Download Chapter 8 - Anna Middle School

Probability
and Statistics
Focal Point
Use statistical procedures
and probability to describe
data and make predictions.
CHAPTER 8
Probability
Apply concepts of
theoretical and experimental
probability to make predictions.
Evaluate predictions
and conclusions based on
statistical data.
CHAPTER 9
Statistics
Use statistical procedures
to describe data.
Evaluate predictions
and conclusions based on
statistical data.
412
Lawrence Lawry/Getty Images
Math and Science
It’s all in the Genes 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 answer this very question. 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 to
begin this adventure.
Log on to tx.msmath3.com to begin.
Unit 4 Probability and Statistics
Lawrence Lawry/Getty Images
413
8
Probability
Knowledge
and Skills
•
Apply concepts of
theoretical and
experimental probability to
make predictions.
TEKS 8.11
•
Evaluate predictions and
conclusions based on
statistical data. TEKS 8.13
Key Vocabulary
dependent events (p. 430)
independent events (p. 429)
outcome (p. 416)
simulation (p. 445)
Real-World Link
Bicycling If several bicyclists are racing toward the finish
line, you can use a tree diagram or other counting methods
to determine the possible finishing order.
Probability Make this Foldable to help you organize your notes. Begin with a plain sheet of
11” × 17” paper.
1 Fold the sheet in half lengthwise. Cut along the fold.
2 Fold each half in quarters along the width.
3 Unfold each piece and tape to form one long piece.
4 Label each page with the lesson number as shown.
Refold to form a booklet.
n‡£
414
Chapter 8 Probability
Elizabeth Kreutz/NewSport/CORBIS
n‡Ó
n‡Î
n‡{
n‡x
n‡È
n‡Ç
GET READY for Chapter 8
Diagnose Readiness You have two options for checking Prerequisite Skills.
Option 2
Take the Online Readiness Quiz at tx.msmath3.com.
Option 1
Take the Quick Quiz below. Refer to the Quick Review for help.
Write each fraction in simplest
form. (Used in Lessons 8-4 and 8-5)
48
1. _
72
35
2. _
60
3.
21
_
of a possible 18 hours. Write this
portion of time spent driving as a
fraction in simplest form.
Evaluate x(x - 1)(x - 2)(x - 3) for
each value of x. (Used in Lessons 8-3 and 8-4)
6. x = 6
7. x = 9
8. x = 7
Evaluate each expression.
(Used in
Lesson 8-4)
7·6·5
9. _
3·2·1
11.
8·7·6·5
_
4·3·2·1
12 · 11
10. _
2·1
12.
_
Write 45 in simplest form.
51
÷3
99
4. TRAVEL Dustin drove 4 hours out
5. x = 11
Example 1
5·4·3
_
3·2·1
15
45
_
=_
Divide the numerator
and denominator by
their GCF, 3.
17
51
÷3
Example 2
Evaluate x(x - 1)(x - 2)(x - 3) for x = 5.
x(x - 1)(x - 2)(x - 3)
= 5(5 - 1)(5 - 2)(5 - 3)
= 5(4)(3)(2)
= 120
Substitute x = 5.
Simplify.
Multiply.
Example 3
__
Evaluate 8 · 7 · 6 .
4·3·2
336
8
·
7
·
6
_=_
Multiply the numerator
24
4·3·2
and denominator.
= 14
Simplify.
13. JOBS Cassie earns $4 an hour
raking leaves and $12 an hour
baby-sitting. If she raked leaves
for 3 hours and baby-sat for 11
hours, how many times greater
is her total earnings for babysitting than for raking leaves?
Chapter 8 Get Ready for Chapter 8
415
8-1
Counting Outcomes
Main IDEA
Count outcomes by using
a tree diagram or the
Fundamental Counting
Principle.
Reinforcement of
TEKS 7.10 The
student recognizes
that a physical or
mathematical model can be
used to describe the
experimental and theoretical
probability of real-life
events. (A) Construct
sample spaces for simple or
composite experiments. Also
addresses TEKS 8.15(A).
BICYCLES Antonio wants to buy a
Dynamo bicycle.
#HOOSEYOUR$YNAMO4ODAY
1. How many different styles
are available? colors? sizes?
2. Make a list showing all of the
different bicycles that are available.
3TYLES-OUNTAINOR2OAD
#OLORS2ED"LACKOR'REEN
3IZESINCHORINCH
An outcome is any one of the possible results of an action. For the action
of selecting a specific type, color, and size of bicycle, there are
12 total outcomes. An event is one outcome or collection of outcomes.
An organized list of outcomes, called a sample space, can help you
determine the total number of possible outcomes for an event. One type
of organized list is a tree diagram.
NEW Vocabulary
outcome
event
sample space
tree diagram
Fundamental
Counting Principle
probability
random
Use a Tree Diagram
1 BICYCLES Draw a tree diagram to determine the number of different
bicycles described in the real-world example above.
Style
Color
Red
Mountain
Black
Green
Red
Road
Black
Green
Size
26 in.
Outcome
Mountain, Red, 26 in.
28 in.
Mountain, Red, 28 in.
26 in.
Mountain, Black, 26 in.
28 in.
Mountain, Black, 28 in.
26 in.
Mountain, Green, 26 in.
28 in.
Mountain, Green, 28 in.
26 in.
Road, Red, 26 in.
28 in.
Road, Red, 28 in.
26 in.
Road, Black, 26 in.
28 in.
Road, Black, 28 in.
26 in.
Road, Green, 26 in.
28 in.
Road, Green, 28 in.
There are 12 different Dynamo bicycles.
a. A dime and a penny are tossed. Draw a tree diagram to determine
the number of outcomes.
416
Chapter 8 Probability
You can also find the total number of outcomes by multiplying. This
principle is known as the Fundamental Counting Principle.
+%9#/.#%043
Fundamental Counting Principle
If event M has m possible outcomes and event N has n possible outcomes,
then event M followed by event N has m · n possible outcomes.
2 COMMUNICATIONS In the United States, radio and television stations
use call letters that start with K or W. How many different station
call letters are possible when four letters are used?
There are 2 choices for the first letter and 26 for each of the others.
Use the Fundamental Counting Principle.
2
×
26
×
×
26
26
=
35,152
There are 35,152 possible call letters.
Real-World Link
On October 27, 1920,
KDKA in Pittsburgh,
Pennsylvania, became
the first licensed radio
station.
b. DINING A restaurant offers a choice of 3 types of pasta with 5 types
of sauce. Each pasta entrée comes with or without a meatball. How
many different entrées are available?
Source: Time Almanac
Personal Tutor at tx.msmath3.com
The probability of an event is the ratio of a specific outcome to the total
number of outcomes. Outcomes occur at random if each outcome is
equally likely to occur.
Find Probability
3 GAMES What is the probability of winning a lottery game where
Look Back
You can review
probability on
page 691.
the winning number is made up of three digits, each from 0 to 9,
chosen at random?
First, find the number of possible outcomes.
10
×
10
×
=
10
1,000
There are 1,000 possible outcomes. There is 1 winning number.
1
P(win) = _
1,000
There is 1 winning number out of 1,000.
This can also be written as a decimal, 0.001, or a percent, 0.1%.
c. Two number cubes are rolled. What is the probability that the sum
of the numbers on the cubes is 12?
Extra Examples at tx.msmath3.com
Bettmann/CORBIS
Lesson 8-1 Counting Outcomes
417
Example 1
(p. 416)
Example 2
(p. 417)
Example 3
(p. 417)
(/-%7/2+ (%,0
For
Exercises
4–7
8–13
14–15
See
Examples
1
2
3
1. The spinner is spun two times. Draw a tree diagram
to determine the number of possible outcomes.
2. FOOD A pizza shop has regular, deep-dish, and
thin crusts; 2 different cheeses; and 4 toppings. How
many different one-cheese and one-topping pizzas
can be ordered?
green yellow
red
3. GOVERNMENT The first 3 digits of a social security number are a geographic
code. The next 2 digits are determined by the year and the state where the
number is issued. The final 4 digits are random numbers. What is the
probability of the last 4 digits being the current year?
Draw a tree diagram to determine the number of possible outcomes.
4. A penny, a nickel, and a dime are tossed.
5. A number cube is rolled and a penny is tossed.
6. A white or red ball cap comes in small, medium, large, or extra large.
7. The Sweet Treats Shoppe offers single-scoop ice cream in chocolate, vanilla,
or strawberry, and two types of cones, regular or sugar.
Use the Fundamental Counting Principle to find the number of possible
outcomes.
8. The day of the week is picked at random and a number cube is rolled.
9. A number cube is rolled 3 times.
10. There are 5 true-false questions on a history quiz.
11. There are 4 choices for each of 5 multiple-choice questions on a science
quiz.
12. SCHOOL Doli can take 4 different classes first period, 3 different classes
second period, and 5 different classes third period. How many different
schedules can she have?
Real-World Link
In a recent year, there
were approximately
14.9 million vehicles
registered in Texas.
Source: U.S. Federal
Highway Administration
418
13. VEHICLES The license plate at the left is issued with 2 letters, followed by
2 numbers and a letter. How many different license plates could the state
issue?
14. CLOTHES 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?
Chapter 8 Probability
Texas Department of Transportation
15. GAMES The winning number in a lottery game is made up of five digits
from 0 to 9 chosen at random. If the digits can repeat, what is the
probability of winning the lottery?
ELECTRONICS For Exercises 16 and 17, use the table that shows various
options for a digital music player.
16. How many different players are
available, based on storage capacity
and color?
17. If an FM radio tuner is also available
as an option, how many players are
available?
For Exercises 18 and 19, each spinner
at the right is spun once. Use a tree
diagram to answer each question.
18. What is the probability that at least
Storage Capacity
Colors
256 megabytes
blue
purple
512 megabytes
red
pink
1 gigabyte
green
silver
2.5 gigabytes
white
black
GREEN
RED
BLUE
YELLOW
RED
WHITE
BLUE
one spinner lands lands on blue?
19. What is the probability that at least
one spinner lands on yellow?
LUNCHES For Exercises 20–24, use the following information.
Parent volunteers made lunches for an 8th-grade field trip. Each lunch had
a peanut butter and jelly or a deli-meat sandwich; a bag of potato chips or
pretzels; an apple, an orange, or a banana; and juice, water, or soda. One
of each possible lunch combinations was made.
20. How many different lunch combinations were made?
21. How many of these combinations contained an apple?
%842!02!#4)#%
See pages 714, 735.
22. If the lunches are handed out randomly, what is the probability that a
student receives a lunch containing a banana?
23. What is the probability of a student receiving a lunch with potato chips
and soda?
Self-Check Quiz at
tx.msmath3.com
H.O.T. Problems
24. Suppose 4 types of meat were used for the deli-meat sandwiches. What is
the probability that a student receives one specific type of sandwich?
25. OPEN ENDED Give an example of a situation that has 15 possible outcomes.
26. 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. Without calculating
the number of possible outcomes, how many more outfits can she make if
she buys a print blouse?
27. CHALLENGE If x coins are tossed, write an algebraic expression for the
number of possible outcomes.
28. WRITING MATH Describe a possible advantage for using a tree diagram
rather than the Fundamental Counting Principle.
Lesson 8-1 Counting Outcomes
419
29. A school cafeteria offers sandwiches with three types of meat and two
types of bread. Which table shows all possible sandwich combinations
available?
