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Lesson 8-1 Counting Outcomes
Lesson 8-2 Permutations
Lesson 8-3 Combinations
Lesson 8-4 Probability of Composite Experiments
Lesson 8-5 Experimental and Theoretical Probability
Lesson 8-6 Problem-Solving Investigation: Act It Out
Lesson 8-7 Simulations
Lesson 8-8 Using Sampling to Predict
Five-Minute Check (over Chapter 7)
Main Idea and Vocabulary
Targeted TEKS
Example 1: Use a Tree Diagram
Key Concept: Fundamental Counting Principle
Example 2: Real-World Example
Example 3: Find Probability
• Count outcomes by using a tree diagram or the
Fundamental Counting Principle.
• Outcome
– A result of an action
• Event
– An outcome or collection of
outcome
• sample space
– Lists all possible outcomes
• tree diagram
– Graphical way of showing a
sample space
• Fundamental Counting
Principle
– Multiply all possibilities
• Probability
– Ratio of SUCCESSFUL
outcome over ALL
outcomes
• Random
– Each outcome is equally
likely to occur
NOTES



Counting Outcomes or Determining Size of Sample
Space
1.
Tree Diagrams
2.
Fundamental Counting Principal
sample space problems – USE THE BOX METHOD!:
1.
Draw boxes for each “possibility”
2.
Figure out how many “things” can go in each box.
3.
Multiply the numbers from #2 together
Probability




P (event) = Number of “positive” outcomes
Total number of possible outcomes
P (event) = POSITIVE
POSSIBLE
Use a Tree Diagram
BOOKS A flea market vendor sells new and used
books for adults and teens. Today she has fantasy
novels and poetry collections to choose from. Draw a
tree diagram to determine the number of categories
of books.
Use a Tree Diagram
Answer: There are 8 different categories.
FASHION A store has spring outfits on sale. You can
choose either striped or solid pants. You can also
choose green, pink, or orange shirts. Finally, you can
choose either long-sleeved shirts or short-sleeved
shirts. Draw a tree diagram to determine the number
of possible outfits.
Answer: 12 different outfits.
RESTAURANTS A manager assigns different codes
to all the tables in a restaurant to make it easier for
the wait staff to identify them. Each code consists of
the vowel A, E, I, O, or U, followed by two digits from 0
through 9. How many codes could the manager
assign using this method?
number of
possible
letters for
the first
place
x
5
x
number of
possible
digits for the x
second
place
10
x
number of
possible
digits for
the third
place
10
number of
= possible
codes
=
500
Answer: There are 500 possible codes.
SCHOOLS A middle school assigns each student a
code to use for scheduling. Each code consists of a
letter, followed by two digits from 0 through 9. How
many codes are possible?
A. 2,600
0%
B. 2,950
C. 3,400
D. 3,800
1.
2.
3.
4.
A
B
C
D
A
B
C
D
Find Probability
COMPUTERS What is the probability that Liana will
guess her friend’s computer password on the first try
if all she knows is that it consists of three letters?
Find the number of possible outcomes. Use the
Fundamental Counting Principle.
choices
for the
first letter
26
x
choices for
the second
letter
x
choices for
the third
letter
x
26
x
26
total
= number of
outcomes
=
17,576
Find Probability
Answer: There are 17,576 possible outcomes. There is
1 correct password. So, the probability of
guessing on the first try is
LOCKER COMBINATIONS What is the probability that
Shauna will guess her friend’s locker combination on
the first try if all she knows is that it consists of three
digits from 0 through 9?
A.
B.
C.
0%
1.
2.
3.
4.
A
D.
A
B
C
D
B
C
D
Five-Minute Check (over Lesson 8-1)
Main Idea and Vocabulary
Targeted TEKS
Example 1: Find a Permutation
Example 2: Use Permutation Notation
Example 3: Use Permutation Notation
Example 4: Find Probability
• Find the number of permutations of objects.
• Permutation
– An arrangement where ORDER IS IMPORTANT!!
– Also means something CAN’T BE USED TWICE!
NOTES
 Permutations
P(m,n) is read as “the number of permutations of
M things when taken N at a time”
 Ex: P(31,3) = 31 * 30 * 29 = 26,970
Start at 31
Need 3 boxes, and each box reduces by 1
 Probability
 P (event) = Number of “positive” outcomes

