# Chapter 8

```CHAPTER 8
Probability and Predictions
Chapter 8: Section 1: Part A
Probability and Simple Events
Book pages 429 - 434
8-1-A: Probability and Simple Events

A cheesecake has four
equal slices of each piece
as shown.
 What
fraction of cheesecake
is chocolate? Write in simplest
form.
 Suppose your friend gives you
the first piece of cheesecake without asking you
which type you prefer. Are your chances of
getting original the same as getting raspberry?
Explain.
8-1-A: Probability and Simple Events


An outcome is any one of the possible results of an
action.
A simple event has one outcome or a collection of
outcomes.


For example, getting a piece of chocolate
cheesecake is a simple event.
The chance of that event happening is called its
Favorable outcomes are
probability.
the outcomes for which you
are finding the probability.
PROBABILITY
If all outcomes are equally likely, the probability of a
simple event is a ratio that compares the number of
favorable outcomes to the number of possible
outcomes.
SYMBOLS:
8-1-A: Probability and Simple Events

Find Probability

What is the probability of drawing an ace from a
deck of cards?


What is the probability of drawing a red card?



So the probability is &frac12;, 0.5, or 50%.
Now you try!
P(drawing a heart)
 P(drawing a face card)

1/4
3/13
8-1-A: Probability and Simple Events


Outcomes occur at random if each outcomes occurs by
chance. For example, the number that results when
rolling a number cube is a random outcome.
The probability that an event will happen can be any
number from 0 to 1, including 0 and 1. Notice that
probabilities can be written as fractions, decimals, and
percents.
8-1-A: Probability and Simple Events


Simone and her three friends were deciding how to pick
the song they will sing for their school’s talent show. They
decide to roll a number cube. The person with the lowest
number chooses the song. If her friends rolled a 6, 5, and
2, what is the probability that Simone will get to choose
the song?
The table shows the number of brass instrument players in
the New York Philharmonic. Suppose one brass player is
randomly selected to be a featured performer. Find the
probability of each event. Write your New York Philharmonic
answer as a fraction in simplest form. Brass Instrument Players




P(trumpet)
P(brass)
P(flute)
P(horn or tuba)
Horn
6
Trombone
4
Trumpet
3
Tuba
1
8-1-A: Probability and Simple Events

In one of the previous problems, either Simone will go first
or she will not go first. These two events are
complementary events. Two events are complementary
events if they are the only two possible outcomes.
The sum of the probability of an event and its complement is
1 or 100%.
 SYMBOLS: P(A) + P(not A) = 1


Example:

Ricardo’s teacher uses a spinner similar to the
one shown at the right to determine the order
in which each group will make its
presentation. Use the spinner to find each
probability. Write as a fraction in simplest
form.
P(not group 4)
 P(not group 1 or group 3)

8-1-A: Probability and Simple Events



Geometric probability deals with the areas of figures
instead of the number of outcomes. For example, the
probability of landing in a specific region of a target is
the ratio of the area of the specific region to the area
of the target.
Try these!
Self Assessment: On your own, complete # 1 – 10 on page 431.
Chapter 8: Section 1: Part B
Sample Spaces
Book pages 435 - 439
8-1-B: Sample Spaces

The set of all possible outcomes in a probability experiment is
called the sample space.


For instance, the sample space for rolling a number cube is 1, 2, 3,
4, 5, and 6 because those are the possible outcomes.
A tree diagram is a display that represents a sample space.
A vendor sells vanilla and chocolate ice cream. Customers can
choose from a waffle or sugar cone. Find the sample space for all
possible orders of
one scoop of ice
cream in a cone.
NOTE: When creating a
tree diagram, you must
start the diagram with
one of the possibilities.
 YOU TRY!
The animal shelter has both male and female Labradors in yellow,
brown, or black. Find the sample space for all possible Labradors
at the shelter.

8-1-B: Sample Spaces

Find the Probability

Delmar tosses three coins. If all three coins show up heads, Delmar wins.
Otherwise, Kara wins. Find the sample space. Then find the probability
that Delmar wins.
