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370 Unit 4 Probability and StatisticsGetty Images

People often base

their decisions about

the future on data

they’ve collected. In

this unit, you will

learn how to make

such predictions using

probability and

statistics.

People often base

their decisions about

the future on data

they’ve collected. In

this unit, you will

learn how to make

such predictions using

probability and

statistics.

Probability

Statistics and Matrices

370 Unit 4 Probability and StatisticsGetty Images

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It’s all in the Genes

Math and Science Mirror, mirror on the wall... why do I look like my parents at all?

You’ve been selected to join a team of genetic researchers to find an answer to this

very question. On this adventure, 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 also make predictions based on an analysis of your findings. So grab

your lab coat and your 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 37

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7/21/2019 chap08


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

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

372 Chapter 8 ProbabilityDUOMO/CORBIS

C HA P T E R

Probability

372 Chapter 8 Probability


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Probability Make thisFoldable to help you organize your notes. Begin with

two sheets of 8

12" 11"

unlined paper.

Chapter 8 Getting Started 37

Diagnose ReadinessTake this quiz to see if you are ready tobegin Chapter 8. Refer to the lesson orpage number in parentheses for review.

Vocabulary ReviewComplete each sentence.

1. The equation

1

6

5

2

5 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.

4782 4.

3650 5.

2919

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

6. x 11 7. x 6

8. x 9 9. x 7

Evaluate each expression. (Lesson 1-2)

10.73

62

51 11.

122

111

12.84

73

62

51 13.

53

42

31

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

14.

23

34 15.

145

57

16.

78

49 17.

35

16

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

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

?

?

▲

Readiness To prepare yourself for thischapter with another quiz, visit

msmath3.net/chapter_readiness

Chapter Notes Each

time you find this logo

throughout the chapter, use your Noteables™:

Interactive Study Notebook with Foldables™

or your own notebook to take notes. Begin your chapter notes with this Foldable activity.

Fold in QuartersFold each sheet in quarters along the width.

TapeUnfold each sheet and tape to form one

long piece.

Label

Label 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 37

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Probability ofSimple 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. When

picking dominoes, one event is picking a double. is thechance that an event will happen.

Probability

simple event

random

sample 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.


impossible

equally likely

certain

somewhat likely not very likely

50%

1

2or 0.5

3

4or 0.75

1

4or 0.25

75% 100%

1

25%0%

0

Double

GAMES The game of double-six dominoesis played with 28 tiles. Seven of the tiles arecalled doubles.

1. Write the ratio that compares the numberof 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 are

doubles.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?

8-1

What You’ll LEARN

Find the probability of asimple event.

NEW Vocabulary

outcomesample spacerandomsimple eventprobability complementary events

REVIEW Vocabulary

percent: a ratio thatcompares a numberto 100 (Lesson 5-1)

Key Concept: Probabilit y

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

NEW YORK Performance Indicator 7.S.8 Int erpret dat a t o provide t he basisfor predict ions and t o est ablish experiment al probabilit ies


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Lesson 8-1 Probability of Simple Events 37

Find Probabilities

A box contains 5 green pens, 3 blue pens, 8 black pens, and 4 red

pens. 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 be

written 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 a

fraction, 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

4

5

1

blue pens red pens

total number of pens

green pens


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

complementary events

Probability of a Complementary Event

PURCHASES A computer company manufactures 2,500 computers

each day. An average of 100 of these computers are returned with

defects. 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,,4500

00 or

2245 There are 2,400 nondefective computer

The probability that your computer is not defective is

2245.

nondefective computers

total number of computers

READING Math

Probability P (green) isread the probability of

green.

Mental Math Theprobability of adefective computer

is21,50000

or 215.

Since defectiveand nondefectivecomputers arecomplementaryevents, the probabilityof a nondefectivecomputer is 1

215

or 2245.

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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 gettinga 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 yellowcards, 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

A beanbag is tossed on the square at the right. It

lands at random in a small square. Write each

probability 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 greaterthan 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 PracticeSee pages 635, 655.