Total number of possible outcomes
Find a Permutation
SOFTBALL There are 10 players on a softball team.
In how many ways can the manager choose three
players for first, second, and third base?
number of
possible
players
for first
base
10
x
number of
possible
players for
second
base
x
x
9
x
number of
possible
players for
third base
8
total
number of
= possible
ways
=
720
Answer: There are 720 different ways the manager can
pick players for first, second, and third base.
STUDENT COUNCIL There are 15 students on
student council. In how many ways can Mrs.
Sommers choose three students for president, vice
president, and secretary?
A. 2,415
B. 2,730
C. 3,150
D. 3,375
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
Use Permutation Notation
Find the value of P(7, 2).
P(7, 2) = 7 ● 6 or 42
Answer: 42
7 things taken 2 at a time
Find the value of P(8, 4).
A. 1,100
B. 1,375
0%
C. 1,420
D. 1,680
1.
2.
3.
4.
A
B
C
D
A
B
C
D
Use Permutation Notation
Find the value of P(13, 7).
P(13, 7) = 13 ● 12 ● 11 ● 10 ● 9 ● 8 ● 7
= 8,648,640
Answer: 8,648,640
13 things
taken 7 at a
time
Find the value of P(12, 5).
A. 72,110
0%
B. 84,800
C. 93,120
D. 95,040
1.
2.
3.
4.
A
A
B
C
D
B
C
D
Find Probability
NUMBERS Consider all of the five-digit numbers
that can be formed using the digits 1, 2, 3, 4, and 5
where no digit is used twice. Find the probability,
expressed as a percent, that one of these numbers
picked at random is an even number.
You are considering all permutations of 5 digits taken 5
at a time. You wish to find the probability that one of
these numbers picked at random is even.
Solve the Test Item
Find the number of possible five-digit numbers.
P(5, 5) = 5!
Find Probability
For a number to be even, the ones digit must be 2 or 4.
number of
ways to
pick the
last digit
2
x
x
number of
ways to
pick the first
four digits
P(4, 4)
=
=
number of
permutations
that are even
2P(4, 4) or 2 x 4 x 3 x 2 x 1
Find Probability
Substitute.
Divide out common
factors.
Simplify.
Answer: 40%
NUMBERS Consider all of the five-digit numbers that
can be formed using the digits 1, 2, 3, 4, and 5 where
no digit is used twice. Find the probability that one of
these numbers picked at random is an odd number.
A. 30%
B. 40%
C. 50%
D. 60%
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
Five-Minute Check (over Lesson 8-2)
Main Idea and Vocabulary
Targeted TEKS
Example 1: Find a Combination
Example 2: Combinations and Permutations
Example 3: Combinations and Permutations
Example 4: Combinations and Permutations
Example 5: Combinations and Permutations
• Find the number of combinations of objects.
• Combination
– An arrangement where order is NOT IMPORTANT!!
– Also means something CAN BE USED TWICE!
• Factorial
– X! means X * X-1 * X-2 * … * 1
– Ex: 5! = 5 * 4 * 3 * 2 *1
NOTES
 ORDER IS NOT IMPORTANT!!
 Combinations =
Number of Permutations
Number of ways they can be arranged
 Another way to say it is
C (x,y) = P(x,y) / y!
Find a Combination
TOURNAMENTS Five teams are playing in a
tournament. If each team plays every other team
once, how many games are played?
Method 1 Let A, B, C, D, and E represent the five teams.
First, list all of the possible permutations of A,
B, C, D, and E taken two at a time. Then cross
out the letter pairs that are the same as one
another.
Team A playing Team
B is the same as Team
AB AC AD AE BA
B playing Team A, so
BC BD BE CA CB
cross off one of them.
CD CE DA DB DC
DE EA EB EC ED
Find a Combination
Method 2
Find the number of permutations of 5
teams taken 2 at a time.
P(5, 2) = 5 ● 4 or 20
Since order is not important, divide the number of
permutations by the number of ways 2 things can be
arranged.
Answer: There are 10 games that can be played.
TOURNAMENTS Six teams are playing each other in
a tournament. If each team plays every other team
once, how many games are played?
A. 10
B. 12
C. 15
D. 20
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
Combinations and Permutations
SCHOOL An eighth grade teacher needs to select 4
students from a class of 22 to help with sixth grade
orientation. Does this represent a combination or a
permutation?
Answer: This is a combination problem since the order
is not important.
SCHOOL A teacher needs to select 5 students from a
class of 26 to help with parent-teacher conferences.
Does this represent a combination or a permutation?
A. combination
0%
0%
B. permutation
1.
2.
A
B
A
B
Combinations and Permutations
SCHOOL An eighth grade teacher needs to select 4
students from a class of 22 to help with sixth grade
orientation. How many possible groups could be
selected to help out the new students?
22 students taken 4 at a
time
Answer: There are 7,315 different groups of eighth
grade students that could help the new
students.
SCHOOL A teacher needs to select 5 students from a
class of 26 to help with parent-teacher conferences.
How many possible groups could be selected to
help?
A. 28,110
B. 46,240
C. 55,070
0%
1.
2.
3.
4.
A
D. 65,780
A
B
C
D
B
C
D
Combinations and Permutations
SCHOOL An eighth grade teacher needs to select 4
students from a class of 22 to help with sixth grade
orientation. One eighth grade student will be assigned
to sixth grade classes on the first floor, another
student will be assigned to classes on the second
floor, another student will be assigned to classes on
the third floor, and still another student will be
assigned to classes on the fourth floor. Does this
represent a combination or a permutation?
Answer: Since it makes a difference which student
goes to which floor, order is important. This is
a permutation.
SCHOOL A teacher needs to select 5 students from a
class of 26 to help with parent-teacher conferences. One
student will be assigned to fifth grade parents, another
student will be assigned to sixth grade parents, another
student will be assigned to seventh grade parents, another
student will be assigned to eighth grade parents, and still
another student will be assigned to ninth grade parents.
Does this represent a combination or a permutation?
0%
A. combination
B. permutation
0%
1.
2.
A
B
A
B
Combinations and Permutations
SCHOOL An eighth grade teacher needs to select 4
students from a class of 22 to help with sixth grade
orientation. One eighth grade student will be
assigned to sixth grade classes on the first floor,
another student will be assigned to classes on the
second floor, another student will be assigned to
classes on the third floor, and still another student
will be assigned to classes on the fourth floor. In
how many possible ways can the eighth graders be
assigned to help with the sixth grade orientation?
P(22, 4) = 22 ● 21 ● 20 ● 19
= 175,560
Definition of P(22, 4)
Combinations and Permutations
Answer: There are 175,560 ways for the eighth grade
students to be selected to help with sixth
grade orientation.
SCHOOL A teacher needs to select 5 students from a class of
26 to help with parent-teacher conferences. One student will be
assigned to fifth grade parents, another student will be
assigned to sixth grade parents, another student will be
assigned to seventh grade parents, another student will be
assigned to eighth grade parents, and still another student will
be assigned to ninth grade parents. In how many possible
ways can the students be assigned to help with the parentteacher conferences?
A. 5,122,200
B. 7,893,600
D. 11,708,240
0%
0%
D
0%
C
A
0%
B
C. 8,346,250
A.
B.
C.
D.
A
B
C
D
Five-Minute Check (over Lesson 8-3)
Main Idea and Vocabulary
Targeted TEKS
Key Concept: Probability of Independent Events
Example 1: Probability of Independent Events
Example 2: Use Probability to Solve a Problem
Key Concept: Probability of Dependent Events
Example 3: Probability of Dependent Events
• Find the probability of independent and dependent
events.
• Composite experiment
– 2 or more events
• independent events
– The outcome of the 1st events does NOT affect the
outcome of the next events.
• dependent events
– The outcome of the 1st events DOES affect the outcome
of the next events.
NOTES
 Probability
 P (A) =
Number of “positive” outcomes