A
Bread
White
Wheat
White
Wheat
Meat
Ham
Turkey
Ham
Turkey
C
Bread
White
White
White
Wheat
Wheat
Wheat
Meat
Ham
Turkey
Beef
Ham
Turkey
Beef
B
Bread
White
White
White
Wheat
Wheat
Wheat
Rye
Rye
Rye
Meat
Ham
Turkey
Beef
Ham
Turkey
Beef
Ham
Turkey
Beef
D
Bread
White
White
White
White
Wheat
Wheat
Wheat
Wheat
Meat
Ham
Turkey
Beef
Bologna
Ham
Turkey
Beef
Bologna
30. The regular prisms at the right are similar.
Find the volume of the smaller prism. Round
to the nearest tenth. (Lesson 7-9)
IN
IN
Find the lateral area and the surface area of each
regular pyramid. Round to the nearest tenth. (Lesson 7-8)
31.
CM
32.
CM
CM
IN
33.
FT
IN
CM
FT
CM
FT
IN
IN
34. INTERNET The number of U.S. households with high-speed Internet
access increased 66% between 2003 and 2004. If 63 million households
had high-speed Internet access in 2004, what was the number of
households with high-speed Internet access in 2003? (Lesson 5-8)
PREREQUISITE SKILL Evaluate n(n - 1)(n - 2)(n - 3) for each value of n.
35. n = 5
420
Chapter 8 Probability
36. n = 10
37. n = 12
(Lesson 1-2)
38. n = 8
IN
8-2
Permutations
Main IDEA
Find the number of
permutations of objects.
Targeted TEKS 8.11
The student applies
concepts of
theoretical and
experimental probability to
make predictions. (B)
Use theoretical probabilities
and experimental results to
make predictions and
decisions. Also addresses
TEKS 8.14(D).
Show all of the ways 4 different
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 game
pieces could you pick for the second place?
4. Use the Fundamental Counting Principle to determine the number
of arrangements for first and second places.
5. How do the numbers in Exercises 1 and 4 compare?
NEW Vocabulary
permutation
When deciding who goes first and who goes second, order is important.
An arrangement or listing in which order is important is called
a permutation.
Find a Permutation
1 FOOD An ice cream shop has 31 flavors. Carlos wants to buy a
three-scoop cone with three different flavors. How many cones
could he buy if the order of the flavors is important?
There are 31 choices for the first scoop, 30 choices for the second
scoop, and 29 choices for the third scoop. Use the Fundamental
Counting Principle.
31
×
30
×
=
29
26,970
Carlos could buy 26,970 different cones.
a. In a race with 7 runners, how many ways can the runners end up
in first, second, and third place?
READING
in the Content Area
The symbol P(31, 3) represents the number of permutations of 31 things
taken 3 at a time.
For strategies in reading
this lesson, visit
tx.msmath3.com.
Extra Examples at tx.msmath3.com
Aaron Haupt
Start with 31.
P(31, 3)
=
31 · 30 · 29
Use three factors.
Lesson 8-2 Permutations
421
Use Permutation Notation
READING Math
P(a, b) P(a, b) is read the
number of permutations of
a things taken b at a time.
Find each value.
3 P(6, 6)
2 P(8, 3)
8 things taken 3 at a time.
6 things taken 6 at a time.
P(8, 3) = 8 · 7 · 6 or 336
P(6, 6) = 6 · 5 · 4 · 3 · 2 · 1 or 720
Find each value.
b. P(12, 2)
c. P(4, 4)
d. P(10, 5)
Permutations can be used when finding probabilities.
Find Probability
4 NUMBERS Consider all of the four-digit numbers that can be formed
using the digits 1, 2, 3, and 4 where no digit is used twice. Find the
probability that one of these numbers picked at random is between
1,000 and 2,000.
You are considering all of the permutations of 4 digits taken 4 at a
time. You wish to find the probability that one of these numbers
picked at random is greater than 1,000, but less than 2,000.
First, find the number of possible four-digit numbers.
P(4, 4) = 4 · 3 · 2 · 1
For a number to be between 1,000 and 2,000, the thousands digit
must be 1.
number of ways
to pick the
thousands digit
1
×
number of ways
to pick the last
three digits
=
number of
permutations between
1,000 and 2,000
×
P(3, 3)
=
P(3, 3)
permutations between 1,000 and 2,000
total permutations
3·2·1
_
=
Substitute.
4·3·2·1
P(between 1,000 and 2,000) = ____
1
1
3·2·1
=_
4·3·2·1
1
1
=_
or 25%
4
Divide out common factors.
1
Simplify.
e. MUSIC Out of 10 songs on a CD, 3 different songs are randomly
selected to play without repeating a song. What is the probability
that the first 3 songs on the CD are selected?
Personal Tutor at tx.msmath3.com
422
Chapter 8 Probability
Example 1
1. SPORTS How many ways can the Wildcats’ coach pick the first 3 batters out
(p. 421)
Examples 2, 3
(p. 422)
Example 4
of the 9 players on the baseball team?
Find each value.
2. P(5, 3)
For
Exercises
7–10
11, 12,
15–18
13, 14
19, 20
See
Examples
1
2
3
4
4. P(12, 5)
5. P(8, 8)
6. PHONE NUMBERS Ivy knows that the last four digits of her friend’s phone
(p. 422)
(/-%7/2+ (%,0
3. P(7, 4)
number are 0, 3, 5, and 6, but she can’t remember the exact order. She
knows that the first digit is 5. What is the probability that Ivy randomly
selects the digits in the correct order?
7. CODES A security system has a keypad with 10 digits. How many four-
number codes are available if no digit is repeated?
8. ENTERTAINMENT 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?
9. MUSIC A disk jockey has 12 songs he plans to play in the next hour. How
many ways can he pick the next 3 different songs?
10. CONSTRUCTION A contractor can build 11 different model homes. She only
has 4 lots. How many ways can she put a different house on each lot?
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)
LETTERS Each arrangement of the letters in the word quilt is placed on a
piece of paper. One paper is selected at random. Find each probability.
19. P(arrangement begins with q)
20. P(arrangement ends with lt)
21. GAMES Tanisha and Eric are playing Tic Tac Toe. Each player takes turns
placing an X or an O in any of the locations that are empty. How many
different ways can the first 3 moves of the game occur?
22. SOCCER The teams of the
%842!02!#4)#%
See pages 715, 735.
Self-Check Quiz at
tx.msmath3.com
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?
MLS
Eastern Conference
Chicago Fire
Columbus Crew
D.C. United
Kansas City Wizards
MetroStars
New England Revolution
Lesson 8-2 Permutations
Jonathan Daniel/Getty Images
423
H.O.T. Problems
23 FIND THE ERROR Daniel and Liana are evaluating P(7, 3). Who is correct?
Explain.
P(7, 3) = 7 · 6 · 5 · 4 · 3
= 2,520
P(7, 3) = 7 · 6 · 5
= 210
Daniel
Liana
24. SELECT A TECHNIQUE If P(9, 9) = 362,880, which technique would you use to
find P(10, 10)? Justify your selection. Then find P(10, 10).
mental math
number sense
estimation
25. CHALLENGE Compare P(n, n) and P(n, n - 1), where n is any whole number
greater than one. Explain your reasoning.
*/ -!4( Write a real-world problem for which you would use
(*/
83 *5*/(
26.
a permutation to solve. Then solve your problem.
27. How many seven-digit phone
28. The school talent show is featuring 13
numbers are available if a digit can be
used only once and the first digit
cannot be 0?
acts. In how many ways can the talent
show coordinator order the first 5 acts?
F 6,227,020,800
H 154,440
A 5,040
C 604,800
G 371,293
J
B 544,320
D 10,000,000
1,287
29. SPORTS The Seven Springs Ski Resort has 6 ski lifts up the mountain and
12 trails down the mountain. How many different ways can a skier take
a ski lift up the mountain and then ski down? (Lesson 8-1)
MM
MM
30. The cylinders at the right are similar.
Find the surface area of the larger cylinder.
Round to the nearest tenth. (Lesson 7-9)
MM
31. If one leg of a right triangle is 5 feet and its
hypotenuse is 13 feet, how long is the other leg?
PREREQUISITE SKILL Evaluate each expression.
32.
424
6·5·4
_
3·2·1
Chapter 8 Probability
(l)Comstock/SuperStock, (r) BananaStock/SuperStock
33.
10 · 9 · 8 · 7
_
4·3·2·1
(Lesson 3-5)
(Lesson 1-2)
34.
20 · 19
_
2·1
35.
6·5·4·3·2
__
5·4·3·2·1
8-3
Combinations
Main IDEA
Find the number of
combinations of objects.
Targeted TEKS 8.16
The student uses
logical reasoning to
make conjectures
and verify conclusions.
(A) Make conjectures from
patterns or sets of examples
and nonexamples.
Each member of the group should shake hands with every other
member of the group. Make a list of each handshake.
1. How many different handshakes did you record?
2. Is the number of handshakes equal to P(6, 2)? Explain.
In the Mini Lab, the order in which you shook hands was not important.
An arrangement or listing where order is not important is called a
combination.
Find a Combination
NEW Vocabulary
combination
A
B
1 GEOMETRY Four points are located on a circle.
How many line segments can be drawn with
these points as endpoints?
C
D
METHOD 1
First list all possible permutations of endpoints A, B, C, and D taken
two at a time. If any segments are the same, cross out one of them.
−− −− −−− −− −− −−
AB AC AD BA BC BD
6 different segments remain.
−− −− −−− −−− −− −−−
CA CB CD DA DB DC
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.
12
_
or 6
2·1
There are 6 line segments that can be drawn.
a. HANDSHAKES If there are 8 people in a room, how many
handshakes will occur if each person shakes hands with every
other person?
Personal Tutor at tx.msmath3.com
Lesson 8-3 Combinations
425
READING Math
C(a, b) C(a, b) is read the
number of combinations of
a things taken b at a time.
The symbol C(4, 3) represents the number of combinations of 4 things
taken 3 at a time.
the number of
combinations
of 4 things taken
3 at a time
P(4, 3)
C(4, 3) = _
3·2·1
the number of permutations
of 4 things taken 3 at a time
the number of ways 3
things can be arranged
Combinations and Permutations
MUSIC The makeup of a symphony is
shown in the table at the right.
2 A group of 3 musicians from the
Source: World Book
Number
Strings
strings section will talk to students
at Madison Middle School. Does
this represent a combination or a
permutation?
Real-World Link
The harp is one of
the oldest stringed
instruments. It is about
70 inches tall and has
47 strings.
Symphony Makeup
Instrument
45
Woodwinds
8
Brass
8
Percussion
3
Harps
2
This is a combination problem since the
order is not important.
3 How many possible groups could talk to the students?
45 · 44 · 43
C(45, 3) = _
3·2·1
15
22
45 · 44 · 43
=_
or 14,190
3·2·1
1
45 musicians taken 3 at a time
Divide out common factors.
1
There are 14,190 different groups that could talk to the students.
4 One member from the strings section will talk to students at Brown
Middle School, another to students at Oak Avenue Middle School,
and another to students at Jefferson Junior High. Does this
represent a combination or a permutation?
Since it makes a difference which member goes to which school,
order is important. This is a permutation.
5 How many possible ways can the strings members talk to the
students?
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.
b. FOOD At a restaurant, customers can select three hamburger
toppings: catsup, mustard, pickles, lettuce, onions, and cheese.
Suppose the layering of the toppings is not important. Does the
possible number of hamburgers with three toppings represent a
combination or permutation? How many three-topping
hamburgers are possible?
426
Chapter 8 Probability
Andy Sacks/Getty Images
Extra Examples at tx.msmath3.com
Example 1
(p. 425)
Examples 2–5
(p. 426)
1. GEOMETRY Eight points are located on a circle. How many line segments
can be drawn with these points as endpoints?
Determine whether each situation is a permutation or a combination. Then
find the number of possible outcomes.
2. writing a four-digit number using the numbers 0 through 9 with no digit
used more than once
3. choosing 3 shirts to pack for vacation from a choice of 7 shirts
(/-%7/2+ (%,0
For
Exercises
4–7
8–11
See
Examples
1
2–5
4. How many different combinations of 2 colors can be chosen as school
colors from a possible list of 8 colors?