For Exercises

11–20, 24–25

21

See Examples

1–3

4

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Lesson 8-1 Probability of Simple Events 37James Balog/Getty Ima

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

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

25. Suppose a person living in the United States in 2000 waspicked at random. What is the probability that the personlived 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, and2 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 unfavorable

outcomes. 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 ifappropriate, 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


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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 andred bouquet.

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

We have all the orders for the Valentine’s

Day bouquets. Each student could choose

any combination of red, pink, white, or

yellow 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.

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

What You’ll LEARN

Solve problems by makingan organized list.

NEW YORK Performance Indicator 7.RP.2 Use mat hemat ical st rat egies t oreach a conclusion


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Lesson 8-2a Problem-Solving Strategy: Make an Organized List 37

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 the

possible 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 thinkingof two integers between 10 and 10 thathave a product of 12. If Derek has one

guess, what is the probability that he willguess the pair of numbers?

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

BASEBALL For Exercises 8–10, use the

following information.

In the World Series, two teams play each other

until one team wins 4 games.8. What is the least number of games needed

to 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 talksin his or her sleep? Write the probabilityas 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 fofishing 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 bestidentifies the pattern inthe table?

y x2

y 2x2

y 0.5x2

y x2D

C

B

A

Age18–2429%

Age25–3423%

Age35–4915% Age

509%

Percent Who Talk in Their Sleep

Source: The Better Sleep Council

Millions of

Grills 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

NEW YORK

Test Practice


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Counting Outcomes

BICYCLES Antonio wants

to buy a Dynamo bicycle.1. How many different

styles are available?

2. How many differentcolors are available?

3. How many differentsizes are available?

4. Make an organized listto determine how manydifferent bicycles are

available.

Choose your Dynamo Thoose your Dynamo Today!dayChoose your Dynamo Today!

Stt

yles: Mountles: ount

ain or 1in or 1

0-Speed- peed

Colors: Red, Black, or Greenolors: Red Black or Green

Sizes: 2izes: 2

6-inch or 2-inch or 2

8-inch-inch

Styles: Mountain or 10-SpeedColors: Red, Black, or GreenSizes: 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 of

different 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 paired with each style andcolor of bicycle.

List of all theoutcomes whenchoosing a bicycle.

List each styleof bicycle.

8-2

What You’ll LEARN

Count outcomes by usinga tree diagram or theFundamental CountingPrinciple.

NEW Vocabulary

tree diagramFundamental

Counting Principle

NEW YORK Performance Indicator 7.S.8 Inter pret data to provide thebasis f or predictions and to establish experimental probabilities


7/21/2019 chap08


Lesson 8-2 Counting Outcomes 38Bettmann/COR

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

Use the Fundamental Counting Principl

COMMUNICATIONS In the United States, radio and televisionstations use call letters that start with K or W. How many differencall 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 of sauce. Each pasta entrée comes with or without a meatball.

atotal

number opossible

call letter

number of possible

letters forthe fourth

letter

number of possible

letters forthe third

letter

number of possible

letters forthe second

letter

number of possible

letters forthe firstletter

Find Probabilit y

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 is1,0

100. This

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

total numberof 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.

Key Concept: 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 are6 2 or 12 possible outcomes.






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1. Describe a possible advantage for using a tree diagramrather than the Fundamental 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. Howmany 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 ondifferent 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 andthe 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

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 ascience quiz.


For Exercises

9–12, 17

13–16, 22–23

18–21

See Examples

1

2

3

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Lesson 8-2 Counting Outcomes 38Getty Ima

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


18. What is the probability that both spinners land on thesame 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 lotterygame where the winning number is made up of five digitsfrom 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 then2 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 tea

listed 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. Courtneyhas a black, a green, a red, and a white sweatshirt. Each girlpicks a sweatshirt at random to wear to the picnic. What isthe 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 inthe 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

mint

orange

peach

raspberry

strawberry


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Permutations

When deciding who goes first and who goes second, order isimportant. An arrangement or listing in which order is importantis 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.

totalnumber of possiblecones

number of possible

flavors for thethird scoop

number of possible

flavors for thesecond scoop

number of possible

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 gamewith 4 different game pieces. Showall of the ways the game pieces can

be chosen first and second. Recordeach arrangement.