COIN 1
COIN 2
COIN 3
 Sample Space:
 Probability:


Look at the last column of the tree diagram.
Here is where you can find the total number of Heads
outcomes. In this case, there are 8. Out of those
8, there is only ONE option with three heads.
 The probability of Delmar winning
is 1/8.
Now you try!


Tails
Tails
Tails
Tails
Elba tosses a quarter, a dime, and a nickel.
Tails
If tails comes up at least twice, Steve wins.
Tails
Otherwise, Elba wins. Find the sample space for the
Tails
game. Then find the probability that Elba will win.
Ming rolls a number cube, tosses a coin, and chooses a card from two
cards marked A and B. If an even number and a heads appears, Ming
wins no matter which card is chosen. Otherwise, Lashonda wins. Find the
sample space. Then find the probability that Ming will win.
8-1-B: Sample Spaces



Another way to describe the chance of an event occurring is with
odds. The odds in favor of an event is a ratio that compares the
number of ways the even can occur to the number of ways that
the event cannot occur.
Find the odds in favor of rolling a 5 or 6 on a number cube.

How many ways can the event occur?


How may ways can the event NOT occur?




4 ways: 1, 2, 3, or 4
What are the odds in favor?


2 ways: a 5 or a 6
2:4  1:2
You try!
A letter from the alphabet is chosen at random. Find the odds in
favor of picking an A, E, U, or O.
What is the difference between finding a probability and
finding odds in favor?
Self Assessment: On your own, complete # 1 – 4 on page 436.
Chapter 8: Section 1: Part C
Count Outcomes
Book pages 440 - 443
8-1-C: Count Outcomes

Off Broadway Shoes sells sandals in different colors
and styles.
1.
2.
3.
4.
According to the
table, how may colors
of sandals are
available?
How many styles are
available?
Draw a tree diagram
to find the number of
different color and style combinations.
Find the product of the two numbers you found in 1
and 2. How does this number compare to the
number of combinations you found in 3.
8-1-C: Count Outcomes

On the previous slide, you discovered that
multiplication, instead of a tree diagram, can be
used to find the number of possible outcomes in a
sample space. This is called the Fundamental
Counting Principle.
THE FUNDAMENTAL COUNTING PRINCIPLE
If event M has m possible outcomes
and event N has n
c
possible outcomes, then event M followed by event N
has m &times; n possible outcomes.
 The
counting principle gives the number of
possible outcomes whereas the tree diagram
gives the entire sample space.
8-1-C: Count Outcomes

Find the Number of Outcomes
Find the total number
of outcomes when a
coin is tossed and a number
cube is rolled.
 You try!
 Find the total number of outcomes when choosing
from bike helmets that come in three colors and two
styles.


6
 Find
the total number of outcomes when a number
from 0 to 9 is picked randomly and when a letter
from A to D is picked randomly.

40.
8-1-C: Count Outcomes

The Gap sells young men’s
jeans in different sizes, styles,
and lengths. Find the number
of jeans available. Then find
the probability of randomly
selecting a size 32 &times; 34 slim fit.
Is it likely or unlikely that the
size would be chosen?


There are 45 different types of jeans to choose from. Out of
the 45 possible outcomes, only one is favorable. So the
probability of selecting 32 &times; 34 slim fit is 1/45 or about 2%. It is
very unlikely that the size would be chosen at random.
8-1-C: Count Outcomes


If the Gap adds relaxed fit
jeans to its selection, find the
number of available jeans.
Then find the probability of
randomly selecting a 36 &times; 30
relaxed fit pair of jeans. Is it
likely or unlikely that the size
would be chosen?
A shoe store sells shoes in different
sizes, widths, and styles, as shown in
the table. Find the number of pairs
of shoes available. Then find the
probability of selecting a size 8 in
medium width sneakers. Is it likely or
unlikely that the size would be
chosen at random?
Self Assessment: On your own, complete # 1 – 3 on page 441.
Chapter 8: Section 1: Part D
Permutations
Book pages 444 - 447
8-1-D: Permutations

In how many different ways can you arrange the
letters in the word DOG? Let’s list them on the board!