Total number of possible outcomes
 Probability of Composite Events
 P (A and B) = P(A) * P(B)
 For INDEPENDENT events
 Just multiply the numbers!
 For DEPENDENT events
 The probability of the second (or later) events
CHANGES with the results of the early events.
 The BOX process also works with prob. problems
 REMEMBER! AND = MULT and OR = ADD
Probability of Independent Events
The two spinners below are spun. What is the
probability that both spinners will show a number
greater than 6?
Probability of Independent Events
Answer:
The two spinners below
are spun. What is the
probability that both
spinners will show a
number less than 4?
0%
0%
0%
0%
D
D.
C
C.
B
B.
A
A.
A.
B.
C.
D.
A
B
C
D
Use Probability to Solve a Problem
TEST EXAMPLE A red number cube and a white
number cube are rolled. The faces of both cubes
are numbered from 1 to 6. What is the probability of
rolling a 3 on the red number cube and rolling the
number 3 or less on the white number cube?
A
B
C
D
Read the Test Item
You are asked to find the probability of rolling a 3 on the
red number cube and rolling a number 3 or less on the
white number cube. The events are independent because
rolling one number cube does not affect rolling the other
cube.
Use Probability to Solve a Problem
Solve the Test Item
First, find the probability of each event.
Then, find the probability of both events occurring.
P(A and B) =
P(A) ● P(B)
Multiply.
Use Probability to Solve a Problem
Answer:
A white number cube and a green number cube are
rolled. The faces of both cubes are numbered from 1
to 6. What is the probability of rolling an even number
on the white number cube and rolling a 3 or a 5 on
the green number cube?
A.
C.
0%
B.
D.
1.
2.
3.
4.
A
B
C
D
A
B
C
D
Probability of Dependent Events
There are 4 red, 8 yellow, and 6 blue socks mixed up
in a drawer. Once a sock is selected, it is not
replaced. Find the probability of reaching into the
drawer without looking and choosing 2 blue socks.
Since the first sock is not replaced, the first event affects
the second event. These are dependent events.
number of blue socks
total number of socks
Probability of Dependent Events
number of blue socks
after one blue sock is
removed
total number of socks
after one blue sock is
removed
Answer:
BrainPop:
Probability: Compound Events
There are 6 green, 9 purple, and 3 orange marbles in
a bag. Once a marble is selected, it is not replaced.
Find the probability that two purple marbles are
chosen.
A.
B.
C.
0%
1.
2.
3.
4.
A
D.
A
B
C
D
B
C
D
Five-Minute Check (over Lesson 8-4)
Main Idea and Vocabulary
Targeted TEKS
Example 1: Theoretical and Experimental
Probability
Example 2: Theoretical and Experimental
Probability
Example 3: Real-World Example
Example 4: Use Probability to Predict
• Find experimental and theoretical probabilities and
use them to make predictions.
• Theoretical Probability
– What SHOULD happen
• Experimental Probability
– What ACTUALLY happened!
– They are FREQUENTLY different!!
NOTES

