.

4. MONEY MATTERS Destiny wants to buy acookie from a vending machine. The cookiecosts 45¢. If Destiny uses exact change, howmany different combinations of nickels,dimes, and quarters can she use?

5. READING Rosa checked out three booksfrom the library. While she was at thelibrary, she visited the fiction, nonfiction,and biography sections. What are thepossible combinations of book types shecould have checked out?

Solve. Use any strategy.

6. GAMES Steven and Derek are playing aguessing game. Steven says he is thinking of two integers between �10 and 10 thathave a product of �12. If Derek has oneguess, what is the probability that he willguess the pair of numbers?

7. COOKING The graph shows the number oftypes of outdoor grills sold. How does thenumber of charcoal grills compare to thenumber of gas grills?

BASEBALL For Exercises 8–10, use thefollowing information.In the World Series, two teams play each otheruntil one team wins 4 games.

8. What is the least number of games neededto determine a winner of the World Series?

9. What is the greatest number of gamesneeded to determine a winner?

10. How many different ways can a team winthe World Series in six games or less? (Hint: The team that wins the series mustwin the last game.)

11. SLEEP What is the probability that aperson between the ages of 35 and 49 talks in his or her sleep? Write the probability as a fraction and as a decimal.

12. MULTI STEP At 2:00 P.M., Cody beganwriting the final draft of a report. At 3:30 P.M., he had written 5 pages. If heworks at the same pace, when should hecomplete 8 pages?

13. MONEY MATTERS Rebecca is shopping forfishing equipment. She has $135 and hasalready selected items that total $98.50. If the sales tax is 8%, will she have enough topurchase a fishing net that costs $23?

14. STANDARDIZEDTEST PRACTICEWhich equation best identifies the pattern in the table?

y � x2

y � 2x2

y � 0.5x2

y � �x2D

C

B

A

Age18–2429%

Age25–3423%

Age35–4915% Age

50�9%

Percent Who Talk in Their Sleep

Source: The Better Sleep Council

Millions ofGrills Sold

Charcoal

GasElectric

7.9 4.3 0.16

Source: Barbecue Industry Association

�2 2

�1 0.5

0 0

1 0.5

2 2

x y

You will use the make an organized list strategy in the next lesson.You will use the make an organized list strategy in the next lesson.

SOL Practice

Count outcomes by using a tree diagram or theFundamental CountingPrinciple.

tree diagramFundamental

Counting Principle

Counting Outcomes

BICYCLES Antonio wants to buy a Dynamo bicycle.

1. How many different styles are available?

2. How many different colors are available?

3. How many different sizes are available?

4. Make an organized list to determine how manydifferent bicycles are available.

Choose your Dynamo TChoose your Dynamo Today!oday!Choose your Dynamo Today!

StStyles: Mountyles: Mountain or 1ain or 10-Speed0-SpeedColors: Red, Black, or GreenColors: Red, Black, or Green

Sizes: 2Sizes: 26-inch or 26-inch or 28-inch8-inch

Styles: Mountain or 10-SpeedColors: Red, Black, or Green

Sizes: 26-inch or 28-inch


An organized list can help you determine the number of possiblecombinations or outcomes. One type of organized list is a .tree diagram

Use a Tree Diagram

BICYCLES Draw a tree diagram to determine the number ofdifferent bicycles described in the real-life example above.

There are 12 different Dynamo bicycles.


Mountain

10-Speed

26 in.28 in.26 in.28 in.26 in.28 in.26 in.28 in.26 in.28 in.26 in.28 in.

Mountain, Red, 26 in.Mountain, Red, 28 in.Mountain, Black, 26 in.Mountain, Black, 28 in.Mountain, Green, 26 in.Mountain, Green, 28 in.10-Speed, Red, 26 in.10-Speed, Red, 28 in.10-Speed, Black, 26 in.10-Speed, Black, 28 in.10-Speed, Green, 26 in.10-Speed, Green, 28 in.

Style Size OutcomeColor

Red

Black

Green

Red

Black

Green

Each color ispaired with eachstyle of bicycle.

Each size is pairedwith each style andcolor of bicycle.

List of all theoutcomes whenchoosing a bicycle.

List each styleof bicycle.

Lesson 8-2 Counting Outcomes 381Bettmann/CORBIS

You can also find the total number of outcomes by multiplying. Thisprinciple is known as the .Fundamental Counting Principle

COMMUNICATIONS OnOctober 27, 1920, KDKA inPittsburgh, Pennsylvania,became the first licensedradio station.Source: Time Almanac

Fundamental Counting Principle

Words If event M can occur in m ways and is followed by event Nthat can occur in n ways, then the event M followed by theevent N can occur in m � n ways.

Example If a number cube is rolled and a coin is tossed, there are 6 � 2 or 12 possible outcomes.

Use the Fundamental Counting Principle

COMMUNICATIONS In the United States, radio and televisionstations use call letters that start with K or W. How many differentcall letters with 4 letters are possible?

Use the Fundamental Counting Principle.

� � � �

2 � 26 � 26 � 26 � 35,152

There 35,152 possible call letters.

Use the Fundamental Counting Principle to findthe number of possible outcomes.

a. A hair dryer has 3 settings for heat and 2 settings for fan speed.

b. A restaurant offers a choice of 3 types of pasta with 5 types ofsauce. Each pasta entrée comes with or without a meatball.

totalnumber ofpossible

call letters

number ofpossible

letters for the fourth

letter

number ofpossible

letters for the third

letter

number ofpossible

letters for the second

letter

number ofpossible

letters for the first

letter

Find Probability

GAMES What is the probability of winning a lottery game wherethe winning number is made up of three digits from 0 to 9 chosenat random?

First, find the number of possible outcomes. Use the FundamentalCounting Principle.

� � �

10 � 10 � 10 � 1,000

There are 1,000 possible outcomes. There is 1 winning number. So, the probability of winning with one ticket is �1,0

100�. This

can also be written as a decimal, 0.001, or a percent, 0.1%.

total number of outcomes

choices forthe third digit

choices for thesecond digit

choices forthe first digit

� � � � �

� � ��

You can also use the Fundamental Counting Principle when there aremore than two events.




1. Describe a possible advantage for using a tree diagram rather than theFundamental Counting Principle.

2. OPEN ENDED Give an example of a situation that has 15 outcomes.

3. NUMBER SENSE Whitney has a choice of a floral, plaid, or striped blouse to wear with a choice of a tan, black, navy, or white skirt. How many more outfits can she make if she buys a print blouse?

The spinner at the right is spun two times.

4. Draw a tree diagram to determine the number of outcomes.

5. What is the probability that both spins will land on red?

6. What is the probability that the two spins will land on different colors?

7. FOOD A pizza parlor has regular, deep-dish, and thin crust, 2 differentcheeses, and 4 toppings. How many different one-cheese and one-topping pizzas can be ordered?

8. GOVERNMENT The first three digits of a social security number are ageographic code. The next two digits are determined by the year and the state where the number is issued. The final four digits are randomnumbers. How many possible ways can the last four digits be assigned?

green yellow

red

Exercises 1 & 2

Draw a tree diagram to determine the number of outcomes.

9. A penny, a nickel, and a dime are tossed.

10. A number cube is rolled and a penny is tossed.

11. A sweatshirt comes in small, medium, large, and extra large. It comes in white or red.

12. The Sweet Treats Shoppe has three flavors of ice cream: chocolate, vanilla, and strawberry; and two types of cones, regular and sugar.

Use the Fundamental Counting Principle to find the number of possible outcomes.

13. The day of the week is picked at random and a number cube is rolled.

14. A number cube is rolled 3 times.

15. There are 5 true-false questions on a history quiz.

16. There are 4 choices for each of 5 multiple-choice questions on a science quiz.


For Exercises

9–12, 17

13–16, 22–23

18–21

See Examples

1

2

3

Lesson 8-2 Counting Outcomes 383PhotoDisc

For Exercises 17–20, each of the spinners at the right is spun once.


18. What is the probability that both spinners land on the same color?

19. What is the probability that at least one spinner lands on blue?

20. What is the probability that at least one spinner lands on yellow?

21. PROBABILITY What is the probability of winning a lottery game where the winning number is made up of five digits from 0 to 9 chosen at random?

22. SCHOOL Doli can take 4 different classes first period, 3 different classessecond period, and 5 different classes third period. How many differentschedules can she have?