5. Ten people come to a party. Everyone shakes hands with everyone else.
How many handshakes will take place?
6. How many three-topping pizzas can
be ordered from the list of toppings at
the right?
7. How many different starting squads
of 6 players can be picked from
10 volleyball players?
Pizza Toppings
anchovies
sausage
onions
bacon
green peppers
black olives
ham
hot peppers
green olives
pepperoni
mushrooms
pineapple
Determine whether each situation is a permutation or a combination. Then
find the number of possible outcomes.
8. choosing a committee of 5 from the 36 members of a class
9. choosing 2 co-captains of the basketball team with 14 players
10. choosing the placement of 9 model cars in a line from a selection of
12 models
11. choosing 4 out of 12 photographs to display at different locations
ROLLER COASTERS For Exercises 12 and 13, use the following information.
An amusement park has 16 roller coasters. Suppose you have time to ride only
half of the coasters.
12. How many ways are there to ride half of the roller coasters if order
is important?
13. How many ways are there to ride half of the roller coasters if order is
%842!02!#4)#%
See pages 715, 735.
Self-Check Quiz at
tx.msmath3.com
not important?
14. RADIO Twenty listeners called into a radio show to request their favorite
songs, and each song was different. The DJ only has time to play 12 of the
songs. Determine whether this situation represents a permutation or a
combination. Then determine the number of possible song selections.
Lesson 8-3 Combinations
427
H.O.T. Problems
15. OPEN ENDED Describe a situation that could be represented by C(15, 5).
16. Which One Doesn’t Belong? Identify the situation that is does not belong with
the other three. Explain your reasoning.
choosing
3 toppings
for the pizzas
to be served
at the party
choosing
3 members
for the
decorating
committee
choosing
3 people
to chair
3 different
committees
choosing
3 desserts
to serve at
the party
17. CHALLENGE Is the value of P(x, y) sometimes, always, or never greater than the
value of C(x, y)? Explain. Assume x and y are positive integers and x ≥ y.
*/ -!4( Give an example of a situation in which you would
(*/
83 *5*/(
18.
use a combination. Then, change the situation so that you need to use a
permutation. Explain the difference between the situations.
19. Which situation is represented by
20. The enrollment for Centerville Middle
C(8, 3)?
School is shown. How many different
four-person committees could be
formed from the students in the
8th grade?
A the number of arrangements of
8 people in a line
B the number of ways to pick 3 out of
8 vegetables to add to a salad
Centerville Middle School
C the number of ways to pick 3 out
of 8 students to be the first, second,
and third contestant in a
spelling bee
D the number of ways 8 people can
sit in a row of 3 chairs
Find each value.
21. P(7, 2)
Class
Boys
Girls
6th grade
42
47
7th grade
55
49
8th grade
49
53
F 211,876
H 4,249,575
G 292,825
J
149,059,680
(Lesson 8-2)
22. P(15, 4)
23. P(20, 5)
24. P(7, 7)
25. SCHOOL At the school cafeteria, students can choose from 4 entrees and
3 beverages. How many different lunches of one entree and one beverage
can be purchased at the cafeteria? (Lesson 8-1 )
PREREQUISITE SKILL Multiply. Write in simplest form.
26.
428
_4 · _3
5
8
Chapter 8 Probability
27.
3 _
_
·5
10
6
28.
(Lesson 2-3)
7 _
_
· 3
12
14
29.
9
_2 · _
3
10
8-4
Probability of
Composite Experiments
Main IDEA
Find the probability
of independent and
dependent events.
GAMES A game uses a number
cube and the spinner shown.
red
2 1
1. A player rolls the number cube.
Targeted TEKS 8.11
The student applies
concepts of
theoretical and
experimental probability to
make predictions. (A) Find
the probabilities of
dependent and independent
events. (B) Use theoretical
probabilities and
experimental results to make
predictions and decisions.
NEW Vocabulary
composite experiment
independent events
dependent events
Vocabulary Link
Independent
Everyday Use not under
the control of others
Math Use not relying on
another quantity or action
blue
green
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 player
will roll an odd number and spin red.
The combined action of rolling a number cube and spinning a spinner is
a composite experiment. In general, a composite experiment consists of
two or more simple events.
The outcome of the spinner does not depend on the outcome of the
number cube. These events are independent. For independent events,
the outcome of one event does not affect the other event.
+%9#/.#%04
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)
Probability of Independent Events
1 The two spinners are spun.
What is the probability that
both spinners will show an
even number?
7
2
6
5
3
P(first spinner is even) = _
8
1
4
3
1
7
2
6
3
5
4
7
1
P(second spinner is even) = _
2
3 _
3
P(both spinners are even) _
· 1 or _
7
2
14
Use the above spinners to find each probability.
a. P(both show a 2)
b. P(both are less than 4)
Lesson 8-4 Probability of Composite Experiments
429
2 A spinner and a number cube are used in a game. The spinner has an
Mental Math You
may wish to simplify
individual probabilities
before multiplying
them.
equal chance of landing on one of five colors: red, yellow, blue, green,
and purple. The faces of the cube are labeled 1 through 6. What is the
probability of a player spinning blue and then rolling a 3 or 4?
3
A _
1
B _
11
1
C _
4
1
D _
30
15
Read the Test Item
You are asked to find the probability of the spinner landing on blue
and rolling a 3 or 4 on a number cube. The events are independent
because spinning the spinner does not affect the outcome of rolling
a number cube.
Solve the Test Item
First, find the probability of each event.
number of ways to spin blue
___
1
P(blue) = _
5
number of possible outcomes
2
1
P(3 or 4) = _
or _
6
3
number of ways to roll 3 or 4
___
number of possible outcomes
Then, find the probability of both events occurring.
1 _
P(blue and 3 or 4) = _
·1
P(A and B) = P(A) · P(B)
5 3
_
= 1
Multiply.
15
1
The probability is _
, which is answer C.
15
c. A game requires players to roll two fair number cubes to move the
game pieces. The faces of the cube are labeled 1 through 6. What is
the probability of rolling a 2 or 4 on the first number cube and then
rolling a 5 on the second?
1
F _
3
1
G _
2
1
H _
12
J
1
_
18
Personal Tutor at tx.msmath3.com
If the outcome of one event affects the outcome of another event, the
compound events are called dependent events.
Vocabulary Link
Dependent
Everyday Use under the
control of others
+%9#/.#%04
Words
If two events, A and B, are dependent, then the probability of
both events occurring is the product of the probability of A and
the probability of B after A occurs.
Symbols
P(A and B) = P(A) · P(B following A)
Math Use relying on
another quantity or action
430
Chapter 8 Probability
Probability of Dependent Events
Probability of Dependent Events
3 There are 2 white, 8 red, and 5 blue marbles
BrainPOP® tx.msmath3.com
in a bag. Once a marble is selected, it is not
replaced. Find the probability that two red
marbles are chosen.
Since the first marble is not replaced, the first
event affects the second event. These are
dependent events.
8
P(first marble is red) = _
number of red marbles
total number of marbles
15
number of red marbles after
one red marble is removed
total number of marbles after
one red marble is removed
7
P(second marble is red) = _
14
4
1
15
14
8 _
4
· 7 or _
P(two red marbles) = _
7
15
1
Refer to the situation above. Find each probability.
d. P(two blue marbles)
e. P(a white marble and then a blue marble)
f. P(a red marble and then a white marble)
g. P(two white marbles)
Example 1
(p. 429)
Example 2
(p. 430)
A penny is tossed and a number cube is rolled. Find each probability.
1. P(tails and 3)
3.
TEST PRACTICE A spinner and a number cube are used in a game. The
spinner has an equal chance of landing on 1 of 3 colors: red, yellow, and
blue. The faces of the cube are labeled 1 through 6. What is the probability
of a player spinning red and then rolling an even number?
2
A _
5
Example 3
(p. 431)
2. P(heads and odd)
1
B _
1
C _
3
6
1
D _
12
A card is drawn from the cards shown and
not replaced. Then, a second card is drawn.
Find each probability.
4. P(two even numbers)
5. P(a number less than 4 and then a
number greater than 4)
Extra Examples at tx.msmath3.com
Lesson 8-4 Probability of Composite Experiments
431
(/-%7/2+ (%,0
For
Exercises
6–11
12, 13
14–19
See
Examples
1
2
3
A number cube is rolled, and the spinner at the right
is spun. Find each probability.
A
6. P(1 and A)
7. P(3 and B)
8. P(even and C)
9. P(odd and B)
10. P(greater than 2 and A)
B
B
C
B
11. P(less than 3 and B)
12. LAUNDRY A laundry basket contains 18 blue socks and 24 black socks. What
is the probability of randomly picking 2 black socks from the basket?
13. GAMES Beth is playing a board game that requires rolling two number
cubes to move a game piece. She needs to roll a sum of 6 on her next turn
and then a sum of 10 to land on the next two bonus spaces. What is the
probability that Beth will roll a sum of 6 and then a sum of 10 on her next
two turns?
A jar contains 3 yellow, 5 red, 4 blue, and 8 green candies. After a candy is
selected, it is not replaced. Find each probability.
14. P(two red candies)
15. P(two blue candies)
16. P(a yellow candy and then
17. P(a green candy and then a
a blue candy)
red candy)
18. P(two candies that are not green) 19. P(two candies that are neither blue
nor green)
20. MARKETING A discount supermarket has found that 60% of their customers
spend more than $75 each visit. What is the probability that the next two
customers will each spend more than $75?
SCHOOL For Exercises 21 and 22, use the
information below and in the table.
Clearview Middle School
At Clearview Middle School, 56% of the
students are girls and 44% are boys.
Art
16%
Language Arts
13%
21. If two students are chosen at random,
Math
28%
Music
7%
what is the probability that the first
student is a girl and that the second
student’s favorite subject is science?
Favorite Subject
Science
21%
Social Studies
15%
22. What is the probability that of two randomly
selected students, one is a boy and the other
is a student whose favorite subject is not art or math?
23. MOVIES You and a friend plan to see 2 movies over the weekend. You
can choose from 6 comedy, 2 drama, 4 romance, 1 science fiction, or
3 action movies. You write the movie titles on pieces of paper and place
them in a bag, and you each randomly select a movie. What is the
probability that neither of you selects a comedy? Is this a dependent or
independent event? Explain.
432
Chapter 8 Probability
%842!02!#4)#% 24. MONEY Donoma had 8 dimes and 6 pennies in her pocket. If she took
out 1 coin and then a second coin without replacing the first, what is the
See pages 715, 735.
probability that both coins were dimes? Is this a dependent or independent
event? Explain.
Self-Check Quiz at
tx.msmath3.com
POPULATION For Exercises 25 and 26,
use the information in the table.
Texas Population
Demographic
Group
Assume that age is not dependent on the
region.
Fraction of the
Population
_3
10
_3
5
_1
10
_4
5
_1
Under age 18
25. If a Texan is picked at random, what is
18 to 64 years old
the probability that the person is under
18 years old or 18 to 64 years old and
from a rural area?
65 years or older
26. What is the probability that a person
Urban Area
from Texas selected randomly is less
than 18 years old or 65 years or older
and from an urban area?
Rural Area
5
Source: U.S. Census Bureau
27. CONTESTS A car dealer is giving away a new car to one of 10
contestants. Each contestant randomly selects a key from 10 keys,
with only 1 winning key. What is the probability that none of the first
three contestants selects the winning key?
28. DOMINOES A standard set of dominoes contains 28 tiles, with each tile
Real-World Link
The game of
dominoes is believed
to have originated in
12th century China.
having two sides of dots from 0 to 6. Of these tiles, 7 have the same number
of dots on each side. If four players each randomly choose a tile, what is
the probability that each chooses a tile with the same number of dots on
each side?