Theoretical prob. – What SHOULD happen

Uses formula we’ve been using



# Positive outcomes

# Possible outcomes
Experimental probability – What DID happen

What “positive” outcomes happened

How many times was it done?
Both can be used to predict future outcomes by
setting up proportions.
1.
Remember: Keep the units
2.
Remember: Keep the SAME
units on TOP and BOTTOM
Interactive Lab: Theoretical and
Experimental Probability
Theoretical and Experimental Probability
Nikki is conducting an experiment to find the
probability of getting various results when three
coins are tossed. The results of her experiment are
given below. What is the theoretical probability of
tossing all heads on the next turn?
Answer:
Marcus is conducting an
experiment to find the probability
of getting various results when
four coins are tossed. The results
of his experiment are given
below. What is the theoretical
probability of tossing all tails on
the next turn?
D.
A
B
0%
C
D
0%
D
C.
A
0%
A.
B.
C.0%
D.
C
B.
B
A.
Theoretical and Experimental Probability
Nikki is conducting an experiment to find the
probability of getting various results when three
coins are tossed. The results of her experiment are
given below. According to the experimental
probability, is Nikki more likely to get all heads or no
heads on her next toss?
Theoretical and Experimental Probability
Examine the table. Out of 80 trials, flipping all heads
occurred 6 times, and flipping no heads occurred 12
times. So, the experimental probability of flipping all
heads is
or
. And the experimental probability of
flipping no heads is
Answer: Nikki is more likely to get no heads on her next
toss.
Marcus is conducting an experiment to find the
probability of getting various results when four coins
are tossed. The results of his experiment are given
below. According to the experimental probability, is
Marcus more likely to get all heads or no heads on
his next toss?
0%
A. all heads
0%
B. no heads
A
1. B
2.
A
B
MARKETING Eight hundred adults were asked
whether they were planning to stay home for winter
vacation. Of those surveyed, 560 said that they
were. What is the experimental probability that an
adult planned to stay home for winter vacation?
There were 800 people surveyed and 560 said that they
were staying home.
Answer:
MARKETING Five hundred adults were asked
whether they were planning to stay home for New
Year’s Eve. Of those surveyed, 300 said that they
were. What is the experimental probability that an
adult planned to stay home for New Year’s Eve?
A.
0%
1.
2.
3.
4.
B.
C.
D.
A
B
C
D
A
B
C
D
Use Probability to Predict
MATH TEAM Over the past three years, the probability
that the school math team would win a meet is
Is this probability experimental or theoretical? If
the team wants to win 12 more meets in the next 3
years, how many meets should the team enter?
This problem can be solved using a proportion.
3 out of 5 meets
were wins
Solve the proportion.
12 out of x meets
should be wins.
Use Probability to Predict
Write the proportion.
Find the cross products.
Multiply.
Divide each side by 3.
Answer: This is an experimental probability since it is
based on what happened in the past. They
should enter 20 meets.
SPEECH AND DEBATE Over the past three years,
the probability that the school speech and debate
team would win a meet is
Is this probability
experimental or theoretical?
0%
0%
A. experimental
B. theoretical
A
1.
2.
A
B
B
SPEECH AND DEBATE Over the past three years,
the probability that the school speech and debate
team would win a meet is
If the team wants to win
20 more meets in the next 3 years, how many meets
should the team enter?
0%
D
D. 25 meets
A
B
C0%
D
C
C. 20 meets
A.
B.
0% C.0%
D.
B
B. 18 meets
A
A. 16 meets
Five-Minute Check (over Lesson 8-5)
Main Idea
Targeted TEKS
Example 1: Problem-Solving Investigation:
Act It Out
• Solve problems by acting it out.
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.
Problem-Solving Investigation:
Act It Out
Melvin paid for a $5 sandwich with a $20 bill. The
cashier has $1, $5, and $10 bills in the register. How
many different ways can Melvin get his change?
Explore
You know that Melvin should receive $20 – $5
or $15 in change. You need to determine how
many different ways the cashier can make $15
in change with $1, $5, and $10 bills.
Plan
Use manipulatives such as play money to act
out the problem. Record the different ways the
cashier can make $15 in change.
Problem-Solving Investigation:
Act It Out
Solve
The cashier can make the change in 6
different ways.
Check
Make sure each method adds up to $15 in
change.
Answer: The cashier can make the change in 6
different ways.
Amanda paid for an $8 CD with a $20 bill. The cashier
has $1, $5, and $10 bills in the register. How many
different ways can Amanda get her change?
A. 3 ways
B. 4 ways
C. 5 ways
D. 6 ways
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
Five-Minute Check (over Lesson 8-6)
Main Idea and Vocabulary
Targeted TEKS
Example 1: Describe a Simulation
Example 2: Real-World Example
Example 3: Real-World Example
• Perform probability simulations to model real-world
situations involving uncertainty.
• Simulation
– An experiment designed to act out a given situation
NOTES