1. How many differentarrangements did you make?

2. How many different game pieces could you pick for thefirst 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

8-3

What You’ll LEARN

Find the number of permutations of objects.

NEW Vocabulary

permutationfactorial

MATH Symbols

P (a, b

) the number of permutations of a things taken bat a time

! factorial

READINGin the Content Area

For strategies in readingthis lesson, visitmsmath3.net /reading.

NEW YORK Performance Indicator 7.S.8 Interpret datato provide the basis for predictions and

to establishexperimentalprobabilities

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Lesson 8-3 Permutations 38

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 6also 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 arandom 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 11 1

number of permutations between 1,000 and 2,000

total number of permutations

number of permutations between

1,000 and 2,000

number of waysto pick the last

three digits

number of ways to pick the first digit

DCBA

READING Math

Permutations P (8, 3) canalso be written 8P 3.

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

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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 valueof P(7, 3).

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

correct? Explain.

Bai ley P(7 , 3) = 7 6 5

= 2 10

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?

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 wayscould 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 play3 different games. In how many orders can he play the 3 games?

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

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


For Exercises

11–22

23–26, 29–3227–28

See Examples

2, 3

14

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http://endmat.pdf/

http://endmat.pdf/

http://endmat.pdf/

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Lesson 8-3 Permutations 38Ronald Martinez/Getty Ima

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 and11 trails down the mountain. How many different ways can a skier takea 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 themissing side of the triangle at the right. Then find themissing length. (Lesson 3-4)

13 ft

5 ft

a

IHGF

DCBA

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

43. 36

52

41 44.

140

39

28

17 45.

202

119 46.

65

54

43

32

21

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

ENTERTAINMENT For Exercises 30–32, use the followinginformation.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 competingin the Rose Bowl. If they cannot be first or second, how many wayscan 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 Fire

Columbus Crew

D.C. United

MetroStars

New England Revolution

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


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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 a

circle. 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 by

the 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 if each person shakes hands with every other person?

A B

C

D


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.

8-4

What You’ll LEARN

Find the number of combinations of objects.

NEW Vocabulary

combination

MATH Symbols

C (a , b) the number of

combinations of a things taken b at a time

NEW YORK Performance Indicator 7.S.8 Inter pret data to provide thebasis f or predictions and to establish experimental probabilities


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Lesson 8-4 Combinations 38Andy Sacks/Getty Ima

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

C(4,2) P(4

2,!2)

the number of combinationsof 4 things taken 2 at a time

the number of permutationof 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 about70 inches tall and has47 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 theorder 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

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

One member from the strings section will talk to students at

Brown Middle School, another to students at Oak Avenue MiddleSchool, and another to students at Jefferson Junior High. Does

this represent a combination or a permutation? How many

possible 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 S ymphony

Instrument Number

Strings 45

Woodwinds 8

Brass 8

Percussion 3

Harps 2

READING Math

Combinations C (7, 4) canalso be written as 7C 4.






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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.

choosing3 dessertsto serve at the party

choosing3 peopleto chair

3 different committees

choosing3 members

for thedecoratingcommittee

choosing3 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?

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 orderedfrom a list of toppings at the right?

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


For Exercises

11–18

19–24, 27–32

25–26

See Examples

2

3, 4

1

Pizza Toppings

anchovies sausage onions

bacon green peppers black olives

ham hot peppers green olives

pepperoni mushrooms pineapple

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Lesson 8-4 Combinations 39Aaron Ha

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 of52 cards?

30. GAMES In the game of cribbage, a player gets 2 points

for each combination of cards that totals 15. How manypoints for totals of 15 are in the hand at the right?

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

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

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 Schoolis given at the right. How many different four-person committeescould 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 Girl

6th grade 42 47

7th grade 55 49

8th grade 49 53

Centerville Middle Schoo


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• paper• pencil


Combinations and Pascal’s TriangleFor many years, mathematicians have been interested in a pattern

called 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 tossinga 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 a

conjecture about how many outcomes have exactly no head,one head, two heads, and so on. Test your conjecture.