23. STATES In 2003, Ohio celebrated its bicentennial. The state issued bicentennial license plates with 2 letters, followed by 2 numbers and then 2 more letters. How many bicentennial license plates could the state issue?

24. CRITICAL THINKING If x coins are tossed, write an algebraic expressionfor the number of possible outcomes.

yellow

blue

red

green

blue

red

white

25. MULTIPLE CHOICE At the café, Dion can order one of the flavors of tealisted at the right. He can order the tea in a small, medium, or large cup.How many different ways can Dion order tea?

5 8 12 15

26. GRID IN Felisa has a red and a white sweatshirt. Courtney has a black, a green, a red, and a white sweatshirt. Each girl picks a sweatshirt at random to wear to the picnic. What is the probability the girls will wear the same color sweatshirt?

Each letter of the word associative is written on 11 identical slips of paper. A piece of paper is chosen at random. Find each probability. (Lesson 8-1)

27. P(s) 28. P(vowel) 29. P(not r) 30. P(d)

31. MEASUREMENT How many significant digits are in the measurement 14.4 centimeters? (Lesson 7-9)

DCBA

PREREQUISITE SKILL Evaluate n(n � 1)(n � 2)(n � 3) for each value of n.(Lesson 1-2)

32. n � 5 33. n � 10 34. n � 12 35. n � 8

Flavors of Tea

mintorangepeachraspberrystrawberry

msmath3.net/self_check_quiz/sol

SOL Practice


Find the number ofpermutations of objects.

permutationfactorial

Aaron Haupt

Permutations

When deciding who goes first and who goes second, order isimportant. An arrangement or listing in which order is important is called a .permutation

Find a Permutation

FOOD An ice cream shop has 31 flavors. Carlos wants to buy athree-scoop cone with three different flavors. How many conescan he buy if order is important?

� � �

31 � 30 � 29 � 26,970

There are 26,970 different cones Carlos can order.

total number ofpossible

cones

number ofpossible

flavors for thethird scoop

number ofpossible

flavors for thesecond scoop

number ofpossible

flavors for thefirst scoop


��

The symbol P(31, 3) represents the number of permutations of 31 things taken 3 at a time.

Start with 31.

P(31, 3) � 31 � 30 � 29

Use three factors.

�

Work with a partner.

Suppose you are playing a game with 4 different game pieces. Show all of the ways the game pieces can be chosen first and second. Record each arrangement.

1. How many different arrangements did you make?

2. How many different game pieces could you pick for the first place?

3. Once you picked the first-place game piece, how many gamepieces could you pick for the second place?

4. Use the Fundamental Counting Principle to determine thenumber of arrangements for first and second places.

5. How do the numbers in Exercises 1 and 4 compare?

• four differentgame pieces

P(a, b) the number ofpermutations ofa things taken bat a time

! factorial

Lesson 8-3 Permutations 385

Use Permutation Notation

Find each value.

P(8, 3)

P(8, 3) � 8 � 7 � 6 or 336 8 things taken 3 at a time.

P(6, 6)

P(6, 6) � 6 � 5 � 4 � 3 � 2 � 1 or 720 6 things taken 6 at a time.

Find each value.

a. P(12, 2) b. P(4, 4) c. P(10, 5)

In Example 3, P(6, 6) � 6 � 5 � 4 � 3 � 2 � 1. The mathematical notation 6!also means 6 � 5 � 4 � 3 � 2 � 1. The symbol 6! is read six . n! means the product of all counting numbers beginning with n andcounting backward to 1. We define 0! as 1.

factorial

Find Probability

MULTIPLE-CHOICE TEST ITEM Consider all of the four-digit numbersthat can be formed using the digits 1, 2, 3, and 4 where no digit isused twice. Find the probability that one of these numbers picked atrandom is between 1,000 and 2,000.

33�13

�% 25% 20% 10%

Read the Test Item

You are considering all of the permutations of 4 digits taken 4 at atime. You wish to find the probability that one of these numberspicked at random is greater than 1,000, but less than 2,000.

Solve the Test Item

Find the number of possible four-digit numbers. P(4, 4) � 4!

In order for a number to be between 1,000 and 2,000, the thousandsdigit must be 1.

� �

1 � P(3, 3) � P(3, 3) or 3!

P(between 1,000 and 2,000)

�

� �34

!!� Substitute.

� Definition of factorial

� �14

� or 25% The probability is 25%, which is B.

1 13 � 2 � 1

��4 � 3 � 2 � 1

1 1

number of permutations between 1,000 and 2,000��

total number of permutations

number ofpermutations between

1,000 and 2,000

number of waysto pick the last

three digits

number ofways to pickthe first digit

DCBA

PermutationsP(8, 3) can also bewritten 8P3.

Be PreparedBefore the day of the test,ask if you can use aidssuch as a calculator. Thencome prepared on the dayof the test. In Example 4,you could find the answerquickly by using thefollowing keystrokes.

3

4

0.25ENTERENTER

PRB�

ENTERPRB

��

msmath3.net/extra_examples

SOLPractice

/sol


386 Chapter 8 ProbabilityCORBIS

1. Tell the difference between 9! and P(9, 5).

2. OPEN ENDED Write a problem that can be solved by finding the value of P(7, 3).

3. FIND THE ERROR Daniel and Bailey are evaluating P(7, 3). Who iscorrect? Explain.

Bai leyP(7 , 3) = 7 � 6 � 5

= 210

DanielP(7, 3) = 7 � 6 � 5 � 4 � 3

= 2,520

Find each value.

4. P(5, 3) 5. P(7, 4) 6. 3! 7. 8!

8. In a race with 7 runners, how many ways can the runners end up in first,second, and third place?

9. How many ways can you arrange the letters in the word equals?

10. SPORTS There are 9 players on a baseball team. How many ways can thecoach pick the first 4 batters?

Exercises 2 & 3

Find each value.

11. P(6, 3) 12. P(9, 2) 13. P(5, 5) 14. P(7, 7)

15. P(14, 5) 16. P(12, 4) 17. P(25, 4) 18. P(100, 3)

19. 2! 20. 5! 21. 11! 22. 12!

23. How many ways can the 4 runners on a relay team be arranged?

24. FLAGS The flag of Mexico is shown at the right. How many ways could the Mexican government have chosen to arrange the three bar colors (green, white, and red) on the flag?

25. A security system has a pad with 9 digits. How many four-number“passwords” are available if no digit is repeated?

26. Of the 10 games at the theater’s arcade, Tyrone plans to play 3 different games. In how many orders can he play the 3 games?

27. MULTI STEP Each arrangement of the letters in the word quilt is written on a piece of paper. One paper is drawn at random. What is the probability that the word begins with q?

28. MULTI STEP Each arrangement of the letters in the word math is written on a piece of paper. One paper is drawn at random. What is the probability that the word ends with th?


For Exercises

11–22

23–26, 29–32

27–28

See Examples

2, 3

1

4

Data Update How many floats, bands, and equestrian groups were in the lastTournament of Roses Parade? Visit msmath3.net/data_update to learn more.

Lesson 8-3 Permutations 387Ronald Martinez/Getty Images

35. MULTIPLE CHOICE How many seven-digit phone numbers are availableif a digit can only be used once and the first number cannot be 0 or 1?

5,040 483,840 544,320 10,000,000

36. MULTIPLE CHOICE The school talent show is featuring 13 acts. In howmany ways can the talent show coordinator order the first 5 acts?

6,227,020,800 371,293 154,440 1,287

37. SPORTS The Silvercreek Ski Resort has 4 ski lifts up the mountain and 11 trails down the mountain. How many different ways can a skier take a ski lift up the mountain and then ski down? (Lesson 8-2)

A number cube is rolled. Find each probability. (Lesson 8-1)

38. P(5 or 6) 39. P(odd) 40. P(less than 10) 41. P(1 or even)

42. Write an equation you could use to find the length of the missing side of the triangle at the right. Then find the missing length. (Lesson 3-4)

13 ft5 ft

a

IHGF

DCBA

PREREQUISITE SKILL Evaluate each expression. (Lesson 1-2)

43. �36 �

�52

��

41

� 44. �140

��39

��28

��17

� 45. �20

2��

11

9� 46. �

65

��

54

��

43

��

32

��

21

�

29. SOCCER The teams of the Eastern Conference of Major League Soccer are listed at the right. If there are no ties for placement in the conference, how many ways can the teams finish the season from first to last place?