To create an experiment that simulates an
event,
1.
Design an experiment with the SAME NUMBER OF
OUTCOMES AND THE SAME PERCENTAGE OF
POSSIBLE OUTCOMES!
2.
Execute the experiment as many times as necessary
Describe a Simulation
SHOPPING A supermarket is issuing 1 of 6 different
in-store discount coupons to each customer who
enters the store. If the coupons are given out
randomly, describe a model that could be used to
simulate which coupons would be given to the first
100 customers.
Choose a method that has 6 possible outcomes, such
as rolling a number cube. Let each outcome represent
a different coupon.
Describe a Simulation
Answer: Roll a number cube to simulate the coupons
that might be given to the first 100 customers.
Repeat 100 times.
1 → coupon 1
2 → coupon 2
3 → coupon 3
4 → coupon 4
5 → coupon 5
6 → coupon 6
An electronics store is issuing 1 of 8 different instore discount coupons to each customer who enters
the store. If the coupons are given out randomly,
describe a model that could be used to simulate
which coupons would be given to the first 50
customers.
Sample answer:
Choose a method that has 8 possible outcomes, such as
flipping a coin 3 times. Let each outcome represent a
different coupon.
Toss a coin 3 times to simulate the coupons that might
be given to the first 50 customers. Repeat 50 times.
HHH → coupon 1
TTT → coupon 5
HHT → coupon 2
TTH → coupon 6
HTH → coupon 3
THT → coupon 7
HTT → coupon 4
THH → coupon 8
ORCHESTRA The conductor of the school orchestra
needs to choose 6 students at random to perform
with the all-city band. If there are 36 students in the
orchestra, describe a model that she could use to
simulate choosing these 6 students.
There are 36 students in the orchestra, so select objects
that combined have 36 outcomes, such as a number cube
and a spinner with 6 spaces. Assign each student one of
the possible outcomes.
6 numbers ● 6 letters = 36 outcomes
Answer: The conductor should roll the number cube
and spin the spinner at least 6 times to choose
the students for the band.
NATURE HIKE The director of a national park needs
to choose 3 rangers at random to lead nature hikes
this weekend. If there are 10 rangers working at the
park this weekend, describe a model that he could
use to simulate choosing these 3 rangers.
Answer: There are 10 rangers working this weekend,
so select objects that combined have 10
outcomes, such as a coin and a spinner with 5
spaces. Assign each ranger one of the
possible outcomes.
2 sides ● 5 letters = 10 outcomes
The park director should flip the coin and spin
the spinner 3 times to choose the rangers for
the nature hikes.
SOFTBALL During the regular season, Keisha had
base hits 40% of her times at bat. Describe an
experiment she could use to simulate her next 20
times at bat.
The probability that Keisha will get a hit is 40% or
,
and the probability that she will not get a hit is 60% or
Answer: She could use a spinner with 5 sectors, 2
representing getting a base hit and 3
representing a failure to get a base hit. She
would spin the spinner 20 times.
.
SALES During the holiday season, 75% of the
customers who enter a retail store make a purchase.
Describe an experiment a store manager could use to
simulate the next 50 customers.
Sample answer:
The probability that a customer will make a purchase is
75% or
and the probability that a customer will not
make a purchase is 25% or
The store manager could
use a spinner with 4 sectors, 3 representing a customer
making a purchase and 1 representing a customer not
making a purchase. The manager would spin the spinner
50 times.
Five-Minute Check (over Lesson 8-7)
Main Idea and Vocabulary
Targeted TEKS
Concept Summary: Unbiased Samples
Concept Summary: Biased Samples
Example 1: Determine Validity of Conclusions
Example 2: Determine Validity of Conclusions
Example 3: Use Sampling to Predict
• Predict the actions of a larger group by using a
sample.
• Population
– A large group of people
• Sample
– Small subgroup used to simulate a total population
• unbiased sample
– Sample is chosen to accurately represent the TOTAL
population
• biased sample
– Sample is chosen so that does NOT represent the TOTAL
population.
NOTES

In order for the results of a survey to be valid,
the sample must be chosen so that:
1.
Selection is RANDOM!
2.
People chosen must represent the total population.

If both are true, results are UNBIASED.