29. WEATHER A weather forecaster states that there is an 80% chance of rain on
Monday and a 30% chance of rain on Tuesday. What is the probability of it
raining on Monday and Tuesday? Assume these are independent events.
Source: infoplease.com
30.
H.O.T. Problems
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would find a
compound probability.
31. OPEN ENDED There are 9 marbles representing 3 different colors. Write a
problem where 2 marbles are selected at random without replacement and
1
the probability is _
.
6
32. 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.
9
_3 · _3 = _
6
_3 · _2 = _
Evita
Tia
5
5
25
5
4
10
Lesson 8-4 Probability of Composite Experiments
David Muir/Masterfile
433
33. CHALLENGE Determine whether the following statement is true or false.
If the statement is false, provide a counterexample.
If two events are independent, then the probability of both events is less than 1.
*/ -!4( Compare and contrast independent events and
(*/
83 *5*/(
34.
dependent events.
35. Jeremy tossed 5 coins What is the
36. The spinners below are each spun
probability that each coin landed on
tails?
once.
1
A _
10
1
B _
25
1
C _
32
1
D _
64
2%$
2%$
7()4%
",5%
What is the probability of spinning
2 and white?
1
F _
16
1
G _
4
2
H _
J
5
_3
5
37. GOVERNMENT Five people are elected to the city council. From those 5
people, how many ways can a 2-person committee be chosen?
(Lesson 8-3)
38. SPORTS There are 10 players on a softball team. How many ways can a
coach pick the first 3 batters?
(Lesson 8-2)
Find the volume of each solid described. Round to the nearest tenth
if necessary. (Lessons 7-5 and 7-6)
39. rectangular pyramid: length, 14 m; width, 12 m; height 7 m
40. cone: diameter, 22 cm; height, 24 cm
41. Graph polygon ABCDE with vertices A(-5, -3), B(-2, 1), C(-3, 4),
D(0, 2), and E(0, -3). Then graph the image of the figure after a reflection
over the y-axis and write the coordinates of its vertices. (Lesson 6-5)
PREREQUISITE SKILL Write each fraction in simplest form.
42.
434
52
_
120
Chapter 8 Probability
43.
33
_
90
44.
49
_
70
45.
24
_
88
CH
APTER
Mid-Chapter Quiz
8
Lessons 8-1 through 8-4
1. BREAKFAST Draw a tree diagram to
14.
determine the number of one-bread and
one-beverage outcomes using the breakfast
choices listed below. (Lesson 8-1)
TEST PRACTICE Roman has ten cards
numbered 1 to 10. What is the probability of
picking two even-numbered cards one after
the other, if the first card picked is replaced?
(Lesson 8-4)
"REAKFAST#HOICES
1
A _
TOAST
COFFEE
2
B _
MUFFIN
MILK
5
BAGEL
JUICE
9
1
C _
4
3
D _
2. FASHION Reina has three necklaces, three
pairs of earrings, and two bracelets. How
many combinations of the three types of
jewelry are possible? (Lesson 8-1)
3. MUSIC Five band members play the flute.
How many ways can these members be
chosen for the first, second, and third chairs
of the flute section? (Lesson 8-2)
8
A box contains 3 purple, 2 yellow, 4 pink,
3 orange, and 2 blue markers. Once a marker
is selected, it is not replaced. Find each
probability. (Lesson 8-4)
15. P(two purple markers)
16. P(two orange markers)
17. P(a pink marker then an orange marker)
Find each value.
(Lessons 8-2 and 8-3)
4. P(5, 3)
5. P(6, 2)
6. P(5, 5)
7. P(4, 1)
8. C(5, 3)
9. C(6, 2)
10. C(5, 5)
11. C(7, 6)
18. P(two markers that are not blue)
19. P(two markers that are neither yellow
nor pink)
20. P(two markers that are neither purple
nor pink)
21.
12. SCHOOL How many ways can 2 student
council members be elected from 7
candidates? (Lesson 8-3)
13. FLOWERS A florist is making flower
arrangements by placing three differentcolored flowers in a vase. The color choices
are red, white, pink, yellow, and orange. If
an arrangement is selected at random, what
is the probability that the flowers are red,
pink, and white? (Lesson 8-3)
TEST PRACTICE A bag contains 4 red,
20 blue, and 6 green marbles. Seth picks one
at random and keeps it. Then Amy picks a
marble. What is the probability that they
each select a red marble? (Lesson 8-4)
1
F _
150
1
G _
15
2
H _
145
J
1
_
870
8-5
Experimental and
Theoretical Probability
Interactive Lab tx.msmath3.com
Main IDEA
Find experimental and
theoretical probabilities
and use them to make
predictions.
Targeted TEKS 8.11
The student applies
concepts of
theoretical and
experimental probability
to make predictions. (A)
Find the probabilities of
dependent and independent
events. (B) Use theoretical
probabilities and
experimental results to
make predictions and
decisions.
NEW Vocabulary
experimental probability
theoretical probability
Draw one marble from a bag containing 10 different-colored marbles.
Record its color, and replace it in the bag. Repeat 50 times.
1. Find the ratio ___ for each color.
number of times color was drawn
total number of draws
2. Is it possible to have a certain color marble in the bag and
never draw that color?
3. Open the bag and count the marbles. Find the ratio
number of each color marble
___
for each color of marble.
total number of marbles
4. Are the ratios in Exercises 1 and 3 the same? Explain.
In the Mini Lab above, you determined a probability by conducting an
experiment. Probabilities that are based on the outcomes obtained by
conducting an experiment are called experimental probabilities.
Probabilities based on known characteristics or facts are called
theoretical probabilities. 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 and Experimental Probability
1 What is the theoretical probability of rolling a double 6 using two
number cubes?
1 _
1
The theoretical probability is _
· 1 or _
.
6
6
36
2 The graph shows the results of
an experiment in which two
number cubes were rolled.
According to the experimental
probability, is a sum of 12 likely
to occur?
Only 1 of the 58 sums is 12. So,
the experimental probability of
2ESULTSOF2OLLING
4WO.UMBER#UBES
.UMBEROF2OLLS
Experimental
Probability
Experimental
probabilities usually
vary depending on
the number of trials
performed or when
the experiment is
repeated.
3UM
1
rolling a sum of 12 is _
. It is not likely that a sum of 12 will occur.
58
a. Refer to the graph above. According to the experimental
probability, which sum is most likely to occur?
436
Chapter 8 Probability
3 MARKETING Two hundred teenagers
were asked whether they purchased
certain items in the past year. What
is the experimental probability that
a teenager bought a photo frame in
the last year?
Item
Number Who
Purchased the Item
candle
110
photo frame
95
There were 200 teenagers surveyed and 95 purchased a photo frame
95
19
or _
.
in the last year. The experimental probability is _
200
Real-World Career
How Does a Marketing
Manager Use Math?
A marketing manager
uses information
from surveys and
experimental
probability to help
make decisions about
changes in products
and advertising.
40
b. What is the experimental probability that a teenager bought a
candle in the last year?
Personal Tutor at tx.msmath3.com
You can use past performance to predict future events.
Use Probability to Predict
For more information,
go to tx.msmath3.com.
4 FARMING Over the last 10 years, the probability that soybean seeds
10
planted by Ms. Diaz produced soybeans is _
.
13
Is this probability experimental or theoretical? Explain.
This is an experimental probability since it is based on what
happened in the past.
If Ms. Diaz wants to have 10,000 soybean-bearing plants, how
many seeds should she plant?
This problem can be solved using a proportion.
10 out of 13 seeds should
produce soybeans.
Mental Math
For every 10
soybean-bearing
plants, Ms. Diaz
must plant 3 extra
seeds. Think:
10,000 ÷ 10 = 1,000
Ms. Diaz must plant
3 × 1,000 or 3,000
extra seeds. She
must plant a total of
10,000 + 3,000 or
13,000 seeds.
10,000
10
_
=_
13
x
10,000 out of x seeds
should produce soybeans.
Solve the proportion.
10,000
10
_
=_
x
13
Write the proportion.
10 · x = 13 · 10,000
Find the cross products.
10x = 130,000
Multiply.
130,000
10x
_
=_
Divide each side by 10.
10
10
x = 13,000
Ms. Diaz should plant 13,000 seeds.
c. SURVEYS In a recent survey of 150 people, 18 responded that they
were left-handed. If an additional 2,500 people are surveyed, how
many would be expected to be left-handed?
Extra Examples at tx.msmath3.com
LWA–Dann Tardif/CORBIS
Lesson 8-5 Experimental and Theoretical Probability
437
Example 1
(p. 436)
For Exercises 1–3, use the table that
shows the results of tossing three
coins, one at a time, 50 times.
1. What is the theoretical
probability of tossing
exactly two heads?
Example 2
(p. 436)
Result
Frequency
Result
Frequency
HHH
6
TTT
3
HHT
5
TTH
6
HTH
10
THT
5
HTT
5
THH
10
2. Find the experimental probability of tossing exactly two heads.
3. How likely is it that a toss will have two heads? Explain.
For Exercises 4 and 5, use the table at the right
showing the results of a survey of cars that
passed the school.
Example 3
(p. 437)
Example 4
(p. 437)
(/-%7/2+ (%,0
For
Exercises
6, 9
8, 11
7, 10
See
Examples
1, 2
3
4
4. What is the probability that the next car will
be white?
5. Out of the next 180 cars, how many would
Cars Passing the School
Color
Number of Cars
white
35
red
23
green
12
other
20
you expect to be white?
SCHOOL For Exercises 6 and 7, use the following information.
In keyboarding class, 4 out of the 60 words Cleveland typed contained an error.
6. What is the probability that his next word will have an error?
7. In a 1,000-word essay, how many errors would you expect Cleveland
to make?
8. BASKETBALL In practice, Crystal made 80 out of 100 free throws. What is the
experimental probability that she will make a free throw?
FOOD For Exercises 9 and 10, use the results of
a survey of 150 people shown at the right.
9. What is the probability that a person’s
favorite fruit was bananas?
Favorite Fruit
Fruit
Number
apples
55
bananas
40
10. Out of 450 people, how many would you
oranges
35
expect to state that bananas are their
favorite fruit?
grapes
15
other
5
11. SCHOOL In the last 40 school days, Esteban’s bus has been late 8 times.
What is the experimental probability that the bus will be late tomorrow?
12. SPORTS In a survey of 90 students at Genoa Middle School, 42 liked to
watch basketball and 24 liked to watch soccer. If there are 300 students in
the middle school, how many would you expect to like to watch soccer?
438
Chapter 8 Probability
For Exercises 13–15, use the table that shows
the results of spinning an equally divided
8-section spinner.
13. Compare the theoretical and experimental
probabilities of the spinner landing on 5.
Number on
Spinner
Frequency
1
8
2
5
3
9
4
4
5
10
6
6
7
5
8
3
14. Based on the experimental probability, how
many times would you expect the spinner
to land on 3 if the spinner is spun 200 times?
15. Jarred predicts that the spinner will land on
4 or 8 on the next spin. Is this a reasonable
prediction? Explain.
BASEBALL For Exercises 16 and 17,
use the table which shows the
batting results of a baseball
player for a season.
3INGLE
$OUBLE
16. Based on the results, how likely
4RIPLE
is it that the player would be out
after his next turn batting?
17. The next time the player is at bat,
how likely is it for him to hit a
single or a double?
2ESULT
(OME2UN
7ALK
/UT
FOOD For Exercises 18 and 19, use the following information.
The manager of a school cafeteria asked selected
Menu Item
students to pick their favorite menu item. The
Hot Dog
results of the survey are shown in the table.
18. If the cafeteria serves 350 lunches, and
students can choose only one lunch, how
many hamburgers could the manager
expect to sell?