ENTERTAINMENT For Exercises 30–32, use the following information.In the 2002 Tournament of Roses Parade, there were 54 floats, 23 bands, and 26 equestrian groups.

30. In how many ways could the first 3 bands be chosen?

31. In how many ways could the first 3 equestrian groups be chosen?

32. Two of the 54 floats were entered by the football teams competing in the Rose Bowl. If they cannot be first or second, how many ways can the first 3 floats be chosen?

33. CRITICAL THINKING If 9! � 362,880, use mental math to find 10!Explain.

34. CRITICAL THINKING Compare P(n, n) and P(n, n � 1), where n is anywhole number greater than one. Explain.

Eastern Conference

Chicago FireColumbus CrewD.C. UnitedMetroStarsNew England Revolution


SOL Practice

/sol

http://msmath3.net/data_update


Find the number ofcombinations of objects.

combination

PhotoDisc

In the Mini Lab, it did not matter whether you shook hands with yourfriend, or your friend shook hands with you. Order is not important.An arrangement or listing where order is not important is called a

. Let’s look at a simpler form of the handshake problem.combination

Find a Combination

GEOMETRY Four points are located on acircle. How many line segments can be drawn with these points as endpoints?

Method 1

First list all of the possible permutations of A, B, C, and D taken two at a time. Then cross out the segments that are the same as one another.

A�B� A�C� A�D� B�A� B�C� B�D�C�A� C�B� C�D� D�A� D�B� D�C�There are only 6 different segments.

Method 2

Find the number of permutations of 4 points taken 2 at a time.

P(4, 2) � 4 � 3 or 12

Since order is not important, divide the number of permutations bythe number of ways 2 things can be arranged.

�122!� � �

21�21

� or 6

There are 6 segments that can be drawn.

a. If there are 8 people in a room, how many handshakes will occur ifeach person shakes hands with every other person?

A B

C

D


C(a, b) the number ofcombinations of a thingstaken b at a time

Combinations

A��B is the same as B��A,so cross off one of them.

Work in a group of 6.

Each member of the group should shake hands with every othermember of the group. Make a list of each handshake.

1. How many different handshakes did you record?

2. Find P(6, 2).

3. Is the number of handshakes equal to P(6, 2)? Explain.

Lesson 8-4 Combinations 389(l)Andy Sacks/Getty Images, (r)Alvis Upitis/Getty Images

The symbol C(4, 2) represents the number of combinations of 4 thingstaken 2 at a time.

C(4, 2) � �P(4

2,!2)

�

CombinationsC(7, 4) can also bewritten as 7C4.

the number of combinationsof 4 things taken 2 at a time

the number of permutationsof 4 things taken 2 at a time

the number of ways 2things can be arranged

Use Combination Notation

Find C(7, 4).

C(7, 4) � �P(7

4,!4)

� Definition of C(7, 4)

� or 35 P(7, 4) � 7 � 6 � 5 � 4 and 4! � 4 � 3 � 2 � 1

12 1

7 � 6 � 5 � 4��4 � 3 � 2 � 11 1 1

MUSIC The harp is one of the oldest stringedinstruments. It is about 70 inches tall and has 47 strings.Source: World Book

Combinations and Permutations

MUSIC The makeup of a symphony is shown in the table at the right.

A group of 3 musicians from the strings section will talk to students at Madison Middle School. Does this represent a combination or a permutation? How many possible groups could talk to the students?

This is a combination problem since the order is not important.

C(45, 3) � �P(4

35!, 3)� 45 musicians taken 3 at a time

� or 14,190 P(45, 3) � 45 � 44 � 43 and 3! � 3 � 2 � 1

There are 14,190 different groups that could talk to the students.

One member from the strings section will talk to students atBrown Middle School, another to students at Oak Avenue MiddleSchool, and another to students at Jefferson Junior High. Doesthis represent a combination or a permutation? How manypossible ways can the strings members talk to the students?

Since it makes a difference which member goes to which school,order is important. This is a permutation.

P(45, 3) � 45 � 44 � 43 or 85,140 Definition of P(45, 3)

There are 85,140 ways for the members to talk to the students.

15 2245 � 44 � 43��

3 � 2 � 11 1

Makeup of the Symphony

Instrument Number

Strings 45

Woodwinds 8

Brass 8

Percussion 3

Harps 2



390 Chapter 8 ProbabilityKS Studios

1. OPEN ENDED Give an example of a combination and an example of apermutation.

2. Which One Doesn’t Belong? Identify the situation that is not the sameas the other three. Explain your reasoning.

choosing 3 desserts to serve atthe party

choosing 3 people to chair

3 differentcommittees

choosing 3 members

for thedecoratingcommittee

choosing 3 toppings

for the pizzasto be servedat the party

Find each value.

3. C(6, 2) 4. C(10, 5) 5. C(7, 6) 6. C(8, 4)

Determine whether each situation is a permutation or a combination.

7. writing a four-digit number using no digit more than once

8. choosing 3 shirts to pack for vacation

9. How many different starting squads of 6 players can be picked from10 volleyball players?

10. How many different combinations of 2 colors can be chosen as schoolcolors from a possible list of 8 colors?

Exercises 1 & 2

Find each value.

11. C(9, 2) 12. C(6, 3) 13. C(9, 8) 14. C(8, 7)

15. C(9, 5) 16. C(10, 4) 17. C(18, 4) 18. C(20, 3)

Determine whether each situation is a permutation or a combination.

19. choosing a committee of 5 from the members of a class

20. choosing 2 co-captains of the basketball team

21. choosing the placement of 9 model cars in a line

22. choosing 3 desserts from a dessert tray

23. choosing a chairperson and an assistant chairperson for a committee

24. choosing 4 paintings to display at different locations

25. How many three-topping pizzas can be ordered from a list of toppings at the right?

26. GEOMETRY Eight points are located on a circle. How many line segments can be drawn with these points as endpoints?


For Exercises

11–18

19–24, 27–32

25–26

See Examples

2

3, 4

1

Pizza Toppings

anchovies sausage onionsbacon green peppers black olivesham hot peppers green olivespepperoni mushrooms pineapple

Lesson 8-4 Combinations 391Aaron Haupt

27. There are 20 runners in a race. In how many ways can the runners takefirst, second, and third place?

28. How many ways can 7 people be arranged in a row for a photograph?

29. How many five-card hands can be dealt from a standard deck of 52 cards?

30. GAMES In the game of cribbage, a player gets 2 pointsfor each combination of cards that totals 15. How many points for totals of 15 are in the hand at the right?

ENTERTAINMENT For Exercises 31 and 32, use the following information.An amusement park has 15 roller coasters. Suppose you only have time to ride 8 of the coasters.

31. How many ways are there to ride 8 coasters if order is important?

32. How many ways are there to ride 8 coasters if order is not important?

33. CRITICAL THINKING Is the value of P(x, y) sometimes, always, or nevergreater than the value of C(x, y)? Explain. Assume x and y are positiveintegers and x � y.

34. MULTIPLE CHOICE Which situation is represented by C(8, 3)?

the number of arrangements of 8 people in a line

the number of ways to pick 3 out of 8 vegetables to add to a salad

the number of ways to pick 3 out of 8 students to be the first,second, and third contestant in a spelling bee

the number of ways 8 people can sit in a row of 3 chairs

35. SHORT RESPONSE The enrollment for Centerville Middle School is given at the right. How many different four-person committees could be formed from the students in the 8th grade?

Find each value. (Lesson 8-3)

36. P(7, 2) 37. P(15, 4) 38. 10! 39. 7!

40. SCHOOL At the school cafeteria, students can choose from 4 entrees and3 beverages. How many different lunches of one entree and one beveragecan be purchased at the cafeteria? (Lesson 8-2)

D

C

B

A

PREREQUISITE SKILL Multiply. Write in simplest form. (Lesson 2-3)

41. �45

� � �38

� 42. �130� � �

56

� 43. �172� � �

134� 44. �

23

� � �190�

Class Boys Girls

6th grade 42 477th grade 55 498th grade 49 53

Centerville Middle School


SOL Practice

/sol


Identify patterns in Pascal’sTriangle.

• paper• pencil


Combinations and Pascal’s TriangleFor many years, mathematicians have been interested in a patterncalled Pascal’s Triangle.

Row Sum

0

1

2

3

4

1 � 20

2 � 21

4 � 22

8 � 23

16 � 24

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

A Follow-Up of Lesson 8-4


Find all possible outcomes if you toss a penny and a dime.