If either one is NOT true, results are biased

Un-biased Examples


simple random sample

stratified random sample

systematic random sample
Biased Examples

convenience sample

voluntary response sample
Determine Validity of Conclusions
Determine whether the conclusion is valid. Justify
your answer. To determine which school lunches
students like most, the cafeteria staff surveyed every
tenth student who walked into the cafeteria. Out of 40
students surveyed, 19 students stated that they liked
the burgers best. The cafeteria staff concluded that
about 50% of the students like burgers best.
Answer: The conclusion is valid. Since the population is
the students of the school, the sample is a
systematic random sample. It is an unbiased
sample.
Determine whether the conclusion is valid. Justify
your answer. To determine what ride is most popular,
every tenth person to walk through the gates of a
theme park is surveyed. Out of 290 customers, 98
stated that they prefer The Zip. The park manager
concludes that about a third of the park’s customers
prefer The Zip.
Answer: The conclusion is valid. Since the population is
the customers of the theme park, the sample
is a systematic random sample. It is an
unbiased sample.
Determine Validity of Conclusions
Determine whether the conclusion is valid. Justify
your answer. To determine what sports teenagers
like, Janet surveyed the student athletes on the girls’
field hockey team. Of these, 65% said that they like
field hockey best. Janet concluded that over half of
teenagers like field hockey best.
Answer: The conclusion is not valid. The students
surveyed probably prefer field hockey. This is
a biased sample. The sample is a
convenience sample because the people are
easily accessed.
Determine whether the conclusion is valid. Justify
your answer. To determine whether people prefer
dogs or cats, a researcher surveys 80 people at a dog
park. Of those surveyed, 88% said that they prefer
dogs, so the researcher concluded that most people
prefer dogs.
Answer: The conclusion is not valid. People at a dog
park probably prefer dogs. This is a biased
sample. The sample is a convenience sample
because the people are easily accessed.
Using Sampling to Predict
BOOKS The student council is trying to decide what
types of books to sell at its annual book fair to help
raise money for the eighth grade trip. It surveyed 40
students at random. The books they prefer are in the
table. If 220 books are to be sold at the book fair, how
many should be mysteries?
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.
0.30 × 220 = 66
Answer: About 66 books should be mysteries.
PENS The student shop sells pens. It surveys 50
students at random. The pens they prefer are in the
table. If 300 pens are to be sold at the student shop,
how many should be gel pens?
A. 96
0%
1.
2.
3.
4.
B. 114
C. 125
D. 132
A
B
C
D
A
B
C
D
Five-Minute Checks
Image Bank
Math Tools
Theoretical and Experimental Probability
Probability: Compound Events
Lesson 8-1 (over Chapter 7)
Lesson 8-2 (over Lesson 8-1)
Lesson 8-3 (over Lesson 8-2)
Lesson 8-4 (over Lesson 8-3)
Lesson 8-5 (over Lesson 8-4)
Lesson 8-6 (over Lesson 8-5)
Lesson 8-7 (over Lesson 8-6)
Lesson 8-8 (over Lesson 8-7)
To use the images that are on the
following three slides in your own
presentation:
1. Exit this presentation.
2. Open a chapter presentation using a
full installation of Microsoft® PowerPoint®
in editing mode and scroll to the Image
Bank slides.
3. Select an image, copy it, and paste it
into your presentation.
(over Chapter 7)
Find the area of a triangle with base 10 in. and
height 15 in.
A. 25 in.2
B. 50 in.2
C. 75 in.2
D. 150 in.
2
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
(over Chapter 7)
Find the area of a circle with diameter 14 m.
A. 153.9 m2
B. 196.0 m2
C. 307.8 m2
D. 615.6 m2
0%
1.
2.
3.
4.
A
B
C
D
A
B
C
D
(over Chapter 7)
Find the volume of a rectangular prism with length
3 yd, width 5 yd, and height 7 yd. Round to the
nearest tenth if necessary.
A. 15 yd3
0%
1.
2.
3.
4.
3
B. 52.5 yd
C. 71 yd3
A
D. 105 yd3
B
A
B
C
D
C
D
(over Chapter 7)
Find the volume of a cone with diameter 9 ft and
height 5 ft. Round to the nearest tenth if necessary.
A. 106 ft3
B. 159 ft3
C. 318 ft3
D. 424 ft3
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
(over Chapter 7)
Find the surface area of a rectangular prism that is
15 cm long, 20 cm wide, and 25 cm tall.
A. 1,175 cm2
B. 2,350 cm
C. 3,750 cm2
D. 7,500 cm
0%
2
2
1.
2.
3.
4.
A
B
C
D
A
B
C
D
(over Chapter 7)