DOG, DGO, ODG, OGD, GOD, GDO

When we first started making the list, how many choices did
we have for our first letter?


Once our first letter was selected, how many choices did we
have for the second letter? For the third?


3
2, 1
A permutation is an arrangement, or listing, of objects
in which order is important. The arrangement of DGO is
a permutation of DOG because the order of the letters
is different.

You can use the Fundamental Counting Principle to find the
number of permutations.
8-1-D: Permutations

Find a Permutation


How many different ways are there to arrange your first three
classes if they are math, science, and language arts?
Using the Counting
Principle, find the
number of possible
arrangements.
You try!

In how many ways can the starting six players of a volleyball team
stand in a row for a picture?


A team of bowlers has five members, who bowl one at a time. In
how many orders can they bowl?


720
120
How many 3-digit numbers can be formed using the digits 9, 3, 4,
7, and 6?

60
8-1-D: Permutations

Find Probability



The finals of the Northwest Swimming League features
8 swimmers. If each swimmer has an equally likely
chance of finishing in the top two, what is the
probability that Yumii will be in first place and
Paquita in second place?
There are 56 possible arrangements of the first and second place finishers. Of those
56, there is only one arrangement in which Yumii is first and Paquita is second. So,
the probability is 1/56.
You try!

Two letters are randomly selected from the letters in the word math. What is the
probability that the first letter selected is m and the second letter is h?


1/12
A school fair holds a raffle with 1st, 2nd, and 3rd prizes. Seven people enter the raffle,
including Mark, Lilly, and Heather. What is the probability that Mark will win the 1st
prize, Lilly will win the 2nd prize, and Heather will win the 3rd prize?

1/210
8-1-D: Permutations

 An arrangement, or listing, of objects in which order is not important is
called a combination. You can find the number of combinations of
objects by dividing the number of permutations of the entire set by the
number of ways the smaller set can be arranged.


This just means that DOG would be the same as DGO, and OGD, etc. because
the same three letters are used.
A one-on-one basketball tournament has 6 players. How many different
pairings are there in the first round.

Find the number of ways 2 players can be chosen from a group of 6.



So there are 15 different pairings in the first round.
You try!

Of 12 websites, in how many ways can you choose
to visit 6?


924
There are 8 members on the debate team. How many different
5-player teams are possible?

56
Self Assessment: On your own, complete # 1 – 4 on page 445.
Chapter 8: Section 2: Part A
EXPLORE Independent and Dependent Events
Book page 449
8-2-A: Explore Independent and Dependent Events

Place two red counters and two white counters in a
paper bag. Then complete the following:

1.
2.
3.

1.
2.
3.

1st Experiment
Without looking, remove a counter from the bag and record its color.
Place the counter back in the bag.
Without looking, remove a second counter from the bag and record its
color. The two colors are one trial. Place the counter back in the bag.
Repeat until you have 50 trials. Count and record the number of times
you choose a red counter followed by a white counter.
2nd Experiment
Without looking, remove a counter from the bag and record its color.
This time, do not replace the counter.
Without looking, remove a second counter and record its color. The two
colors are one trial. Place both counters back in the bag.
Repeat until you have 50 trials. Count and record the number of times
you choose a red counter followed by a white counter.
We will discuss your findings as a class.
Chapter 8: Section 2: Part B
Independent and Dependent Events
Book pages 450 - 455
8-2-B: Independent and Dependent Events

A sale advertises that if you
buy an item from the column
of the left, you get a tote bag
items at random.
1.
2.
3.

Type of Item
T-shirt
jacket
hat
beach towel
visor
polo shirt
Tote Bag
Colors
green
red
white
What is the probability of
buying a beach towel? receiving a red tote bag?
What is the product of the probabilities in question 1?
Draw a tree diagram to determine the probability
red tote bag.
The combined action of buying an item and
receiving the free tote bag is a compound event.
8-2-B: Independent and Dependent Events

A compound event consists of two or more
simple events. These events are independent
events because the outcome of one event
does not affect the other event.