Copy and complete the tree diagram shown below.

In the tree diagram above, how many outcomes haveexactly no heads? one head? two heads?

Use a tree diagram to determine the outcomes of tossing a penny, a nickel, and a dime. How many outcomes haveexactly no head, one head, two heads, three heads?

Heads, Heads ?

Heads Tails

Heads

? ?

Heads ?

TailsPenny

Dime

Outcomes

1. Describe the pattern in the numbers in Pascal’s Triangle. Use thepattern to write the numbers in Rows 5, 6, and 7.

2. Explain how your tree diagrams are related to Pascal’s Triangle.

3. Suppose you toss a penny, nickel, dime, and quarter. Make aconjecture about how many outcomes have exactly no head, one head, two heads, and so on. Test your conjecture.

Lesson 8-4b Hands-On Lab: Combinations and Pascal’s Triangle 393

Pascal’s Triangle can also be used to find probabilities of events forwhich there are only two possible outcomes, such as heads-tails, boy-girl, and true-false.

4. Suppose you guess on a five-item true-false test. What is theprobability of getting all of the right answers?

5. There are ten true-false questions on a quiz. What is the probabilityof guessing at least six correct answers and passing the quiz?

6. If you toss eight coins, you would expect there to be four heads andfour tails. What is the probability this will happen?

For Exercises 7–9, use the following information.The Band Boosters are selling pizzas. You can choose to add onions,pepperoni, mushrooms, and/or green pepper to the basic cheese pizza.

7. Find each number of combinations of toppings.

a. C(4, 0) b. C(4, 1) c. C(4, 2) d. C(4, 3) e. C(4, 4)

8. How many different combinations are there in all?

9. Suppose the Boosters decide to offer hot peppers as an additionalchoice. How many combinations of pizzas are available?


In a five-item true-false quiz, what is the probability of gettingexactly three right answers by guessing?

Since there are five items, look at Row 5.

There are 10 ways to get exactly three right answers.

Find the total possible outcomes.

1 + 5 + 10 + 10 + 5 + 1 = 32

Find the probability.

� �3120� or �

156�

So, the probability of guessing exactly three right answers is �156�.

number of ways to guess 3 right answers��

number of outcomes

0 1 2 3 4 5

1 5 10 10 5 1

Number Right

Row 5


XX

XX

CHAPTER

1. Draw a spinner where P(green) is �14

�. (Lesson 8-1)

2. Write a problem that is solved by finding the value of P(8, 3). (Lesson 8-3)

There are 6 purple, 5 blue, 3 yellow, 2 green, and 4 brown marbles in a bag. One marble is selected at random. Write each probability as a fraction, a decimal, and a percent. (Lesson 8-1)

3. P(purple) 4. P(blue) 5. P(not brown)

6. P(purple or blue) 7. P(not green) 8. P(blue or green)

For Exercises 9–11, a penny is tossed, and a number cube is rolled. (Lesson 8-2)


10. What is the probability that the penny shows heads and the number cube shows a six?

11. What is the probability that the penny shows heads and the number cube shows an even number?

Find each value. (Lessons 8-3 and 8-4)

12. P(5, 3) 13. P(6, 2) 14. P(5, 5)

15. C(5, 3) 16. C(6, 2) 17. C(5, 5)

18. SCHOOL How many ways can 2 student council members be elected from 7 candidates? (Lesson 8-4)

19. MULTIPLE CHOICE A pizza shopadvertises that it has 3 differentcrusts, 3 different meat toppings,and 5 different vegetables. IfCarlotta wants a pizza with onemeat and one vegetable, howmany different pizzas can sheorder? (Lesson 8-2)

11 15

45 90

20. GRID IN The spinner below is used for a game. Find theprobability that the spinner will not land on yellow. (Lesson 8-1)

Y

B

RGW

RWY

G

B

DC

BA

SOL Practice

The Game Zone: Probability 395John Evans

Players: threeMaterials: 15 index cards, scissors, markers,

3 paper bags

• Cut each index card in half, making 30 cards.

• Give each player 10 cards.

• Each player writes one number from 0 to 9 on each card.

• Each player takes a different bag and places his or her cards in the bag.

• Each player writes down three numbers each between 0 and 9.Repeat numbers are allowed.

• Each player draws a card from his or her paper bag without looking.These are the winning numbers.

• Each player scores 2 points if one number matches, 16 points if twonumbers match, and 32 points if all three numbers match. Order isnot important.

• Replace the cards in the paper bags. Repeat the process.

• Who Wins? The first person to get a total of 100 points is the winner.

0

1

Winning Numbers

Find the probability ofindependent anddependent events.

compound eventindependent eventsdependent events

Probability of Compound Events

GAMES A game uses a number cube and the spinner shown at the right.

1. A player rolls the number cube. What is P(odd number)?

2. The player spins the spinner. What is P(red)?

3. What is the product of the probabilities in Exercises 1 and 2?

4. Draw a tree diagram to determine the probability that the playerwill get an odd number and red.

5. Compare your answers for Exercises 3 and 4.


The combined action of rolling a number cube and spinning a spinneris a compound event. In general, a consists of two ormore simple events.

The outcome of the spinner does not depend on the outcome of thenumber cube. These events are independent. For ,the outcome of one event does not affect the other event.

independent events

compound event

Probability of Independent Events

The two spinners are spun. What is the probability that both spinners will show an even number?

P(first spinner is even) � �37

�

P(second spinner is even) � �12

�

P(both spinners are even) � �37

� � �12

� or �134�

5 3

6 2

4

17

6 3

7 2

45

18


2 1 green

red

blue

Probability of Two Independent Events

Words The probability of two independent events can be found bymultiplying the probability of the first event by the probabilityof the second event.

Symbols P(A and B) � P(A) � P(B)

Lesson 8-5 Probability of Compound Events 397Sylvain Grandadam/Getty Images

If the outcome of one event affects the outcome of another event, thecompound events are called .dependent events

Use Probability to Solve a Problem

POPULATION Use the information in the table. In the United States, what is the probability that a person picked at random will be under the age of 18 and live in an urban area?

P(younger than 18) � �14

�

P(urban area) � �45

�

P(younger than 18 and urban area)

� �14

� � �45

� or �15

�

The probability that the two events will occur is �

15

�.

Probability of Dependent Events

There are 2 white, 8 red, and 5 blue marbles in a bag. Once amarble is selected, it is not replaced. Find the probability that twored marbles are chosen.

Since the first marble is not replaced, the first event affects thesecond event. These are dependent events.

P(first marble is red) � �185�

P(second marble is red) � �174�

P(two red marbles) � � or

Find each probability.

a. P(two blue marbles) b. P(a white marble and then ablue marble)

4�15

17

�1471

48

�15

number of red marbles after one red marble is removedtotal number of marbles afterone red marble is removed

number of red marblestotal number of marbles

Demographic Fraction of theGroup Population

Under age 18 �41

�

18 to 64 years�58

�old

65 years or�81

�older

Urban �45

�

Rural �51

�


Mental MathYou may wish to

simplify �174� to �

12

�

before multiplyingthe probabilities.

Probability of Two Dependent Events

Words If two events, A and B, are dependent, then the probability ofboth events occurring is the product of the probability of Aand the probability of B after A occurs.

Symbols P(A and B) � P(A) � P(B following A)

POPULATION Thepopulation of the UnitedStates is getting older. In 2050, the fraction of the population 65 years and older is expected to

be about �15

�.Source: U.S. Census Bureau

United States

←←

←← �

�




1. Compare and contrast independent events and dependent events.

2. OPEN ENDED Give an example of dependent events.

3. FIND THE ERROR The spinner at the right is spun twice. Evita and Tia are finding the probability that both spins will result in an odd number. Who is correct? Explain.

Tia

�53

� � �42

� = �260� or �

130�

Ev i ta

�35� � �

35� = �

295�

4

35

21

A penny is tossed, and a number cube is rolled. Find each probability.

4. P(tails and 3) 5. P(heads and odd)

Two cards are drawn from a deck of ten cards numbered 1 to 10. Once a card is selected, it is not returned. Find each probability.

6. P(two even cards) 7. P(a 6 and then an odd number)

8. MARKETING A discount supermarket has found that 60% of theircustomers spend more than $75 each visit. What is the probability thatthe next two customers will spend more than $75?