Find the volume of a sphere with a diameter of
8 cm. Round your answer to the nearest hundredth.
A. 33.51 cm3
0%
B. 67.02 cm3
1.
2.
3.
4.
C. 268.08 cm3
D. 2144.66 cm
3
A
B
A
B
C
D
C
D
(over Lesson 8-1)
Determine the number of outcomes using a tree
diagram. Two number cubes are rolled.
A. 12 outcomes
B. 24 outcomes
C. 36 outcomes
D. 72 outcomes
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
(over Lesson 8-1)
Determine the number of outcomes using a tree
diagram. Four kinds of candy come in either red,
blue, or yellow wrappers.
A. 7 outcomes
0%
B. 12 outcomes
C. 16 outcomes
1.
2.
3.
4.
A
B
C
D
A
D. 24 outcomes
B
C
D
(over Lesson 8-1)
Use the Fundamental Counting Principle to find the
number of possible outcomes. A month of the year
is picked at random and a quarter is flipped.
A. 24 outcomes
0%
1.
2.
3.
4.
B. 36 outcomes
C. 72 outcomes
A
D. 144 outcomes
B
A
B
C
D
C
D
(over Lesson 8-1)
Use the Fundamental Counting Principle to find the
number of possible outcomes. A 4-digit code is
created using the numbers 0–6.
A. 28 outcomes
B. 784 outcomes
D. 16,384 outcomes
0%
D
A
B
0%
C
D
C
A
0%
B
C. 2,401 outcomes
A.
B.
0%
C.
D.
(over Lesson 8-1)
A university gives each student an ID number with
2 letters (A–Z) followed by 3 digits (0–9). How many
possible ID numbers are there?
A. 352 possible ID numbers
B. 1,676 possible ID numbers
C. 15,600 possible ID numbers
D. 676,000 possible ID numbers
1.
2.
3.
4.
A
B
C
D
0%
A
B
C
D
(over Lesson 8-1)
Lindsey and Barbara are going to a pizza shop.
They can order a pepperoni, sausage, Canadian
bacon, or hamburger pizza. The pizzas can be
made thin, regular, or thick crust. How many
different pizzas can they order?
A. 3
B. 4
0%
1.
2.
3.
4.
A
B
C
D
C. 12
A
D. 64
B
C
D
(over Lesson 8-2)
Find P(10, 5).
A. 3,003
B. 18,900
C. 30,240
D. 151,200
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
(over Lesson 8-2)
Find P(32, 3).
A. 6,545
B. 29,760
C. 39,270
D. 863,040
0%
1.
2.
3.
4.
A
B
C
D
A
B
C
D
(over Lesson 8-2)
Find P(17, 2).
A. 272
0%
B. 3,400
1.
2.
3.
4.
C. 57,120
A
B
C
D
D. 742,560
A
B
C
D
(over Lesson 8-2)
A pizzeria has a selection of 27 different toppings
to put on a pizza. Ming wants to buy a pizza with 4
different toppings. How many different pizzas could
she buy?
A. 108 pizzas
B. 421,200 pizzas
D. 303,600 pizzas
0%
D
A
B
0%
C
D
C
A
0%
B
C. 702 pizzas
A.
B.
0%
C.
D.
(over Lesson 8-2)
How many 9-digit social security numbers can be
made if no digit can be repeated and the first digit
is always 4?
A. 45
0%
B. 81
C. 6,561
1.
2.
3.
4.
A
B
C
D
A
D. 362,880
B
C
D
(over Lesson 8-3)
Find the value of C(8, 4).
A. 70
B. 280
C. 1,680
D. 11,880
0%
0%
A
B
A. A
B. 0% B
C. C
C
D. D
0%
D
(over Lesson 8-3)
Find the value of C(16, 3)
A. 5,814
B. 3,360
C. 1,120
D. 560
0%
1.
2.
3.
4.
A
B
C
D
A
B
C
D
(over Lesson 8-3)
Determine whether the situation is a permutation or
a combination. Choosing 5 places in a tournament
A. permutation
B. combination
A
B
0%
B
A
1.
0%2.
(over Lesson 8-3)
Determine whether the situation is a permutation or
a combination. Choosing 3 people from your class
to go to the mall with you
A. permutation
B. combination
A
B
0%
B
A
1.
0%
2.
(over Lesson 8-3)
Five points are located on a circle. How many line
segments can be drawn with these points as end
points?
A. 5
0%
B. 10
C. 25
1.
2.
3.
4.
A
B
C
D
A
D. 50
B
C
D
(over Lesson 8-3)
Which situation is represented by C(52, 5)?
A. the number of ways to select a
5-card hand from a deck of 52
B. the number of ways to order the
first 5 cards from a deck of 52
C. the number of ways to arrange 5
cards in a row from a deck of 52
D. the number of ways to arrange
the 5’s from a deck of 52 cards
1.
2.
3.
4.
A
B
C
D
A
0%
B
C
D
(over Lesson 8-4)
A day of the week is picked at random and a number
cube is tossed. Find P(begins with S and 4).
A.
B.
0%
D
A
B
0%
C
D
C
D.
A
0%
A.
B.
0%
C.
D.
B
C.
(over Lesson 8-4)
A day of the week is picked at random and a
number cube is tossed. Find P(Wednesday and 3)
A.
0%
B.
C.
D.
1.
2.
3.
4.
A
B
C
D
A
B
C
D
(over Lesson 8-4)
A bag of pencils has 3 red, 5 blue, and 8 yellow
pencils. Find P(red then blue) if each pencil
selected is not returned to the bag.
A.
0%
1.
2.
3.
4.
B.
C.
A
D.
B
A
B
C
D
C
D
(over Lesson 8-4)
A bag of pencils has 3 red, 5 blue, and 8 yellow
pencils. Find P(2 yellows) if each pencil selected is
not returned to the bag.
A.
B.
D.
0%
D
A
B
0%
C
D
C
A
0%
B
C.
A.
B.
0%
C.
D.
(over Lesson 8-4)
Jordan makes 75 percent of his basketball free
throws. What is the probability that he will make 4
free throws in a row? Write your answer as a