8-4b

What You’ll LEARN

Identify patterns inPascal’s Triangle.

NEW YORK Performance Indicator 7.S.8 Inter pret data to provide the basisf or predictions and to establish experimental probabilities


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Lesson 8-4b Hands-On Lab: Combinations and Pascal’s Triangle 39

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 getting

exactly 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


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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 numbercube shows a six?

11. What is the probability that the penny shows heads and the numbercube 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 electedfrom 7 candidates? (Lesson 8-4)

19. MULTIPLE CHOICE A pizza shopadvertises that it has 3 differentcrusts, 3 different meat toppings,and 5 different vegetables. If Carlotta 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 belowis used for a game. Find theprobability that the spinner willnot land on yellow. (Lesson 8-1)

Y

B

RGW

RWY

G

B

DC

BA



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The Game Zone: Probability 39John Ev

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 to9 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 pointsis the winner.

0

1

Winning Numbers


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Probability ofCompound Events

GAMES A game uses a numbercube and the spinner shownat the right.

1. A player rolls thenumber cube. What isP(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 player

will 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

8-5

What You’ll LEARN

Find the probability of independent anddependent events.

NEW Vocabulary

compound eventindependent eventsdependent events

Key Concept: Probability of Independent Events

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

Symbols P ( A and B ) P ( A ) P ( B )

NEW YORK Performance Indicator 7.S.10 Predict the outcome of anexperiment


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Lesson 8-5 Probability of Compound Events 39Sylvain Grandadam/Getty Ima

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

Use Probabilit y to Solve a Problem

POPULATION Use the informationin the table. In the United States,what is the probability that a personpicked at random will be underthe age of 18 and live in anurban area?

P(younger than 18) 14

P(urban area)

45

P(younger than 18 and urban area)

or

15 Simplify.

The probability that the two events

will occur is

15.

1

4

54

4

Probabilit y of Dependent EventsThere 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)

1

7

4

P(two red marbles) or

Find each probability.

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

4

15

17

1471

4

8

15

number of red marbles afterone red marble is removed

total 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

5

8

old

65 years or

81

older

Urban

45

Rural

51


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

be about 15.


United States

←

←

←

←

Key Concept: Probabilit y of 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 )

Mental MathYou may wish to

simplify 174 to

12

before multiplying theprobabilities.

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1. Compare and contrast independent 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 andTia are finding the probability that both spins will result in an

odd number. Who is correct? Explain.

Tia

5

3

4

2 =

2

6

0 or

1

3

0

Evita

3

5

3

5 =

2

9

5

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?

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 headseach time?

16. What is the probability of rolling a number cube 3 times and gettingnumbers 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 PracticeSee pages 636, 655.

For Exercises

9–16

17–22

23–24

See Examples

1

3

2

http://endmat.pdf/

http://endmat.pdf/

http://endmat.pdf/


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Lesson 8-5 Probability of Compound Events 39

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 mixerand 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 deckof 52 cards. Getting a 5 or getting a 6 aremutually exclusive events. To find theprobability of two mutually exclusiveevents, 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 ofmarbles. 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 TODA

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 cardswithout 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 acoach 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.

3930 37.

4790 38.

2848


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Experimental Probability

In the Mini Lab above, you determined a probability by conductingan experiment. Probabilities that are based on frequencies obtained

by conducting an experiment are called .Experimental probabilities usually vary when the experimentis repeated.

Probabilities based on known characteristics or facts are called

. For example, you can compute thetheoretical probability of picking a certain color marble from a bag.Theoretical probability tells you what should happen in an experiment.

theoretical probabilities

experimental probabilities

Experimental Probability

Michelle is conducting an

experiment to find the probability

of getting various sums when two

number cubes are rolled. The

results of her experiment are

given at the right.According to the experimental

probability, 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 number

cubes?

There are 6 6 or 36 possible outcomes.