%842!02!#4)#%
See pages 716, 735.
&REQUENCY
Students
22
Hamburger
19
Pizza
30
Taco
16
Chicken Strips
13
19. Is the next student more likely to buy a
Self-Check Quiz at
hot dog or a hamburger, or is the student
more likely to buy pizza? Explain.
tx.msmath3.com
H.O.T. Problems
20. OPEN ENDED Two hundred fifty people are surveyed about their favorite
color. Make a table of possible results if the experimental probability that
the favorite color is blue is 40%.
21. CHALLENGE An inspector found that 15 out of 250 cars had a loose front
door and that 10 out of 500 cars had headlight problems. What is the
probability that a car has both a loose door and a headlight problem?
22.
*/ -!4( Explain why you would not expect the theoretical
(*/
83 *5*/(
probability of an event and the experimental probability of the same event
to always be the same.
Lesson 8-5 Experimental and Theoretical Probability
439
23. Two number cubes are rolled and the
24. Shannon spun the spinner shown and
difference is recorded. The graph
shows the results of several rolls.
recorded her results.
Number
on
Frequency
Spinner
.UMBEROF2OLLS
$IFFERENCEOF2OLLING
4WO.UMBER#UBES
1
20
2
10
3
2
4
40
5
8
$IFFERENCE
What is the experimental probability
Based on past results, what is the
probability that the difference is 2?
7
A _
of landing on the number five?
1
F _
11
C _
20
1
B _
20
3
H _
10
_
G 1
5
50
1
D _
25
10
2
J _
5
A jar contains 3 red marbles, 4 green marbles, and 5 blue marbles.
Once a marble is selected, it is not replaced. Find each probability.
25. 2 green marbles
(Lesson 8-4)
26. a blue marble and then a red marble
27. FOOD Pepperoni, mushrooms, onions, and green peppers can be added to
a basic cheese pizza. How many 2-item pizzas can be prepared?
Find the area of each figure. Round to the nearest tenth.
28.
29.
FT
(Lesson 8-3)
(Lesson 7-2)
M
M
M M
M
FT
M
30. SAVINGS Shala’s savings account earned $4.56 in 6 months at a simple
interest rate of 4.75%. How much was in her account at the beginning of
that 6-month period? (Lesson 5-9)
31. PREREQUISITE SKILL Lawanda was assigned some math exercises for
homework. She answered half of them in study period. After school, she
answered 7 more exercises. If she still has 11 exercises to complete, how
many exercises were assigned? Use the work backward strategy. (Lesson 1-8)
440
Chapter 8 Probability
Extend
8-5
Main IDEA
Use experimental and
theoretical probabilities
to decide whether a
game is fair.
Targeted TEKS 8.11
The student applies
concepts of
theoretical and
experimental probability to
make predictions. (B) Use
theoretical probabilities and
experimental results to
make predictions and
decisions.
Probability Lab
Fair Games
Mathematically speaking, a two-player game is fair if each player has
an equally-likely chance of winning. In this lab, you will analyze two
simple games and determine whether each game is fair.
1 In a counter-toss game, players toss three
two-color counters. The winner of each
game is determined by how many counters
land with either the red or yellow side
facing up. Play this game with a partner.
Player 1 tosses the counters. If 2 or 3 chips land red-side
up, Player 1 wins. If 2 or 3 chips land yellow-side up,
Player 2 wins. Record the results in a table like the one
shown below. Place a check in the winner’s column for
each game.
Game
Player 1
Player 2
1
2
Player 2 then tosses the counters and the results are
recorded.
Continue alternating the tosses until each player has
tossed the counters 10 times.
ANALYZE THE RESULTS
1. Make an organized list all of the possible outcomes resulting from
one toss of the 3 counters. Explain your method.
2. Calculate the theoretical probability of each player winning. Write
each probability as a fraction and as a percent.
3. MAKE A CONJECTURE Based on the theoretical probabilities of each
player winning, is this a fair game? Explain your reasoning.
4. Calculate the experimental probability of each player winning. Write
each probability as a fraction and as a percent.
5. Compare the probabilities in Exercises 2 and 4.
6. GRAPH THE DATA Make a graph of the experimental probabilities of
Player 1 winning for 5, 10, 15, and 20 games. Graph the ordered pairs
(games played, Player 1 wins) using a blue pencil, pen, or marker.
Describe how the points appear on your graph.
Extend 8-5 Probability Lab: Fair Games
441
7. Add to the graph you created in Exercise 7 the theoretical
probabilities of Player 1 winning for 5, 10, 15, and 20 games. Graph
the ordered pairs (games played, Player 1 wins) using a red pencil,
pen, or marker. Connect these red points and describe how they
appear on your graph.
8. As the number of games played increases, how does the experimental
probability compare to the theoretical probability?
9. MAKE A PREDICTION Predict the number of times Player 1 would win
if the game were played 100 times.
2 In a number-cube game, players
roll two number cubes. Play this
game with a partner.
Player 1 rolls the number cubes. Player 1 wins if the total
of the numbers rolled is 5 or if a 5 is shown on one
number cube. Otherwise, Player 2 wins. Record the results
in a table like the one shown below.
Game
Player 1
Player 2
1
2
Player 2 then rolls the number cubes and the results are
recorded.
Continue alternating the rolls until each player has rolled
the number cubes 10 times.
ANALYZE THE RESULTS
10. Make an organized list of all the possible outcomes resulting from
one roll. Explain your method.
11. Calculate the theoretical probability of each player winning and
the experimental probability of each player winning. Write each
probability as a fraction and as a percent. Then compare these
probabilities.
12. MAKE A CONJECTURE Based on the theoretical and experimental
probabilities of each player winning, is this a fair game? Explain your
reasoning.
13.
*/ -!4( If the game is fair, explain how you could
(*/
83 *5*/(
change the game so that it is not fair. If the game is not fair,
explain how you could change the game to make it fair. Explain.
442
Chapter 8 Probability
8-6
Problem-Solving Investigation
MAIN IDEA: Solve problems by acting it out.
Targeted TEKS 8.14 The student applies Grade 8 mathematics to solve problems connected to everyday experiences,
investigations in other disciplines, and activities in and outside of school. (C) Select or develop an appropriate problem-solving
strategy from a variety of different types, including … acting it out, … to solve a problem.
e-Mail:
ACT IT OUT
YOUR MISSION: Act it out to solve the problem.
THE PROBLEM: Is tossing a coin a good way to
answer a true–false quiz?
EXPLORE
PLAN
SOLVE
CHECK
▲
Bonita: I wonder if tossing a coin would
be a good way to answer a 5–question true–
false quiz.
You know there are five true-false questions on the quiz. You can carry out an
experiment to test if tossing a coin would be a good way to answer the
questions and get a good grade.
Toss a coin 5 times. If the coin shows tails, the answer is T. If the coin shows
heads, the answer is F. Do three trials.
Suppose the correct answers
Number
Answers T
F
F
T
F
are T, F, F, T, F. Let’s circle
Correct
them in each trial.
Trial 1
T
T
F
F
T
2
Trial 2
F
F
T
T
F
3
Trial 3
T
F
T
F
T
2
Since the experiment produced 2–3 correct answers on a 5-question quiz, it
shows that tossing a coin to answer a true-false quiz is not the way to get a
good grade.
Check by doing several more trials.
1. Explain an advantage of using the act it out strategy to solve a problem.
2.
*/ -!4( Write a problem that could be solved by acting it out.
(*/
83 *5*/(
Then use the strategy to solve the problem. Explain your reasoning.
Lesson 8-6 Problem-Solving Investigation: Act it Out
Laura Sifferlin
443
For Exercises 3–5, solve using the act it out
strategy.
3. COINS Nina wants to buy a granola bar from
a vending machine. The granola bar costs
$0.45. If Nina uses exact change, in how
many different ways can she use nickels,
dimes, and quarters?
4. FITNESS The length of a basketball court is
84 feet long. Hector runs 20 feet forward
and then 8 feet back. How many more times
will he have to do this until he reaches the
end of the basketball court?
5. PHOTOGRAPHS Omar is taking a picture of
the French Club’s five officers. The club
secretary will always stand on the left and
the treasurer will always stand on the right.
How many different ways can he arrange
the officers in a single row for the picture?
8. MONEY Carmen received money for a
birthday gift. She loaned $5 to her sister
Emily and spent half of the remaining
money. The next day she received $10 from
her uncle. After spending $9 at the movies,
she still had $11 left. How much money did
she receive for her birthday?
9. UNIFORMS Nick has to wear a uniform
to school. He can wear either navy blue,
black, or khaki pants with a green, white,
or yellow shirt. How many uniform
combinations can Nick wear?
10. COOKING The graph below shows the
number of types of outdoor grills sold. How
does the number of charcoal grills compare
to the number of gas grills?
Charcoal
Use any strategy to solve Exercises 6–10. Some
strategies are shown below.
7.9
G STRATEGIES
PROBLEM-SOLVIN
tep plan.
• Use the four-s
.
• Work backward
rn.
• Look for a patte
ing.
• Logical reason
• Act it out.
6. DESIGN Mrs. Lopez is designing her garden
in the shape of a rectangle. The width of her
1
garden is 2_
times greater than the width of
4
the rectangle shown. Find the perimeter of
Mrs. Lopez’s garden.
Gas
4.3
Millions of
Grills Sold
Electric
0.16
Source: Barbecue Industry Association
For Exercises 11–13, select the appropriate
operation(s) to solve the problem. Justify your
selection(s) and solve the problem.
11. SHOPPING Rita is shopping for fishing
equipment. She has $135 and has already
selected items that total $98.50. If the sales
tax is 8%, will she have enough money to
purchase a fishing net that costs $23?
12. TIME At 2:00 P.M., Cody began writing the
FT
FT
7. PATTERNS Complete the pattern.
100, 98, 94,
444
, 80,
.
Chapter 8 Probability
final draft of a report. At 3:30 P.M., he had
written 5 pages. If he works at the same
pace, when should he complete 8 pages?
13. GEOMETRY The length of a rectangle is
8 inches longer than its width. What are the
length and width of the rectangle if the area
is 84 square inches?
8-7
Simulations
Main IDEA
Perform probability
simulations to model
real-world situations
involving uncertainty.
Targeted TEKS 8.11
The student applies
concepts of
theoretical and
experimental probability to
make predictions. (C) Select
and use different models to
simulate an event.
You can act out rolling a number cube
50 times by using the random number
generator on a TI-83/84 Plus graphing
calculator. Enter 1 as the lower bound
and 6 as the upper bound for 50 trials.
Keystrokes:
50
5 1
,
6
,
ENTER
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.
1. Determine the experimental probability of each number generated
NEW Vocabulary
simulation
on the graphing calculator.
2. Compare the experimental probabilities found in Exercise 1 to the
theoretical probabilities of rolling an actual number cube.
In the Mini Lab above, you used a simulation to find the experimental
probability of rolling a number cube 50 times. A simulation is an
experiment that is designed to act out a given situation. Simulations
often use models to act out an event that would be impractical to perform.
Describe a Simulation
1 PRIZES A cereal company is placing one of eight different trading
Simulations The
objects you choose
for a simulation
should have the
same number of
outcomes as the
number of possible
outcomes of the
problem.
cards in its boxes of cereal. If each card is equally likely to appear
in a box of cereal, describe a model that could be used to simulate
the cards you would find in 15 boxes of cereal.
Choose a method that has 8 possible outcomes, such as tossing a coin
3 times. Let each outcome represent a different card.
Toss a coin 3 times to simulate
the cards that might be in 15
boxes of cereal. Repeat 15 times.