Exercises 1 & 3

A number cube is rolled, and the spinner at the right is spun. Find each probability.

9. P(1 and A) 10. P(3 and B)

11. P(even and C) 12. P(odd and B)

13. P(greater than 2 and A)

14. P(less than 3 and B)

15. What is the probability of tossing a coin 3 times and getting heads each time?

16. What is the probability of rolling a number cube 3 times and getting numbers greater than 4 each time?

There are 3 yellow, 5 red, 4 blue, and 8 green candies in a bag. Once a candy is selected, it is not replaced. Find each probability.

17. P(two red candies) 18. P(two blue candies)

19. P(a yellow candy and then a blue candy)

20. P(a green candy and then a red candy)

21. P(two candies that are not green)

22. P(two candies that are neither blue nor green)

B

BA

BC Extra Practice See pages 636, 655.

For Exercises

9–16

17–22

23–24

See Examples

1

3

2

Lesson 8-5 Probability of Compound Events 399

KITCHENS For Exercises 23 and 24, use the table at the right. Round to the nearest tenth of a percent.

23. What is the probability that a householdpicked at random will have both an electricfrying pan and a toaster?

24. What is the probability that a householdpicked at random will use both a mixer and a drip coffee maker?

EXTENDING THE LESSON If two events cannothappen at the same time, they are said to bemutually exclusive. For example, suppose yourandomly select a card from a standard deck of 52 cards. Getting a 5 or getting a 6 are mutually exclusive events. To find the probability of two mutually exclusive events, add the probabilities.

P(5 or 6) � P(5) � P(6)

� �113� � �

113� or �

123�

Consider a standard deck of 52 cards. Find each probability.

25. P(face card or an ace) 26. P(club or a red card)

27. CRITICAL THINKING There are 9 marbles in a bag having 3 colors of marbles. The probability of picking 2 red marbles at random and without replacement is �

16

�. How many red marbles are in the bag?

Mixer

Electric frying pan

Drip coffee maker

Toaster

Blender

99%

81%

81%

85%

79%

17%

19%

19%

14%

10%

% use% own

Now, where’s that electric pan?Although we own many electric kitchenappliances we rarely use them.

USA TODAY Snapshots®

By Cindy Hall, USA TODAY and Karl Gelles for USA TODAY

Source: NFO Research for Kraft Kitchens

28. MULTIPLE CHOICE Jeremy tossed a coin and rolled a number cube.What is the probability that he will get tails and roll a multiple of 3?

�12

� �13

� �14

� �16

�

29. GRID IN Suppose you pick 3 cards from a standard deck of 52 cards without replacement. What is the probability all of the cards will be red?

Find each value. (Lesson 8-4)

30. C(8, 5) 31. C(7, 2) 32. C(6, 5) 33. C(9, 3)

34. SPORTS There are 10 players on a softball team. How many ways can a coach pick the lineup of the first 3 batters? (Lesson 8-3)

DCBA

PREREQUISITE SKILL Write each fraction in simplest form. (Page 611)

35. �15220

� 36. �39

30� 37. �

47

90� 38. �

28

48�

msmath3.net/self_check_quiz/sol

SOL Practice


Find experimentalprobability.

experimental probability

theoretical probability

Experimental Probability

In the Mini Lab above, you determined a probability by conducting an experiment. Probabilities that are based on frequencies obtained by conducting an experiment are called .Experimental probabilities usually vary when the experiment is repeated.

Probabilities based on known characteristics or facts are called. For example, you can compute the

theoretical probability of picking a certain color marble from a bag.Theoretical probability tells you what should happen in an experiment.

theoretical probabilities

experimental probabilities


Michelle is conducting an experiment to find the probability of getting various sums when twonumber cubes are rolled. The results of her experiment are given at the right.

According to the experimentalprobability, is Michelle likely to get a sum of 12 on the next roll?

Based on the results of the rolls so far, a sum of 12 is not very likely.

How many possible outcomes are there for a pair of numbercubes?

There are 6 � 6 or 36 possible outcomes.

2 3 4 5 6 7 8 9 10 11 12

Results of RollingTwo Number Cubes

Num

ber o

f Rol

ls

Sum

16

12

8

4

0



Draw one marble from the bag, record itscolor, and replace it in the bag. Repeat this 50 times.

1. Compute the ratio for each

color of marble.

2. Is it possible to have a certain color marble in the bag and neverdraw that color?

3. Open the bag and count the marbles. Find the ratio

for each color of marble.

4. Are the ratios in Exercises 1 and 3 the same? Explain why or why not.

number of each color marble��

total number of marbles

number of times color was drawn��

total number of draws

• paper bagcontaining 10 coloredmarbles

proportion: a statementof equality of two ormore ratios, �

ab

� � �dc

�,

b 0, d 0 (Lesson 4-4)

Virginia SOL Standard 8.11 The studentwill analyze problem situations, including games of chance, board games, orgrading scales, and makepredictions, using knowl-edge of probability.

Lesson 8-6 Experimental Probability 401Victoria Pearson/Getty Images

Use Probability to Predict

FARMING Over the last 8 years, the probability that corn seedsplanted by Ms. Diaz produced corn is �

56

�.

Is this probability experimental or theoretical? Explain.

This is an experimental probability since it is based on whathappened in the past.

If Ms. Diaz wants to have 10,000 corn-bearing plants, how manyseeds should she plant?

This problem can be solved using a proportion.

�56

� � �10,

x000�

Solve the proportion.

�56

� � �10,

x000� Write the proportion.

5 � x � 6 � 10,000 Find the cross products.

5x � 60,000 Multiply.

�55x� � �

60,5000� Divide each side by 5.

x � 12,000 Ms. Diaz should plant 12,000 seeds.

Mental MathFor every 5 cornbearing plants, Ms. Diaz must plantan extra seed.Think: 10,000 5 � 2,000Ms. Diaz must plant2,000 extra seeds.She must plant atotal of 10,000 �2,000 or 12,000seeds. The answeris correct.

You can use past performance to predict future events.

5 out of 6 seeds should produce corn.

10,000 out of x seeds should produce corn.

How Does a MarketingManager Use Math?A marketing manager usesinformation from surveysand experimental probabilityto help make decisionsabout changes in productsand advertising.

Online ResearchFor information about a careeras a marketing manager, visit:msmath3.net/careers

Theoretical Probability

What is the theoretical probability of rolling a double six?

The theoretical probability is �16

� � �16

� or �316�.

The experimental probability and the theoretical probability seem to be consistent.


MARKETING Two hundred teenagerswere asked whether they purchasedcertain household items in the past year. The table gives the results of the survey. What is the experimentalprobability that a teenager bought a photo frame in the last year?

There were 200 teenagers surveyed and 95 purchased a photo framein the last year. The experimental probability is �2

9050

� or �14

90�.

a. What is the experimental probability that a teenager bought acandle in the last year?

Item Number WhoPurchased the Item

candles 110

photo frames 95


http://msmath3.net/careers


402 Chapter 8 ProbabilityRobert Thayer

1. Explain why you would not expect the theoretical probability and theexperimental probability of an event to always be the same.

2. OPEN ENDED Two hundred fifty people are surveyed about theirfavorite color. Make a possible table of results if the experimentalprobability that the favorite color is blue is �

25

�.

For Exercises 3–7, use the table that shows the results of tossing a coin.

3. Based on your results, what is the probability of getting heads?

4. Based on the results, how many heads would you expect to occur in 400 tries?

5. What is the theoretical probability of getting heads?

6. Based on the theoretical probability, how many heads would you expectto occur in 400 tries?

7. Compare the theoretical probability to your experimental probability.

For Exercises 8 and 9, use the table at the right showing the results of a survey of cars that passed the school.

8. What is the probability that the next car will be white?

9. Out of the next 180 cars, how many would you expect to be white?

Exercises 1 & 2

SCHOOL For Exercises 10 and 11, use the following information.In keyboarding class, Cleveland made 4 typing errors in 60 words.

10. What is the probability that his next word will have an error?

11. In a 1,000-word essay, how many errors would you expect Cleveland to make?

12. SCHOOL In the last 40 school days, Esteban’s bus has been late 8 times. What is the experimental probability the bus will be late tomorrow?

FOOD For Exercises 13 and 14, use the survey results at the right.

13. What is the probability that a person’s favorite snack while watching television is corn chips?

14. Out of 450 people, how many would you expect to have corn chips as their favorite snack with television?

15. SPORTS In practice, Crystal made 80 out of 100 free throws. What is the experimental probability that she will make a free throw?