2 3 4 5 6 7 8 9 10 11 12

Results of RollingTwo Number Cubes

N

u m b e r o f R o l l s

Sum

16

12

8

4

0



Draw one marble from the bag, record itscolor, and replace it in the bag. Repeat this50 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 orwhy not.

number of each color marble

total number of marbles

number of times color was drawn

total number of draws

• paper bagcontaining

10 coloredmarbles

8-6

What You’ll LEARN

Find experimentalprobability.

NEW Vocabulary

experimental probability theoretical probability

REVIEW Vocabulary

proportion:

a statementof equality of two or

more ratios, a b

d c ,

b 0, d 0(Lesson 4-4)

7.S.8 Inter pret data toprovide the basis f or

predictions and toestablish experimentalprobabilities

7.S.10 Predict theoutcome of anexperiment

7.S.12 Compare actualresults to predictedresults

NEW YORK Performance Indicator


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Lesson 8-6 Experimental Probability 40LWA-Dann Tardif/COR

Use Probabilit y to Predict

FARMING Over the last 8 years, the probability that corn seeds

planted 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,x

000

Solve the proportion.

56

10,x

000 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.

You can use past performance to predict future events.

5 out of 6 seeds shouldproduce corn.

10,000 out of x seedsshould produce corn.

How Does a MarketingManager Use Math?

A marketing manager usesinformation from surveysand experimental probabilityto help make decisionsabout changes in productsand advertising.

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

Theoretical Probabilit y

What is the theoretical probability of rolling a double six?

The theoretical probability is

16

16 or

316.

Experimental Probabilit y

MARKETING Two hundred teenagerswere asked whether they purchasedcertain items in the past year. Whatis the experimental probability thata teenager bought a photo frame inthe last year?

There were 200 teenagers surveyed and 95 purchased a photo fram

in the last year. The experimental probability is

29050 or

1490.

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

ItemNumber Who

Purchased the Item

candles 110

photo frames 95

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 a totalof 10,000 2,000 or12,000 seeds. Theanswer is correct.


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402 Chapter 8 ProbabilityMark Thayer

1. Explain why you would not expect the theoreticalprobability and the experimental probability of an event to always bethe same.

2. OPEN ENDED Two hundred fifty people are surveyed about theirfavorite color. Make a possible table of results if the experimental

probability 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 occurin 400 tries?

5. What is the theoretical probability of getting heads?

6. Based on the theoretical probability, how many heads would you expect

to occur in 400 tries?7. Compare the theoretical probability to your experimental probability.

For Exercises 8 and 9, use the table at the right showingthe 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 expectto be white?

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 expectCleveland to make?

12. SCHOOL In the last 40 school days, Esteban’s bus has been late8 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 whilewatching television is corn chips?

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

15. SPORTS In practice, Crystal made 80 out of 100 free throws. Whatis 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

ResultNumberof 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

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Lesson 8-6 Experimental Probability 40

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 gettwo 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 numbercubes 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 of every 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 each

outcome. (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 beadded 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 of Participants

basketball 42

volleyball 26

soccer 24

football 16

Sports Participation by Teens

Difference of RollingTwo Number Cubes

N u m b e r o f R o l l s

Difference

40

35302520

151050

0 1 2 4 53

21

35

22

5 4

13


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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-83/84 Plus graphingcalculator. Enter 1 as the lower bound and 6 as the upper boundfor 50 trials.

Keystrokes: 5 1 6

50

A set of 50 numbers ranging from 1 to 6appears. Use the right arrow key to seethe next number in the set. Record all50 numbers on a separate sheet of paper.

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

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

ENTER),

,


A company is placing one of 8 different cards of action heroes in

its boxes of cereal. If each card is equally likely to appear, what is

the experimental probability that a person who buys 12 boxes of

cereal will get all 8 cards?

Let the numbers 1 through 8 representthe cards. Use the random numbergenerator on a graphing calculator.Enter 1 as the lower bound and 8 asthe upper bound for 12 trials.

Keystrokes: 5

1 8 12

Record whether all of the numbers are represented.