HHH
HHT
HTH
HTT
card 1
card 2
card 3
card 4
TTT
TTH
THT
THH
card 5
card 6
card 7
card 8
a. TOYS A restaurant is giving away 1 of 5 different toys with its
children’s meals. If the toys are given out randomly, describe a
model that could be used to simulate which toys would be given
with 6 children’s meals.
Extra Examples at tx.msmath2.com
Lesson 8-7 Simulations
445
2 SCHOOL Mr. Hawkins needs to choose 5 students at random from
his homeroom to serve on the school’s activity committee. If there
are 24 students in his homeroom, describe a model that he could
use to simulate choosing these 5 students.
There are 24 students, so select
objects that combined have 24
outcomes, such as a number cube
and a spinner divided into 4 equal
sections. Assign each student one
of the possible outcomes.
GREEN BLUE
) (
RED YELLOW
6 numbers · 4 colors = 24 outcomes
Mr. Hawkins should roll the number cube and spin the spinner at
least 5 times to choose the students.
b. CLOTHING Rodolfo must wear a dress shirt and a tie when he
works at the mall on Friday, Saturday, and Sunday. Each day he
picks one of his 5 dress shirts and 2 ties at random. Describe a
model that Rodolfo could use to simulate his selection of a shirt
and tie.
Simulations can also be used to model events in which the outcomes are
not equally likely.
3 WEATHER The weather forecast states that there is a 60% chance of
rain for each of the next two days. Describe a method you could use
to find the experimental probability of having rain on both of the
next two days. Assume the events are independent.
3
The probability of rain is 60% or _
, and the probability of no rain is
5
2
. Let 3 red marbles represent rain, and 2 blue marbles
40% or _
5
Real-World Link
A forecast stating that
there is a 60% chance
of rain means that
there is a 60% chance
of rain somewhere in
the forecast area, not
necessarily the
entire area.
Source: The National
Weather Service
represent no rain. Place the five marbles in a bag and randomly pick
one marble to simulate the first day. Replace the marble and pick
again to simulate the second day. Then, multiply the probabilities to
find the probability of rain on both days.
c. BASKETBALL During the regular season, Jason made 80% of his
free throws. Describe an experiment Jason could use to find the
experimental probability of making his next two free throws.
Personal Tutor at tx.msmath3.com
446
Chapter 8 Probability
Jeff Greenberg/Alamy Images
Example 1
(p. 445)
Example 2
(p. 446)
Example 3
(p. 446)
(/-%7/2+ (%,0
For
Exercises
4, 5
6, 7
8, 9
See
Examples
1
2
3
1. ACTIVITIES Jordan can choose among going to the football game, the
movies, shopping, or her friend’s house on the weekend. Describe a model
she could use to simulate her activities for the next 5 weekends.
2. ICE CREAM An ice cream store offers 6 premium flavors in a choice of waffle
cone or sugar cone. If each flavor and cone type is equally likely to be
chosen by a customer, describe a model that could be used to simulate the
orders of the next four customers.
3. SALES An electronics store has determined that of customers who buy a
television, 45% buy a wide-screen television. Describe a model that you
could use to find the experimental probability that the next three televisionbuying customers will buy a wide screen television.
4. TESTING The questions on a multiple-choice test each have 4 answer
choices. Describe a model that you could use to simulate the outcome of
guessing the correct answers to a 50-question multiple-choice test.
5. GAMES A game requires drawing balls numbered 0 though 9 for each of
four digits to determine the winning number. Describe a model that could
be used to simulate the selection of the number.
6. SNACKS A container of mixed nuts contains 18 different types of nuts.
Describe a model that could be used to simulate randomly selecting a
certain type of nut if each type is equally likely to be chosen.
7. PICNICS A cooler contains 15 bottles of lemonade, 12 bottles of water, and
9 bottles of fruit punch. Describe a model that you could use to simulate
randomly picking one type of drink.
8. CARNIVALS Players at a carnival game win about 30% of the time. Describe
a model that could be used to find the experimental probability that the
next four players will win.
9. WEATHER On average, 75% of the days in Waco, Texas, are sunny, with
little or no cloud cover. Describe a model that you could use to find the
experimental probability of sunny days each day for a week.
10. SCIENCE Suppose a mouse is placed in the
%842!02!#4)#%
See pages 716, 735.
Self-Check Quiz at
tx.msmath3.com
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.
)N
/UT
Lesson 8-7 Simulations
447
H.O.T. Problems
11. OPEN ENDED Describe a situation that could be represented by a simulation.
What objects could be used in this simulation?
12. CHALLENGE A simulation uses cards numbered 0 through 9 to generate
five 2-digit numbers. A card is selected for the tens digit and not replaced.
Then a card for the ones digit is drawn and not replaced. The process is
repeated until all the cards are used. If the simulation is performed 10
times, about how many times could you expect a 2-digit number to begin
with a 5? Explain.
*/ -!4( Explain how using a simulation is related to
(*/
83 *5*/(
13.
experimental probability.
14. Marcus placed 8 blue tiles and 12 red
15. Claire tosses a coin and rolls a number
tiles in a container. He plans to draw
a tile, record its color and replace it in
the container before drawing another.
If he does this 50 times, how many
times should he expect to draw a
red tile?
A 8
C 20
B 12
D 30
cube 100 times. How many times
should she expect to have the coin
show heads and roll a 1 or a 2?
F 17
G 33
H 50
J
66
16. FITNESS Roberto runs 10 meters forward and then 5 meters backward.
How many times will he need to repeat this to reach the end of a 100meter field? (Lesson 8-6)
17. GIVE-A-WAYS A local video store has advertised that one out of every
four customers will receive a free box of popcorn with their video
rental. So far, 15 out of 75 customers have received popcorn. Compare
the experimental and theoretical probabilities of receiving popcorn.
(Lesson 8-5)
18. FARMING When filled to capacity, a cylindrical silo can hold 8,042 cubic
feet of grain. The circumference C of the silo is approximately 50.3 feet.
Find the height h of the silo to the nearest foot. (Lesson 7-5)
19. GEOMETRY Copy the figure at the right onto graph paper. Then draw
the image of the figure after it is translated 4 units left and 2 units up.
(Lesson 6-7)
PREREQUISITE SKILL Solve each problem.
20. Find 35% of 90.
448
Chapter 8 Probability
(Lessons 5-3 and 5-7)
21. Find 42% of 340.
Extend
8-7
Main IDEA
Graphing Calculator Lab
Simulations
You can use a TI-83/84 Plus graphing calculator to simulate real-world
probability experiments.
Use technology to perform
probability simulations.
Targeted TEKS 8.11
The student applies
concepts of
theoretical and
experimental probability to
make predictions. (C) Select
and use different models to
simulate an event.
1 TESTING Ms. Mendez creates a test with 20 multiple choice
questions that each have 4 answer choices, A, B, C, or D. If the
correct answer choices are randomized, design a probability
simulation using a graphing calculator to determine the
probability that 60% or more of the correct answers will be C.
60
3
3
Since 60% = _
or _
, you want to find the probability that _
of 20 or
100
5
5
12 or more of the 20 correct answers are C. You can use a spinner
divided into 4 equal-size sections to simulate the randomizing of the
correct answer choices on the test.
Access the probability simulator by pressing APPS ALPHA
[P] and
until you find Prob Sim. Then press ENTER twice.
Press 4 to select Spin Spinner. Notice that the spinner is
already divided into 4 equal-size sections. For the purposes
of this simulation, let 1 represent a correct answer choice
of A, 2 represent a correct answer choice of B, 3 represent
a correct answer choice of C, and 4 represent a correct
answer choice of D.
Press F3 to change the setting from 1 to 20 trials. Then
press F5 to confirm this change and F2 to spin. Use the
and
keys to scroll through the results shown in the bar
graph. Record the frequency of the number of times the
number 3, answer choice C, was selected.
Return to the settings screen by pressing F3 and use
the , , and ENTER keys to select Yes next to ClearTbl.
Press F5 to confirm and F2 to spin again. Record the results
as before.
Repeat Step 4 until a total of 20 tests have been generated.
Extend 8-7 Graphing Calculator Lab: Simulations
449
ANALYZE THE RESULTS
1. What is the experimental probability of choice C being the correct
answer choice for 12 or more questions?
2. Based on your experimental probability, if a student taking
Ms. Mendez’s test hopes to pass with a score of 60%, would you
recommend guessing answer choice C for every problem? Explain.
2 TRAVEL The plane for a certain flight seats 72 passengers.
Based on previous experience, the airline knows that customers
who purchase tickets for this flight are 90% likely to check in.
If the airline overbooks, selling 76 tickets, design a probability
simulation to determine the probability that at most 72 passengers
will check in.
You need a simulation in which one outcome is 90% likely (checking
in) and another outcome is 10% likely (not checking in).
Access the probability simulator and select Toss Coins.
Press F3 to change the settings from 1 to 76 trials. Press F2
to select the advanced settings. Weight the coin tossed by
changing the probability of the coin landing on tails to 0.9.
Press F5 twice to confirm these changes.
Press F2 to begin the coin tosses. Use the
and
keys
to scroll through the results. Record the frequency of the
number of times the coin landed on tails.
Repeat this simulation 20 more times, clearing the table
before each set of coin tosses as you did in Activity 1.
ANALYZE THE RESULTS
3. What is the experimental probability that at most 72 passengers will
check in for this flight?
4. MAKE A CONJECTURE How many tickets should be sold if the airline
wants the probability of filling the flight (at least 72 check-ins) to be
95%? Use the probability simulation to check your conjecture.
450
Chapter 8 Probability
Other Calculator Keystrokes at tx.msmath3.com
8-8
Using Sampling
to Predict
Main IDEA
Predict the actions of a
larger group by using a
sample.
Targeted TEKS 8.13
The student
evaluates predictions
and conclusions
based on statistical
data. (A) Evaluate methods
of sampling to determine
validity of an inference
made from a set of data.
8.14 The student applies
Grade 8 mathematics to
solve problems connected
to everyday experiences,
investigations in other
disciplines, and activities in
and outside of school. (A)
Identify and apply
mathematics to everyday
experiences, to activities in
and outside of school, with
other disciplines, and with
other mathematical topics.
NEW Vocabulary
sample
population
unbiased sample
simple random sample
stratified random sample
systematic random sample
biased sample
convenience sample
voluntary response sample
ENTERTAINMENT The manager of a
television station wants to conduct a
survey to determine what type of
sports people like to watch.
1. Suppose she decides to survey a
group of people at a basketball
game. Do you think the results
would represent all of the people
in the viewing area? Explain.
What Type of Sports
Do You Like to Watch?
Baseball
Basketball
Football
Lacrosse
Soccer
2. Suppose she decides to survey students at your middle school.
Do you think the results would represent all of the people in the
viewing area? Explain.
3. Suppose she decides to call every 100th household in the telephone
book. Do you think the results would represent all of the people in
the viewing area? Explain.
The manager of the radio station cannot survey everyone in the
listening area. A smaller group called a sample is chosen. A sample is
representative of a larger group called a population.
For valid results, a sample must be chosen very carefully. An unbiased
sample is selected so that it is representative of the entire population.
Three ways to pick an unbiased sample are listed below.
#/.#%043UMMARY
Unbiased Samples
Type
Description
Example
Simple
Random
Sample
Each item or person in the
population is as likely to be
chosen as any other.
Each student’s name is
written on a piece of paper.
The names are placed in a
bowl, and names are picked
without looking.
Stratified
Random
Sample
The population is divided into
similar, non-overlapping
groups. A simple random
sample is then selected from
each group.
Students are picked at
random from each grade level
at a school.
Systematic
Random
Sample
The items or people are
Every 20th person is chosen
selected according to a specific from an alphabetical list of all
time or item interval.
students attending a school.
Lesson 8-8 Using Sampling to Predict
Chuck Savage/CORBIS
451
Vocabulary Link
Bias
Everyday Use a tendency
or prejudice.