For Exercises

10, 12–13,15–16, 18

11, 14, 17, 19

20–21

See Examples

1, 4, 5

6

2, 3

heads 26

tails 24

Result Numberof Times

white 35

red 23

green 12

other 20

Cars Passing the School

Color Number of Cars

Snack Number

potato chips 55

corn chips 40

popcorn 35

pretzels 15

other 5

Favorite Snack WhileWatching Television

Lesson 8-6 Experimental Probability 403

SPORTS For Exercises 16 and 17, use the results of a survey of 90 teens shown at the right.

16. What is the probability that a teen plays soccer?

17. Out of 300 teens, how many would you expect to play soccer?

For Exercises 18–22, toss two coins 50 times and record the results.

18. What is the experimental probability of tossing two heads?

19. Based on your results, how many times would you expect to get two heads in 800 tries?

20. What is the theoretical probability of tossing two heads?

21. Based on the theoretical probability, how many times would you expectto get two heads in 800 tries?

22. Compare the theoretical and experimental probability.

23. CRITICAL THINKING An inspector found that 15 out of 250 cars had aloose front door and that 10 out of 500 cars had headlight problems.What is the probability that a car has both problems?


24. MULTIPLE CHOICE Kylie and Tonya are playing agame where the difference of two rolled number cubes determines the outcome of each play. The graphshows the results of rolls of the number cubes so far inthe game. Kylie needs a difference of 2 on her next rollto win the game. Based on past results, what is theprobability that Kylie will win on her next roll?

�270� �

5101� �

210� �

215�

25. SHORT RESPONSE A local video store has advertised that one out ofevery four customers will receive a free box of popcorn with their videorental. So far, 15 out of 75 customers have won popcorn. Compare theexperimental and theoretical probability of getting popcorn.

There are 3 red marbles, 4 green marbles, and 5 blue marbles in a bag.Once a marble is selected, it is not replaced. Find the probability of eachoutcome. (Lesson 8-5)

26. 2 green marbles 27. a blue marble and then a red marble

28. FOOD Pepperoni, mushrooms, onions, and green peppers can be added to a basic cheese pizza. How many 2-item pizzas can be prepared?(Lesson 8-4)

DCBA

PREREQUISITE SKILL Solve each problem. (Lessons 5-3 and 5-6)

29. Find 35% of 90. 30. Find 42% of 340. 31. What is 18% of 90?

Sport Number ofParticipants

basketball 42

volleyball 26

soccer 24

football 16

Sports Participation by Teens

Difference of RollingTwo Number Cubes

Num

ber o

f Rol

ls

Difference

4035302520151050

0 1 2 4 53

21

35

22

5 4

13

/sol

SOL Practice



A Follow-Up of Lesson 8-6

SimulationsA simulation is an experiment that is designed to act out a givensituation. You can use items such as a number cube, a coin, a spinner,or a random number generator on a graphing calculator. From thesimulation, you can calculate experimental probabilities.


Simulate rolling a number cube 50 times.

Use the random number generator on a TI-83 Plus graphingcalculator. Enter 1 as the lower bound and 6 as the upper bound for 50 trials.

Keystrokes: 5 1 6

50

A set of 50 numbers ranging from 1 to 6 appears. Use the right arrow key to see the next number in the set. Record all 50 numbers on a separate sheet of paper.

a. Use the simulation to determine the experimental probability ofeach number showing on the number cube.

b. Compare the experimental probabilities found in Step 2 to thetheoretical probabilities.

ENTER) ,

,

SimulationsRepeating asimulation mayresult in differentprobabilities sincethe numbersgenerated aredifferent each time.


A company is placing one of 8 different cards of action heroes inits boxes of cereal. If each card is equally likely to appear, what isthe experimental probability that a person who buys 12 boxes ofcereal will get all 8 cards?

Let the numbers 1 through 8 represent the cards. Use the random number generator on a graphing calculator. Enter 1 as the lower bound and 8 as the upper bound for 12 trials.

Keystrokes: 5

1 8 12

Record whether all of the numbers are represented.

ENTER) ,,

msmath3.net/other_calculator_keystrokes

Use a graphing calculator tosimulate probabilityexperiments.

• graphing calculator• paper• pencil

http://msmath3.net/other_calculator_keystrokes

Lesson 8-6b Graphing Calculator Investigation: Simulations 405

c. Repeat the simulation thirty times.

d. Use the simulation to find the experimental probability that aperson who buys 12 boxes of cereal will get all 8 cards.

EXERCISES1. A hypothesis is a statement to be tested that describes what you

expect to happen in a given situation. State your hypothesis as tothe results of repeating the simulation in Activity 1 more than 50times. Then test your hypothesis.

2. Explain how you could use a graphing calculator to simulatetossing a coin 40 times.

3. CLOTHING Rodolfo must wear a tie when he works at the mallon Friday, Saturday, and Sunday. Each day, he picks one of his 6 ties at random. Create a simulation to find the experimentalprobability that he wears a different tie each day of the weekend.

4. TOYS A fast food restaurant is putting 3 different toys in theirchildren’s meals. If the toys are placed in the meals at random,create a simulation to determine the experimental probabilitythat a child will have all 3 toys after buying 5 meals.

5. SCIENCE Suppose a mouse is placed in the maze at the right. If each decision about direction is made at random, create a simulation to determine the probability that the mouse will find its way out before coming to a dead end or going out the In opening.

6. WRITE A PROBLEM Write a real-life problem that could beanswered by using a simulation.

For Exercises 7–9, use the following information.Suppose you play a game where there are three containers, eachwith 10 balls numbered 0 to 9. One number is randomly pickedfrom each container. Pick three numbers each between 0 and 9.Then use the random number generator to simulate the game.Score 2 points if one number matches, 16 points if two numbersmatch, and 32 points if all three numbers match. Notice thatnumbers can appear more than once.

7. Play the game if the order of the numbers does not matter. Totalyour score for 10 simulations.

8. Now play the game if order of the numbers does matter. Totalyour score for 10 simulations.

9. With which game rules did you score more points?

In Out

Predict the actions of alarger group by using asample.

samplepopulationunbiased samplesimple random samplestratified random samplesystematic random

samplebiased sampleconvenience samplevoluntary response

sample

Cooperphoto/CORBIS

Using Sampling to Predict

ENTERTAINMENT The manager of a radio station wants to conduct a survey to determine what type of music people like.

1. Suppose she decides to survey a group ofpeople at a rock concert. Do you think theresults would represent all of the people inthe listening area? Explain.

2. Suppose she decides to survey students at your middle school. Do you think theresults would represent all of the people in the listening area? Explain.

3. Suppose she decides to call every 100th household in the telephone book. Do you think the resultswould represent all of the people in the listening area? Explain.


The manager of the radio station cannot survey everyone in the listening area. A smaller group called a is chosen. A sample isrepresentative of a larger group called a .

For valid results, a sample must be chosen very carefully. Anis selected so that it is representative of the entire

population. Three ways to pick an unbiased sample are listed below.unbiased sample

populationsample


What Type of Music Do You Like?

CountryAlternativeRockOldiesTop 40UrbanAdult Contemporary

Type Definition Example

SampleRandomSystematic

SampleRandomStratified

SampleRandomSimple A simple random sample is

a sample where each itemor person in the populationis as likely to be chosen asany other.

In a stratified randomsample, the population isdivided into similar, non-overlapping groups. A simplerandom sample is thenselected from each group.

In a systematic randomsample, the items or peopleare selected according to aspecific time or item interval.

The name of eachstudent attending aschool is written on apiece of paper. Thenames are placed in abowl, and names arepicked without looking.

Students are picked atrandom from each gradelevel at a school.

From an alphabetical listof all students attendinga school, every 20thperson is chosen.

Unbiased Samples

Virginia SOL Standard 8.11 The student will analyze problem situations, includinggames of chance, board games, or grading scales, and make predictions, using knowledge of probability.

Lesson 8-7 Using Sampling to Predict 407Doug Martin

In a , one or more parts of the population are favoredover others. Two ways to pick a biased sample are listed below.

biased sample

Type Definition Example

SampleResponseVoluntary

SampleConvenience A convenience sample

includes members of apopulation that are easilyaccessed.

A voluntary responsesample involves onlythose who want toparticipate in thesampling.

To represent all thestudents attending aschool, the principalsurveys the students inone math class.