percent.
A. 12 percent
0%
B. 24 percent
C. 32 percent
1.
2.
3.
4.
A
B
C
D
A
D. 75 percent
B
C
D
(over Lesson 8-4)
Josh flips a coin and draws a card from a deck of
52. What is the probability that he will get heads
and a seven?
A.
0%
B.
C.
1.
2.
3.
4.
A
D.
A
B
C
D
B
C
D
(over Lesson 8-5)
D.
A
B
0%
C
D
0%
D
C.
A.
B.
0% C.
0%
D.
C
B.
B
A.
A
The figure shows a spinner
and the table that displays
the results of spinning the
spinner. Based on the
results, what is the
probability of spinning
yellow?
(over Lesson 8-5)
The figure shows a spinner
and a table that displays
the results of spinning the
spinner. Based on the
results, how many times
would you expect to spin a
yellow in 300 spins?
A. 278 Times
B.
240 Times
C. 190 Times
D.
110 Times
1.
2.
3.
4.
A
B
C
D
A
0%
B
C
D
(over Lesson 8-5)
The figure shows a spinner
and a table that displays
the results of spinning the
spinner. What is the
theoretical probability of
spinning yellow?
A.
B.
C.
D.
1.
2.
3.
4.
A
B
C
D
0%
A
B
C
D
(over Lesson 8-5)
The figure shows a spinner
and a table that displays
the results of spinning the
spinner. Based on the
theoretical probability, how
many times would you
expect to spin yellow in 300
spins?
0%
0%
D
D. 120 times
A
B
0%
C
D
C
C. 110 times
A.
B.
C.0%
D.
B
B. 75 times
A
A. 60 times
(over Lesson 8-5)
Roseanna recorded the results in
the table shown using a spinner
with 8 equal-sized sections. How
many of the sections do you
predict are labeled Win 1?
A. 1
B. 3
C. 5
D. 7
1.
2.
3.
4.
A
B
C
D
0%
A
B
C
D
(over Lesson 8-6)
Roberto wants to buy a soda from the concession
stand at a football game. The soda costs $0.50. If
Roberto uses exact change, in how many different
ways can he use nickels, dimes, and quarters? Solve
using the act it out strategy.
A. 8 combinations
D. 14 combinations
0%
D
A
B
0%
C
D
C
0%
A
C. 12 combinations
A.
B.
0%
C.
D.
B
B. 10 combinations
(over Lesson 8-6)
Latasha has 5 different pictures that she wants to
display on her bookshelf from left to right. She
always wants to have the picture of her dog to be
the first picture on the left. How many different
ways can she arrange the pictures? Solve using
the act it out strategy.
A. 90 combinations
B. 25 combinations
C. 20 combinations
1.
2.
3.
4.
A
B
C
D
A
D. 24 combinations
0%
B
C
D
(over Lesson 8-6)
A coffee shop sells 4 different specialty coffees. If
they sell their coffee in 3 different sizes, how many
different cups of coffee can they sell? Solve using
the act it out strategy.
A. 6 cups
B. 7 cups
C. 12 cups
0%
1.
2.
3.
4.
A
D. 14 cups
A
B
C
D
B
C
D
(over Lesson 8-6)
The dimensions of a swimming pool are
50 m x 25 m. James decides to swim around the
perimeter of the pool and then one lap lengthwise
across the pool. If he continues this pattern, how
far will he have swum after 12 rotations? Solve
using the act it out strategy.
A. 1200 m
B. 2400 m
C. 3000 m
D. 3600 m
1.
2.
3.
4.
A
B
C
D
0%
A
B
C
D
(over Lesson 8-7)
Tonya placed 4 green marbles, 7 red marbles, and 9
white marbles in a bag. She plans to pick a marble,
record its color and replace it in the bag before
drawing another. If she does this 75 times, how many
times should she draw a green or red marble?
A. 41 times
D. 63 times
0%
D
A
B
0%
C
D
C
A
C. 55 times
A.
B.
0% C.0%
D.
B
B. 28 times
(over Lesson 8-7)
Coach Riggins needs to select 4 players at random from his
football team to go to a football camp. If there are 36 players
on his team, describe a model that he could use to simulate
choosing these 4 players.
A.
There are 36 players, so select objects that
combined have 4 outcomes, such as 2 coins.
Assign each player one of the possible outcomes
and then flip the coin at least 4 times to choose
the players.
B.
There are 36 players, so select objects that
combined have 6 outcomes, such as a coin and a
spinner divided into 3 equal sections. Assign each
player one of the possible outcomes and then flip
the coin at least 4 times to choose the players.
C.
There are 36 players, so select objects that
combined have 36 outcomes, such as 2 number
cubes. Assign each player one of the possible
outcomes and then roll the cubes 4 times to
choose the players.
0%
1.
2.
3.
A
B
C
A B C
(over Lesson 8-7)
Juan rolls a number cube and spins a spinner
divided into 4 equal parts labeled A, B, C, and D
100 times. How many times should he expect to
have the number cube show a 2, 3, or 4 and the
spinner show a B or D?
A. 4 times
B. 10 times
1.
2.
3.
4.
A
B
C
D
0%
C. 16 times
A
D. 25 times
B
C
D
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