Math Use error introduced
by selecting or encouraging
a specific outcome.
In a biased sample, one or more parts of the population are favored
over others. Two ways to pick a biased sample are listed below.
#/.#%043UMMARY
Biased Samples
Type
Description
Example
Convenience
Sample
A convenience sample
includes members of a
population that are easily
accessed.
To represent all the
students attending a
school, the principal
surveys the students
in one math class.
Voluntary
Response
Sample
A voluntary response
sample involves only those
who want to participate in
the sampling.
Students at a school
who wish to express their
opinion are asked to
complete an online survey
Determine Validity of Conclusions
Determine whether each conclusion is valid. Justify your answer.
1 To determine what videos their customers like, every tenth person
to walk into the video store is surveyed. Out of 150 customers, 70
stated that they prefer comedies. The manager concludes that about
half of all customers prefer comedies.
The conclusion is valid. Since the population is the customers of
the video store, the sample is a systematic random sample. It is an
unbiased sample.
2 To determine what people like to do in their leisure time, the
customers of a video store are surveyed. Of these, 85% said that
they like to watch movies, so the store manager concludes that
most people like to watch movies in their leisure time.
The conclusion is not valid. The customers of a video store probably
like to watch videos in their leisure time. This is a biased sample. The
sample is a convenience sample since all of the people surveyed are
in one specific location.
Determine whether each conclusion is valid. Justify your answer.
a. A radio station asks its listeners to call one of two numbers to
indicate their preference for two candidates for mayor in an
upcoming election. 72% of the listeners who responded preferred
candidate A, so the radio station announced that candidate A
would win the election.
b. To award prizes at a sold-out hockey game, four seat numbers are
picked from a barrel containing individual papers representing
each seat number. Tyler concludes that he has as good a chance as
everyone else to win a prize.
452
Chapter 8 Probability
If a sampling method is valid, you can use the results to make predictions.
Using Sampling to Predict
3 SCHOOL The school bookstore sells
Misleading
Probabilities
Probabilities based
on biased samples
can be misleading.
If the students
surveyed were all
boys, the probabilities
generated by the
survey would not
be valid, since both
girls and boys
purchase binders
at the store.
Color
sweatshirts in 4 different colors; red, black,
white, and gold. The students who run the
store survey 50 students at random. The
colors they prefer are indicated at the right.
If 450 sweatshirts are to be ordered to sell
in the store, how many should be white?
Number
red
25
black
10
white
13
gold
2
First, determine whether the sample method is valid. The sample is a
simple random sample since students were randomly selected. Thus,
the sample is valid.
13
_
or 26% of the students prefer white sweatshirts. So, find 26% of 450.
50
0.26 × 450 = 117
About 117 sweatshirts should be white.
c. RECREATION A swimming instructor at a community pool
asked her students if they would be interested in an advanced
swimming course, and 60% stated that they would. If there are
870 pool members, how many people can the instructor expect
to take the course?
Personal Tutor at tx.msmath3.com
Examples 1, 2
(p. 452)
Determine whether each conclusion is valid. Justify your answer.
1. To determine how much money the average family in the United States
spends to cool their home, a survey of 100 households from Alaska are
picked at random. Of the households, 85 said that they spend less than $75
a month on cooling. The researcher concluded that the average household
in the United Stated spends less than $75 on cooling per month.
2. To determine the benefits that employees consider most important, one
person from each department of the company is chosen at random.
Medical insurance was listed as the most important benefit by 67% of
the employees. The company managers conclude that medical insurance
should be provided to all employees.
Example 3
(p. 453)
3. ELECTIONS Three students are running for class
president. Jonathan randomly surveyed some
of his classmates and recorded the results at the
right. If there are 180 students in the class, how
many do you think will vote for Della?
Extra Examples at tx.msmath3.com
Candidate
Number
Luke
7
Della
12
Ryan
6
Lesson 8-8 Using Sampling to Predict
453
(/-%7/2+ (%,0
For
Exercises
4–9
10, 11
See
Examples
1, 2
3
Determine whether each conclusion is valid. Justify your answer.
4. To evaluate the quality of their product, a manufacturer of cell phones
pulls every 50th phone off the assembly line to check for defects. Out of 200
phones tested, 4 are defective. The manager concludes that about 2% of the
cell phones produced will be defective.
5. To determine whether the students will attend a spring music concert at the
school, Rico surveys his friends in the chorale. All of his friends plan to
attend, so Rico assumes that all the students at his school will also attend.
6. To determine the most popular television stars, a magazine asks its readers
to complete a questionnaire and send it back to the magazine. The majority
of those who replied liked one actor the most, so the magazine decides to
write more articles about that actor.
7. To determine what people in Texas think
about a proposed law, 2 people from
each county in the state are surveyed at
random. Of those surveyed, 42% said that
they do not support the proposal. The
legislature concludes that the law should
not be passed.
Do You Support
Proposed Law?
Yes
30%
No
42%
Not sure
28%
8. Two students need to be chosen to represent the 28 students in a science
class. The teacher decides to use a computer program to randomly pick
2 numbers from 1 to 28. The students whose names are next to those
numbers in his grade book will represent the class.
9. 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 the delivery
truck. None of the oranges are bad, so the manager concludes that all of the
oranges are fresh.
10. COMMUNICATION The Student Council advisor
asked every tenth student in the lunch line
how they preferred to be contacted with
school news. The results are shown in the
table. If there are 680 students at the school,
how many can be expected to prefer E-mail?
Real-World Link
63% of teens prefer
to use a telephone to
talk to their friends.
Source: Pew Internet &
American Life Project
Method
Number
Announcement
5
Newsletter
12
E-mail
16
Telephone
3
11. SALES A random survey of shoppers at a grocery store shows that
19 prefer whole milk, 44 prefer low-fat milk, and 27 prefer skim milk.
If 800 containers of milk are ordered, how many should be skim milk?
12. 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
your reasoning.
13. ACTIVITIES Brett wants to conduct a survey about who stays for
after-school activities. Describe a valid sampling method he could use.
454
Chapter 8 Probability
Michael Newman/PhotoEdit
14. Based on this survey, if the
manager orders 2,500 CDs,
how many pop/rock CDs
should be ordered?
15. Based on the survey results,
NUMBEROFRESPONSES
MUSIC For Exercises 14 and 15, use the following information.
The manager of a music store
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sent out 1,000 survey forms to
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households near her store. The
results of the survey are shown
Îää
in the graph at the right.
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the manager concludes that
25% of customers will buy
either rap/hip-hop or R&B/
urban CDs. Is this a valid
conclusion? Explain.
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4YPE
HOMEWORK A survey is to be conducted to find out how many hours
students at a school spend on homework each weekday. Describe the sample
and explain why each sampling method might not be valid.
16. A questionnaire is handed out to all students taking a world language.
17. The students from one homeroom from each grade level are asked to keep
a log for one week.
18. Students in a randomly selected Language Arts class are asked to discuss
their study habits in an essay.
19. Randomly selected parents are sent a questionnaire and asked to return it.
COLLECT THE DATA For Exercises 20–23, conduct a survey of the students in
your math class to determine whether they prefer hamburgers or pizza.
20. What percent prefer hamburgers?
21. Use your survey to predict how many students in your school prefer
hamburgers.
22. Is your survey a good way to determine the preferences of the students in
your school? Explain.
%842!02!#4)#% 23. How could you improve your survey?
See pages 717, 735.
24.
Self-Check Quiz at
tx.msmath3.com
H.O.T. Problems
FIND THE DATA Refer to the Texas Data File on pages 16–19. Choose
some data and write a real-world problem in which you would make a
prediction based on samples.
25. CHALLENGE How could the wording of a question or the tone of voice of the
interviewer affect a survey? Give at least two examples.
26.
*/ -!4( Compare taking a survey and finding an experimental
(*/
83 *5*/(
probability.
Lesson 8-8 Using Sampling to Predict
455
27. Maci surveyed all the members of
28. Ms. Hernandez determined that 60%
her softball team about their favorite
sport.
Sport
Softball
Basketball
Soccer
Volleyball
of the students in her classes brought
an umbrella to school when the
weather forecast predicted rain. If
she has a total of 150 students, which
statement does NOT represent
Ms. Hernandez’s data?
Number of Members
12
5
3
8
F On days when rain is forecast, less
2
than _
of her students bring an
5
umbrella to school.
From these results, Maci concluded
that softball was the favorite sport
among all her classmates. Which is
the best explanation for why her
conclusion might NOT be valid?
G On days when rain is forecast, 90
of her students bring an umbrella
to school.
A The softball team meets only on
weekdays.
B She should have asked only people
who do not play sports.
C The survey should have been done
daily for a week.
H On days when rain is forecast,
1
more than _
of her students bring
2
an umbrella to school.
J
On days when rain is forecast, 60
of her students do not bring an
umbrella to school.
D The sample was not representative
of all of her classmates.
29. PETS Sasha’s dog is expecting a litter of 6 puppies. If the probability that
a puppy is male or female is equally likely, describe a model that could
be used to simulate the mixture of male and female puppies. (Lesson 8-7)
30. MANUFACTURING An inspector finds that 3 out of the 250 DVD players he
checks are defective. What is the experimental probability that a DVD
player is defective? (Lesson 8-5)
Find the volume of each sphere. Round to the nearest tenth.
31.
M
32.
(Lesson 7-6)
33.
FT
IN
34. HEALTH Many health authorities recommend that a healthy diet contains
no more than 30% of its Calories from fat. If Jennie consumes 1,500
Calories each day, what is the maximum number of Calories she should
consume from fat? (Lesson 5-4)
456
Chapter 8 Probability
CH
APTER
8
Study Guide
and Review
Download Vocabulary
Review from tx.msmath3.com
Key Vocabulary
Be sure the following Key Concepts are noted in
your Foldable.
n‡£
n‡Ó
n‡Î
n‡{
n‡x
n‡È
n‡Ç
biased sample (p. 452)
population (p. 451)
combination (p. 425)
probability (p. 417)
composite experiments
sample (p. 451)
(p. 429)
sample space (p. 416)
convenience sample
(p. 452)
Key Concepts
Counting Outcomes
(Lessons 8-1 to 8-3)
• If event M can occur in m ways and is followed
by event N that can occur in n ways, then the
event M followed by the event N can occur in
m · n ways.
simple random sample
(p. 451)
dependent events (p. 430)
simulation (p. 445)
event (p. 416)
stratified random sample
experimental probability
(p. 436)
Fundamental Counting
Principle (p. 417)
(p. 451)
systematic random
sample (p. 451)
theoretical probability
• A permutation is an arrangement or listing in
which order is important.
independent events
• A combination is an arrangement where order is
not important.
outcome (p. 416)
unbiased sample (p. 451)
random (p. 417)
Probability
permutation (p. 421)
voluntary response
sample (p. 452)
(Lessons 8-4, 8-5, and 8-7)
• The probability of two independent events can be
found by multiplying the probability of the first
event by the probability of the second event.
• If two events, A and B, are dependent, then the
probability of both events occurring is the product
of the probability of A and the probability of B
after A occurs.
Statistics
(Lesson 8-8)
• An unbiased sample is representative of an entire
population.
• A biased sample favors one or more parts of a
population over others.
(p. 429)
(p. 436)
tree diagram (p. 416)
Vocabulary Check
Choose the correct term to complete each
sentence.
1. A list of all possible outcomes is called the
(sample space, event).
2. A (combination, permutation) is an
arrangement where order matters.
3. A (combination, composite experiment)
consists of two or more simple events.
4. For (independent, dependent) events, the
outcome of one does not affect the other.
5. (Theoretical, Experimental) probability is
based on known characteristics or facts.
6. A (simple random sample, convenience
sample) is a biased sample.