Students at a schoolwho wish to expresstheir opinion are askedto come to the officeafter school.

Biased Samples

Describe Samples

Describe each sample.

To determine what videos their customers like, every tenth personto walk into the video store is surveyed.

Since the population is the customers of the video store, the sampleis a systematic random sample. It is an unbiased sample.

To determine what people like to do in their leisure time, thecustomers of a video store are surveyed.

The customers of a video store probably like to watch videos in theirleisure time. This is a biased sample. The sample is a conveniencesample since all of the people surveyed are in one location.

Using Sampling to Predict

SCHOOL The school bookstore sells 3-ringbinders in 4 different colors; red, green, blue,and yellow. The students who run the storesurvey 50 students at random. The colors theyprefer are indicated at the right.

What percent of the students prefer blue binders?

13 out of 50 students prefer blue binders.

13 50 � 0.26 26% of the students prefer blue binders.

If 450 binders are to be ordered to sell in the store, how manyshould be blue?

Find 26% of 450.

0.26 � 450 � 117 About 117 binders should be blue.

MisleadingProbabilitiesProbabilities basedon biased samplescan be misleading.If the studentssurveyed were allboys, theprobabilitiesgenerated by thesurvey would notbe valid, since bothgirls and boyspurchase binders atthe store.

Color Number

red 25

green 10

blue 13

yellow 2



408 Chapter 8 ProbabilityAaron Haupt

1. Compare taking a survey and finding an experimental probability.

2. OPEN ENDED Give a counterexample to the following statement.The results of a survey are always valid.


3. To determine how much money the average family in the United States spends to heat their home, a survey of 100 households from Arizona are picked at random.

4. To determine what benefits employees consider most important, oneperson from each department of the company is chosen at random.

ELECTIONS For Exercises 5 and 6, use the following information.Three students are running for class president. Jonathan randomly surveyed some of his classmates and recorded the results at the right.

5. What percent said they were voting for Della?

6. If there are 180 students in the class, how many do you think will vote for Della?

Exercises 1 & 2


7. To evaluate the quality of their product, a manufacturer of cell phones pulls every 50th phone off the assembly line to check for defects.

8. To determine whether the students will attend a spring music concert at the school, Rico surveys her friends in the chorale.

9. To determine the most popular television stars, a magazine asks itsreaders to complete a questionnaire and send it back to the magazine.

10. To determine what people in Texas think about a proposed law, 2 peoplefrom each county in the state are picked at random.

11. To pick 2 students to represent the 28 students in a science class, theteacher uses the computer program to randomly pick 2 numbers from 1 to 28. The students whose names are next to those numbers in hisgrade book will represent the class.

12. To determine if the oranges in 20 crates are fresh, the produce manager at a grocery store takes 5 oranges from the top of the first crate off thedelivery truck.

13. SCHOOL Suppose you are writing an article for the school newspaperabout the proposed changes to the cafeteria. Describe an unbiased way toconduct a survey of students.


For Exercises

7–12, 19–20

14–18

See Examples

1, 2

3, 4

Luke 7

Della 12

Ryan 6

Candidate Number

Lesson 8-7 Using Sampling to Predict 409CORBIS

SALES For Exercises 14 and 15, use the following information.A random survey of shoppers shows that 19 prefer whole milk, 44 preferlow-fat milk, and 27 prefer skim milk.

14. What percent prefer skim milk?

15. If 800 containers of milk are ordered, how many should be skim milk?

16. MARKETING A grocery store is considering adding a world foods area. They survey 500 random customers, and 350 customers agree the world foods area is a good idea. Should the store add this area?Explain.

FOOD For Exercises 17–20, conduct a survey of the students in your mathclass to determine whether they prefer hamburgers or pizza.

17. What percent prefer hamburgers?

18. Use your survey to predict how many students in your school preferhamburgers.

19. Is your survey a good way to determine the preferences of the students in your school? Explain.

20. How could you improve your survey?

21. CRITICAL THINKING How could the wording of a question or the tone of voice of the interviewer affect a survey? Give at least two examples.

22. MULTIPLE CHOICE The Star Theater records the numberof food items sold at its concessions. If the manager orders5,000 food items for next week, approximately how manytrays of nachos should she order?

1,025 850 800 400

23. MULTIPLE CHOICE Brett wants to conduct a survey aboutwho stays for after-school activities at his school. Who should he ask?

his friends on the bus members of the football team

community leaders every 10th student entering school

24. MANUFACTURING An inspector finds that 3 out of the 250 DVD playershe checks are defective. What is the experimental probability that a DVDplayer is defective? (Lesson 8-6)

Each spinner at the right is spun once. Find each probability. (Lesson 8-5)

25. P(3 and B) 26. P(even and consonant)

A

BE

CD

21

4 3

IH

GF

DCBA

Item Number

popcorn 620

nachos 401

candy 597

slices of pizza 336

Food Items Sold at MovieConcessions During the Past Week


SOL Practice

/sol


Probability of Simple Events (pp. 374–377)8-18-1

CHAPTER

biased sample (p. 407)

combination (p. 388)

complementary events (p. 375)

compound events (p. 396)

convenience sample (p. 407)

dependent events (p. 397)

experimental probability (p. 400)

factorial (p. 385)

Fundamental Counting Principle (p. 381)

independent events (p. 396)

outcome (p. 374)

permutation (p. 384)

population (p. 406)

probability (p. 374)

random (p. 374)

sample (p. 406)

sample space (p. 374)

simple event (p. 374)

simple random sample (p. 406)

stratified random sample (p. 406)

systematic random sample (p. 406)

theoretical probability (p. 400)

tree diagram (p. 380)

unbiased sample (p. 406)

voluntary response sample (p. 407)

Lesson-by-Lesson Exercises and Examples

Choose the correct term to complete each sentence.1. A list of all the possible outcomes is called the ( , event).2. (Outcome, ) is the chance that an event will happen.3. The Fundamental Counting Principle says that you can find the total number

of outcomes by ( , dividing).4. A (combination, ) is an arrangement where order matters.5. A (combination, ) consists of two or more simple events.6. For ( , dependent events), the outcome of one does not

affect the other.7. ( , Experimental probability) is based on known

characteristics or facts.8. A (simple random sample, ) is a biased sample.convenience sample

Theoretical probability

independent eventscompound eventpermutationmultiplying

Probabilitysample space

Vocabulary and Concept Check

msmath3.net/vocabulary_review

A bag contains 6 white, 7 blue, 11 red,and 1 black marbles. A marble is pickedat random. Write each probability as afraction, a decimal, and a percent.9. P(white) 10. P(blue)

11. P(not blue) 12. P(white or blue)13. P(red or blue) 14. P(yellow)

15. If a month is picked at random, whatis the probability that the month willstart with M?

Example 1 A box contains 4 green, 7 blue, and 9 red pens. Write theprobability that a pen picked at random is green.There are 4 � 7 � 9 or 20 pens in the box.

P(green) �

� �240� or �

15

�There are 4 green pens out of 20 pens.

The probability the pen is green is �15

�.

green pens��total number of pens


http://msmath3.net/vocabulary_review

A penny is tossed and a 4 sided numbercube with sides of 1, 2, 3, and 4 is rolled.16. Draw a tree diagram to show the

possible outcomes.17. Find the probability of getting a head

and a 3.18. Find the probability of getting a tail

and an odd number.19. Find the probability of getting a head

and a number less than 4.

20. FOOD A restaurant offers 15 mainmenu items, 5 salads, and 8 desserts.How many meals of a main menuitem, a salad, and a dessert are there?

Counting Outcomes (pp. 380–383)

Find each value.21. P(6, 1) 22. P(4, 4)23. P(5, 3) 24. P(7, 2)25. P(10, 3) 26. P(4, 1)

27. NUMBER THEORY How many 3-digitwhole numbers can you write usingthe digits 1, 2, 3, 4, 5, and 6 if no digitcan be used twice?

Permutations (pp. 384–387)

Example 3 Find P(4, 2).P(4, 2) represents the number ofpermutations of 4 things taken 2 at a time.

P(4, 2) � 4 � 3 or 12

Find each value.28. C(5, 5) 29. C(4, 3)30. C(12, 2) 31. C(9, 5)32. C(3, 1) 33. C(7, 2)

34. PETS How many different pairs ofpuppies can be selected from a litter of 8?

Combinations (pp. 388–391)

Example 4 Find C(4, 2).C(4, 2) represents the number ofcombinations of 4 things taken 2 at a time.