ENTER),,

• graphing calculator• paper• pencil

8-6b

What You’ll LEARNUse a graphing calculatorto simulate probability experiments.

SimulationsRepeating a simulationmay result in differentprobabilities since thenumbers generatedare different eachtime.

msmath3.net/other_calculator_keystrokes

NEW YORK Performance Indicator 7.S.11 Design and conduct an experimentto test predictions, 7.S.12 Compare actual results to predicted results

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Lesson 8-6b Graphing Calculator Investigation: Simulations 40

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.

EXERCISES

1. A hypothesis is a statement to be tested that describes what youexpect to happen in a given situation. State your hypothesis as tthe 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 his6 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 isplaced in the maze at the right.If each decision about directionis made at random, create asimulation to determine theprobability that the mouse willfind its way out before comingto a dead end or going outthe 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


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Statistics: UsingSampling to Predict

ENTERTAINMENT The manager of a radiostation wants to conduct a survey todetermine what type of music people like.

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

2. Suppose she decides to survey studentsat 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 thelistening 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. An

is selected so that it is representative of the entirepopulation. Three ways to pick an unbiased sample are listed below.unbiased sample

populationsample


Type Definition Example

Sample

Random

Systematic

Sample

Random

Stratified

Sample

Random

Simple A simple random sample isa 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.

Each student’s name is written on a piece of paper. The names areplaced in a bowl, andnames are picked

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

8-7

What You’ll LEARN

Predict the actions of alarger group by using asample.

NEW Vocabulary

samplepopulationunbiased samplesimple random samplestratified random samplesystematic random

samplebiased sampleconvenience sample voluntary response

sample

What Type of MusicDo You Like?

Country

Alternative

Rock

Oldies

Top 40

Urban

Adult Contemporary

NEW YORK Performance Indicator 7.S.9Determine the validit y of samplingmethods to predict outcomes


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Lesson 8-7 Statistics: Using Sampling to Predict 40Doug Ma

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

Sample

Response

Voluntary

Sample

Convenience A convenience sampleincludes members of a

population that are easily accessed.

A voluntary responsesample involves only those who want toparticipate in thesampling.

To represent all thestudents attending a

school, the principalsurveys the students inone math class.

Students at a school who 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 persoto 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 theleisure time. This is a biased sample. The sample is a convenience

sample 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.

Color Number

red 25

green 10

blue 13

yellow 2

MisleadingProbabilitiesProbabilities based onbiased samples can

be misleading. If thestudents surveyedwere all boys, theprobabilities generatedby the survey wouldnot be valid, sinceboth girls and boyspurchase binders atthe store.


Statistics can sometimesbe misleading whensamples are taken fromlarge populations. For alesson on therelationships betweenrelative error andmagnitude when dealing with large numbers, go tony.msmath3.net.

NEW YORKStudy Tip





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408 Chapter 8 ProbabilityAaron Haupt

1. Compare taking a survey and finding an experimentalprobability.

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 UnitedStates spends to heat their home, a survey of 100 householdsfrom 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 randomlysurveyed 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 willvote for Della?


7.

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

8. To determine whether the students will attend a spring musicconcert 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 from1 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 managerat 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

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Lesson 8-7 Statistics: Using Sampling to Predict 40

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 foodsarea. They survey 500 random customers, and 350 customers agreethe 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 math

class 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 studentsin your school? Explain.

20. How could you improve your survey?

21. CRITICAL THINKING How could the wording of a question or the toneof 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 orders

5,000 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. Whoshould 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


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

C HAPT E R

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 numberof 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 notaffect the other.

7. ( , Experimental probability) is based on knowncharacteristics or facts.

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

Theoretical probability

independent events

compound event

permutation

multiplying

Probability

sample space

Vocabulary and Concept Check

A bag contains 6 white, 7 blue, 11 red,

and 1 black marbles. A marble is picked

at 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 the

probability that a pen picked at randomis 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


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A penny is tossed and a 4 sided number

cube with sides of 1, 2, 3, and 4 is rolled.