Vocabulary Review at tx.msmath3.com
Chapter 8 Study Guide and Review
457
CH
APTER
8
Study Guide and Review
Lesson-by-Lesson Review
8-1
Counting Outcomes
(pp. 416–420)
A penny is tossed and a 4-sided number
pyramid with sides labeled 1, 2, 3 and 4 is
rolled.
7. Draw a tree diagram to show the
possible outcomes.
8. Find the probability of getting a head
and a 3.
9. Find the probability of getting a tail
and an odd number.
10. FOOD A restaurant offers 15 main
Example 1 A car manufacturer makes
8 different models in 12 different colors.
They also offer manual or automatic
transmission. How many choices does
a customer have?
number number
number
total
of × of ×
of
= number
models
colors
transmissions
of cars
8
×
12
×
2
=
192
The customer has 192 choices.
menu items, 5 salads, and 8 desserts.
How many meals of a main menu
item, a salad, and a dessert are there?
8-2
Permutations
(pp. 421–424)
Example 2
Find each value.
11. P(6, 1)
12. P(4, 4)
13. P(5, 3)
14. P(7, 2)
15. P(10, 3)
16. P(4, 1)
17. NUMBER THEORY How many 3-digit
whole numbers can you write using
the digits 1, 2, 3, 4, 5, and 6 if no digit
can be used twice?
8-3
Combinations
P(35, 4) represents the number of
permutations of 35 things taken 4 at a
time.
P(35, 4) = 35 · 34 · 33 · 32 or
= 1,256,640
(pp. 425–428)
Example 3
Find each value.
18. C(5, 5)
19. C(4, 3)
20. C(12, 2)
21. C(9, 5)
22. C(3, 1)
23. C(7, 2)
24. PETS How many different pairs of
puppies can be selected from a litter of
8 puppies?
458
Find P(35, 4).
Chapter 8 Probability
Find C(4, 2).
C(4, 2) represents the number of
combinations of 4 things taken 2 at a time.
4·3
C(4, 2) = _
2·1
4 things taken 2 at a time
2
4·3
=_
or 6
2·1
1
Divide out
common factors.
Mixed Problem Solving
For mixed problem-solving practice,
see page 735.
8-4
Probability of Composite Experiments
(pp. 429–434)
Example 4 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?
A number cube is rolled and a penny
is tossed. Find each probability.
25. P(2 and heads)
26. P(even and heads)
3
P(first marble is blue) = _
10
27. P(1 or 2 and tails)
2
P(second marble is blue) = _
28. P(odd and tails)
9
3 _
P(two blue marbles) = _
·2
10
29. TIES Mr. Dominguez has 4 black ties,
3 gray ties, 2 maroon ties, and 1 brown
tie. If he selects two ties without
looking, what is the probability that he
will pick two black ties?
8-5
Experimental and Theoretical Probability
9
6
1
=_
or _
90
15
(pp. 436–440)
A spinner has four equal-sized 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.
30. P(red)
31. P(green)
32. P(red or blue)
33. Compare the theoretical and
experimental probabilities of the
spinner landing on red.
Example 5 A nickel and a dime are
tossed. What is theoretical probability of
tossing two tails?
1 _
1
The theoretical probability is _
· 1 or _
.
2
4
Example 6 In an experiment, the same
two coins are tossed 50 times. Ten of
those times, tails were both showing.
Find the experimental probability of
tossing two tails.
Since tails were showing 10 out of the
50 tries, the experimental probability is
10
1
_
or _
.
50
SPELLING For Exercises 34 and 35, use the
following information.
On a spelling test, Angie misspells 2 out of
the first 10 words.
34. What is the probability that she will
misspell the next spelling word?
2
5
Example 7 Compare the theoretical
and experimental probabilities of
tossing two tails.
1
The theoretical probability _
is greater
4
1
.
than the experimental probability _
5
35. If the spelling test has 25 words on it,
how many words would you expect
Angie to misspell?
Chapter 8 Study Guide and Review
459
CH
APTER
8
Study Guide and Review
8-6
PSI: Act it Out
(pp. 443–444)
Solve. Use the act it out strategy.
36. READING In English class, each student
must select 4 short stories from a list of
5 short stories to read. How many
different combinations of short stories
could a student read?
_1
37. CARPENTRY Jaime has 14 feet of
4
7
feet for a
lumber. She uses 2_
8
bookshelf. Does Jaime have enough
lumber for four more identical shelves?
Explain.
8-7
Simulations
Example 8 The Spirit Club is making
a banner using three sheets of paper.
How many different banners can they
make using their school colors of black,
orange, and white.
Use three index cards labeled black,
orange, and white to model the different
banners.
There are six different combinations they
can make.
(pp. 445–448)
38. DRINKS There are 12 bottles of apple
juice and 8 bottles of grape juice in a
cooler. Describe a model that could be
used to simulate choosing 2 bottles of
juice.
Example 9 One out of every 6 people
buying a ticket for a movie is given a
pass free. Describe a model to simulate
whether each of the next 5 people
buying tickets receives a pass.
Use a number cube where rolling a 1
represents receiving a coupon and rolling
a 2, 3, 4, 5, or 6 represents not receiving a
coupon.
8-8
Using Sampling to Predict
(pp. 451–456)
CONCERTS For Exercises 39 and 40, use the
following information.
A radio station is taking a survey to
determine how many people would attend
a music festival.
Example 10 In a survey, 25 out of 40
students in the school cafeteria preferred
chocolate milk rather than white milk.
How much chocolate milk should the
school order for 400 students each day?
39. Describe the sample if the station asks
25 out of 40 or 62.5% of the students
prefer chocolate milk.
listeners to call in a response to the
survey.
40. Suppose 12 out of 80 people surveyed
said they would attend the festival.
How many out of 800 people would be
expected to attend the festival?
460
Chapter 8 Probability
Find 62.5% of 400.
0.625 × 400 = 250
The school should order about 250 cartons
of chocolate milk.
CH
APTER
Practice Test
8
1. 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.
14. What is the experimental probability of
no tails?
Categories for Box Lunches
15. What is the experimental probability of
5 types of sandwiches
one tail?
3 types of fresh fruits
16. Draw a tree diagram to show the outcomes
2 types of cookies
of tossing two coins.
Find each value.
17. Compare the experimental probability with
2. C(10, 5)
3. P(6, 3)
4. P(5, 2)
5. C(7, 4)
the theoretical probability of getting no tails
when two coins are tossed.
6. TROPHIES Venus has 8 field hockey trophies.
18. MUSIC Berea Music Store has 20 different
CDs on a sale table. These CDs are priced 3
for $30. How many combinations of 3 CDs
can be purchased?
How many ways can she arrange 4 of them
on a shelf?
7.
TEST PRACTICE Ms. Hawthorne
randomly selects 2 students from
6 volunteers to be on the school activities
committee. If Roberto and Joel volunteer,
what is the probability that they will both
be selected?
1
A _
3
1
B _
15
1
C _
30
19. VOLUNTEERING Student Council surveyed
four homerooms to find out how many
hours students volunteer each year. The
results are shown in the table. If there are
864 students at the school, how many can be
expected to volunteer 21–40 hours?
1
D _
60
Number of Hours
8. BASKETBALL How many teams of 5 players
can be chosen from 15 players?
A jar contains 4 blue, 7 red, 6 yellow, 8 green,
and 3 white tiles. Once a tile is selected, it is
not replaced. Find each probability.
9. P(2 blue)
10. P(red, then white)
11. P(white, then green)
12. P(two tiles that are neither yellow nor red)
13. SOFTBALL Miranda gets a hit 30% of the
times she is at bat. Describe a model to find
the experimental probability that she will
get a hit at each of her next three at-bats.
Chapter Test at tx.msmath3.com
0–10
38
11–20
26
21–40
10
40 or more
20.
Number of Students
6
TEST PRACTICE The Centerville School
Board wants to know if it has community
support to build a new school. How should
they conduct a valid survey?
F Ask parents at a school open house.
G Ask people at the Senior Center.
H Call every 50th number in the phone
book.
J Ask people to call with their opinions.
Chapter 8 Practice Test
461
CH
APTER
8
Texas
Test Practice
Cumulative, Chapters 1–8
Read each question. Then fill in the
correct answer on the answer document
provided by your teacher or on a sheet
of paper.
1. Michael surveyed all the members of the
middle school football team to determine
the favorite sport among students in his
middle school. The results are shown in the
table below.
Favorite Sport
Sport
Number of Votes
Basketball
7
Baseball
6
Football
11
Soccer
4
Based on these results, Michael concluded
that soccer is the least favorite sport among
the students in his middle school. Which is
the best explanation for why his conclusion
might not be valid?
A The sample was not representative
of all the students at the school.
B The survey should have had more
questions.
C The sample should have included
the football coaches.
D The survey should have been done
each day for a month.
2. There are 4 children in the Smith family.
Steve is 1 year older than Pam, and he
is 4 years older than Meredith. Kyle is
5 years old, which is 3 years younger
than Meredith. How old is Pam?
F 8
H 11
G 10
J 12
3. GRIDDABLE Marcy can type 440 words in
8 minutes. What is her typing rate, in words
per minute?
462
Chapter 8 Probability
4. Of the 32 students surveyed in J.T.’s
homeroom, 14 recycle at home. How many
students would you expect to recycle at
home if a total of 880 students were
surveyed?
A 495
B 385
C 281
D 123
5. A car tire travels about 100 inches in 1 full
rotation. What is the radius of the tire, to the
nearest inch?
F 32 inches
G 28 inches
H 24 inches
J 16 inches
6. Janelle wants to buy a new television.
She will finance the total cost of $600 by
making 12 equal monthly payments to pay
back this amount plus interest. What other
information is needed to determine the
amount of Janelle’s monthly payment?
A the interest rate being charged
B the amount of money Janelle has in her
savings account
C the brand of the computer
D the amount of Janelle’s weekly income
7. The probability that Maryanne gets a hit in
3
softball is _
. How many hits would you
5
expect her to get in her next 60 at bats?
F 50
G 36
H 30
J 24
Texas Test Practice at tx.msmath2.com
Get Ready
for the Texas Test
For test-taking strategies and more practice,
see pages TX1–TX23
8. The net below forms a cylinder when
12. A sporting goods company ships
folded. What is the surface area of the
cylinder?
basketballs in cube-shaped boxes. Which of
the following is closest to the surface area of
the box?
IN
IN
IN
A 6.3 in 2
C 21.3 in 2
B 18.8 in 2
D 42.6 in 2
IN
IN
F 85 in 2
H 475 in 2
G 320 in 2
J 510 in 2
9. If three coins are tossed, what is the
Pre-AP
probability that they all show tails?
1
F _
Record your answers on a sheet of paper.
Show your work.
16
_
G 1
8
_
H 1
4
_
J 1
2
13. Tiffany has a bag of 10 yellow, 10 red, and
10 green marbles. Tiffany picks two marbles
at random and gives them to her sister.
a. What is the probability of choosing
2 yellow marbles?
10. GRIDDABLE A rectangular prism has a length
b. Of the marbles left, what is the
of 7.5 inches, a width of 1.4 inches, and a
volume of 86.4 cubic inches. What is the
height of the rectangular prism in inches?
Round to the nearest tenth.
probability of choosing a green marble
next?
Question 13 Extended response
questions often involve several parts.
When one part of the question involves
the answer to a previous part of the
question, make sure to check your
answer to the first part before moving
on. Also, remember to show all of your
work. You may be able to get partial
credit for your answers, even if they are
not entirely correct.
11. A truck used 6.3 gallons of gasoline to travel
107 miles. How many gallons of gasoline
would it need to travel an additional 250
miles?
A 8.4 gal
C 18.9 gal
B 14.7 gal
D 21.0 gal
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463