C(4, 2) � �P(4

2,!2)

� Definition of C(4, 2)

2

� �42

��

31

� or 6 P(4, 2) � 4 � 3 and

12! � 2 � 1

8-28-2

8-38-3

8-48-4

Example 2 BUSINESS A carmanufacturer makes 8 different modelsin 12 different colors. They also offerstandard or automatic transmission. How many choices does a customerhave?

number number number totalof � of � of � number

models colors transmissions of cars

8 � 12 � 2 � 192The customer can choose from 192 cars.

��

Chapter 8 Study Guide and Review 411

A number cube is rolled, and a penny istossed. Find each probability. 35. P(2 and heads) 36. P(even and heads)37. P(1 or 2 and tails) 38. P(odd and tails)39. P(divisible by 3 and tails)40. P(less than 7 and heads)

41. GAMES A card is picked from astandard deck of 52 cards and is not replaced. A second card is picked.What is the probability that both cards are red?

Probability of Compound Events (pp. 396–399)

Example 5 A bag of marbles contains 7 white and 3 blue marbles. Onceselected, the marble is not replaced.What is the probability of choosing 2 blue marbles?

P(first marble is blue) � �130�

P(second marble is blue) � �29

�

P(two blue marbles) � �130� � �

29

�

� �960� or �

115�

Station WXYZ is taking a survey todetermine how many people wouldattend a rock festival.46. Describe the sample if the station

asks listeners to call the station.47. Describe the sample if the station asks

people coming out of a rock concert.48. If 12 out of 80 people surveyed said

they would attend the festival, whatpercent said they would attend?

49. Use the result in Exercise 48 todetermine how many out of 800 people would be expected toattend the festival.

Using Sampling to Predict (pp. 406–409)

Example 7 In a survey, 25 out of 40 students in the school cafeteriapreferred chocolate to white milk. a. What percent preferred chocolate

milk?25 40 � 0.62562.5% of the students prefer chocolate milk.

b. How much chocolate milk should theschool buy for 400 students?Find 62.5% of 400.0.625 � 400 � 250About 250 cartons of chocolate milkshould be ordered.

8-58-5

A spinner has four sections. Each sectionis a different color. In the last 30 spins,the pointer landed on red 5 times, blue10 times, green 8 times, and yellow 7 times. Find each experimentalprobability.42. P(red) 43. P(green)44. P(red or blue) 45. P(not yellow)

Experimental Probability (pp. 400–403)

Example 6 In an experiment, 3 coinsare tossed 50 times. Five times no tailswere showing. Find the experimentalprobability of no tails.Since no tails were showing 5 out of the 50 tries, the experimental probability is�550� or �

110�.

8-68-6

8-78-7


CHAPTER

Chapter 8 Practice Test 413

1. Write a probability problem that involves dependent events.

2. Describe the difference between biased and unbiased samples.

In a bag, there are 12 red, 3 blue, and 5 green candies. One is picked atrandom. Write each probability as a fraction, a decimal, and a percent.

3. P(red) 4. P(no green) 5. P(red or green)

Find each value.

6. C(10, 5) 7. P(6, 3) 8. P(5, 2) 9. C(7, 4)

10. In how many ways can 6 students stand in a line?

11. How many teams of 5 players can be chosen from 15 players?

There are 4 blue, 3 red, and 2 white marbles in a bag. Once selected, it is notreplaced. Find each probability.

12. P(2 blue) 13. P(red, then white) 14. P(white, then blue)

15. Are these events in Exercises 12–14 dependent or independent?

16. FOOD Students at West Middle School can purchase a box lunch to take on their field trip. They choose one item from each category. How many lunches can be ordered?

Two coins are tossed 20 times. No tails were tossed 4 times, one tail was tossed 11 times, and 2 tails were tossed 5 times.

17. What is the experimental probability of no tails?

18. Draw a tree diagram to show the outcomes of tossing two coins.

19. Use the tree diagram in Exercise 18 to find the theoretical probability ofgetting no tails when two coins are tossed.

20. MULTIPLE CHOICE A school board wants to know if it has community support for a new school. How should they conduct a valid survey?

Ask parents at a school open house.

Ask people at the Senior Center.

Call every 50th number in the phone book.

Ask people to call with their opinions.D

C

B

A

ham apple chocolateroast beef banana oatmealtuna orange sugarturkey

Sandwich Fruit Cookie

msmath3.net/chapter_test

SOL Practice

/sol

http://msmath3.net/chapter_test/sol


CHAPTER

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

1. Which of these would be the next number inthe following pattern? (Lesson 1-1)

4, 12, 22, 34, …

40 44

46 48

2. Ms. Yeager asked the students in math classto tell one thing they did during the summer.

What fraction of the class said they went tocamp or worked a summer job? (Lesson 2-1)

�25

� �185�

�1151� �

65

�

3. Find the length of side FH. (Lesson 3-4)

14 m

16 m

17 m

18 m

4. What is the area of the circle? (Lesson 7-2)

540 in2

907.5 in2

1,017.9 in2

1,105.1 in2

5. In the spinner below, what color should the blank portion of the spinner be so that the probability of landing on this color is �

38

�? (Lesson 8-1)

red blue

yellow green

6. Ed, Lauren, Sancho, James, Sofia, Tamara,and Haloke are running for president, vice-president, secretary, and recorder of thestudent council. Each of them would behappy to take any of the 4 positions, andnone of them can take more than oneposition. How many ways can the officesbe filled? (Lesson 8-3)

28 210

840 2,520

7. Alonso surveyed people leaving a pizzaparlor to determine whether people in hisarea like pizza. Explain why this might nothave been a valid survey. (Lesson 8-7)

The survey is biased because Alonsoshould have asked people coming outof an ice cream parlor.

Alonso should have mailed surveyquestionnaires to people.

The survey is biased because Alonsowas asking only people who hadchosen to eat pizza.

Alonso should have conducted thesurvey on a weekend.

D

C

B

A

IH

GF

DC

BA

green

blue red

yellowblue

blue

yellow

I

H

G

F18 in.

D

C

B

20 m 12 m

F H

GA

IH

GF

DC

BA

traveled with family 12

went to camp 6

worked on a summer job 10

other 2

Activity Number ofStudents

SOL Practice

Chapters 1–8 Standardized Test Practice 415

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

8. The first super computer, the Cray-1, wasinstalled in 1976. It was able to perform 160 million different operations in a second.Use scientific notation to represent thenumber of operations the Cray-1 couldperform in one day. (Lesson 2-9)

9. What is the value of x if x is a wholenumber? (Lesson 5-5)

34�13

�% of 27 � x � 75% of 16

10. Find the coordinates of the fourth vertex ofthe parallelogram in Quadrant IV.(Lesson 6-4)

11. Ling knows the circumference of a circleand wants to find its radius. After shedivides the circumference by �, whatshould she do next? (Lesson 7-2)

12. The eighth-grade graduation party is being catered. The caterers offer 4 appetizers, 3 salads, and 2 main courses for each eighth-grade student to choose for dinner. If the caterers wouldlike 48 different combinations of dinners,how many desserts should they offer?(Lesson 8-2)

13. There are 15 glass containers of differentflavored jellybeans in the candy store. IfJordan wants to try 4 different flavors, howmany different combinations of flavors can he try? (Lesson 8-4)

Record your answers on a sheet ofpaper. Show your work.

14. A red number cube and a blue numbercube are tossed. (Lesson 8-2)

a. Make a tree diagram to show theoutcomes.

b. Use the Fundamental CountingPrinciple to determine the number ofoutcomes. What are the advantages of using the Fundamental CountingPrinciple? of using a tree diagram?

c. What is the probability that the sum ofthe two number cubes is 8?

15. Tiffany has a bag of 10 yellow, 10 red, and 10 green marbles. Tiffany picks twomarbles at random and gives them to her sister. (Lesson 8-5)

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

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

c. Of the marbles left, what color has a probability of �

13

� of being picked?Explain how you determined youranswer.

y

xO

Question 15 Extended response questionsoften involve several parts. When one partof the question involves the answer to aprevious part of the question, make sure you check your answer to the first partbefore moving on. Also, remember to showall of your work. You may be able to getpartial credit for your answers, even if theyare not entirely correct.

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chapter 8: probability...374 chapter 8 probability probability words the probability of an event is...

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