16. Draw a tree diagram to show thepossible outcomes.

17. Find the probability of getting a headand a 3.

18. Find the probability of getting a tailand an odd number.

19. Find the probability of getting a headand 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 of permutations of 4 things taken 2 ata 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 of puppies can be selected from a litterof 8?

Combinations (pp. 388–391)

Example 4 Find C (4, 2).

C(4, 2) represents the number of combinations of 4 things taken 2 ata 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 car

manufacturer makes 8 different models

in 12 different colors. They also offerstandard or automatic transmission.

How many choices does a customer

have?

number number number totalof of of number

models colors transmissions of cars

8 12 2 192

The customer can choose from 192 cars.

Chapter 8 Study Guide and Review 41


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Study Guide and Review continued

Mixed Problem Solving For mixed problem-solving practice,

see page 655.

A number cube is rolled, and a penny is

tossed. 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 isnot replaced. A second card is picked.What is the probability that bothcards are red?

Probability of Compound Events (pp. 396–399)

Example 5 A bag of marbles contains

7 white and 3 blue marbles. Once

selected, 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 to

determine how many people would

attend a rock festival.

46. Describe the sample if the stationasks 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 of800 people would be expected toattend the festival.

Statistics: Using Sampling to Predict (pp. 406–409)

Example 7 In a survey, 25 out of

40 students in the school cafeteria

preferred chocolate to white milk.

a. What percent preferred chocolate

milk?

25 40 0.625

62.5% of the students preferchocolate milk.

b. How much chocolate milk should the

school buy for 400 students?

Find 62.5% of 400.

0.625 400 250

About 250 cartons of chocolate milkshould be ordered.

8-58-5

A spinner has four sections. Each section

is a different color. In the last 30 spins,

the pointer landed on red 5 times, blue

10 times, green 8 times, and yellow

7 times. Find each experimental

probability.

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 coins

are tossed 50 times. Five times no tails

were showing. Find the experimental

probability of no tails.

Since no tails were showing 5 out of the50 tries, the experimental probability is

550 or

110.

8-68-6

8-78-7


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C HAPT E R

Chapter 8 Practice Test 41

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 at

random. 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, the

marble is not replaced. 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 purchasea box lunch to take on their field trip. They choose

one item from each category. How many lunchescan 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 of getting no tails when two coins are tossed.

20. MULTIPLE CHOICE A school board wants to know if it has communitysupport 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 chocolate

roast beef banana oatmeal tuna orange sugar turkey

Sandwich Fruit Cookie

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C HAPT E R

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

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 to

camp 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 thecircle? (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 sothat 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 offices

be 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

yellow

blue

blue

yellow

I

H

G

F18 in.

D

C

B

20 m 12 m

F H

G A

IH

GF

DC

BA

traveled with family 12

went to camp 6

worked on a summer job 10

other 2

ActivityNumber of Students



7/21/2019 chap08


Preparing for Standardized Tests

For test-taking strategies and more practicesee pages 660–677.

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

8. The first super computer, the Cray-1, was

installed in 1976. It was able to perform160 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 of the 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 offer4 appetizers, 3 salads, and 2 maincourses for each eighth-grade studentto choose for dinner. If the caterers would

like 48 different combinations of dinners,how many desserts should they offer?(Lesson 8-2)

13. There are 15 glass containers of different

Record your answers on a sheet of

paper. Show your work.14. A red number cube and a blue number

cube are tossed. (Lesson 8-2)

a. Make a tree diagram to show theoutcomes.

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

c. What is the probability that the sum of the 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 toher sister. (Lesson 8-5)

a. What is the probability of choosing2 yellow marbles?

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

c. Of the marbles left, what color has

a probability of

13 of being picked?

Explain how you determined youranswer.

y

x O

Question 15 Extended response questionsoften involve several parts. When one part

of the question involves the answer to a

previous part of the question, make sure

you check your answer to the first part

before moving on. Also, remember to show

http://endmat.pdf/

http://endmat.pdf/

http://endmat.pdf/

http://endmat.pdf/

http://endmat.pdf/

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chap08