Probability of Independent Events
The probability of two independent events can be
found by multiplying the cprobability of the first
event by the probability of the second event.
P(A and B) = P(A) ∙ P(B)
8-2-B: Independent and Dependent Events

One letter tile is selected and the
spinner is spun. What is the
probability that both will be a
vowel?




P(selecting a vowel) = 2/7
P(spinning a vowel) = 2/6 or 1/3
P(both letters are vowels) =
A spinner and a number cube are used in a game. The
spinner has an equal chance of landing on one of five
colors: red, yellow, blue, green, or 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?



What is P(spinning blue)?
What is P(rolling a 3 or 4)?
Then what is P(blue then 3 of 4)?
8-2-B: Independent and Dependent Events

You try!
A
game requires players to roll two number cubes
to move the game pieces. The faces of the cubes
are labeled 1 through 6. What is the probability of
rolling a 2 or 4 on the first number cube
and then rolling a f5 on the second?
 1/18
 The
two spinners are spun. What is the
probability that both spinners will show
a number greater than 6?
 9/100
8-2-B: Independent and Dependent Events

If the outcome of one event affects the
outcome of another event, the events are
called dependent events.
Probability of Dependent Events
If two events A and B are dependent, then
the probability of both
events occurring is
c
the product of the probability of A and
the probability of B after A occurs.
P(A and B) = P(A) ∙ P(B following A)


Independent: outcome does not rely on another event
Dependent: relying on another quantity or action
8-2-B: Independent and Dependent Events

There are 4 oranges, 7 bananas, and
5 apples in a fruit basket. Ignacio
selects a piece of fruit at random and
then Terrance selects a piece of fruit
at random. Find the probability that
two apples are chosen.
Since the first piece of fruit is not
replaced, the first event affects the second event. These are dependent
events.
There are 4 red, 8
 P(first piece is an apple) = 5/16
 P(second piece is an apple) = 4/15 yellow, and 6 blue socks
mixed up in a drawer.
 P(two apples) =
Once a sock is
You try!
selected, it is not
 P(two bananas)
7/40
replaced. Find the
 P(orange then apple)
1/12
probability of reaching
into the drawer without
 P(apple then banana)
7/48
looking at choosing two
1/20
 P(two oranges)
blue socks.
5/51


8-2-B: Independent and Dependent Events



A number cube is rolled. What is the probability of rolling a
multiple of 3 or a 1?



Sometimes, two events cannot happen at the same
time. For example, when a coin is tossed, either heads
or tails will turn up. Tossing heads and tossing tails are
examples of disjoint events, or events that cannot
happen at the same time. Disjoint events are also
called mutually exclusive events.
There are three favorable outcomes, 3, 6, or 1.
So the probability of rolling a multiple of 3 or a 1 is &frac12;.
You try!

A number cube is rolled. Find each probability. Write as a fraction
in simplest form.



1/3
P(4 or 5)
2/3
P(3 or even number)
2/3
P(1 or multiple of 2)
Self Assessment: On your own, complete # 1 – 4 on page 445.
Chapter 8: Section 3: Part A
Theoretical and Experimental Probability
Book pages 458 - 462
8-3-A: Theoretical and Experimental Probability

EXPLORE Follow the steps to determine how many times doubles are
expected to turn up when two number cubes are rolled.
1.
2.
expected number of times doubles
should turn up when rolling two
number cubes 36 times. The top
row represents one number cube,
and the left column represents the
other number cube.
Roll two number cubes 36 times.
Record the number of times doubles
turn up.
experiment.

1.
2.
Compare the number of times you expected to roll doubles with the
number of times you actually rolled doubles.
Write the probability of rolling doubles out of 36 rolls using the number
of times you expected to roll doubles from Step 1. Then write the
probability of rolling doubles out of 36 rolls using the number of times
you actually rolled doubles from Step 2.
8-3-A: Theoretical and Experimental Probabilityc

Theoretical probability is based on what should
happen when conducting an experiment.


Experimental probability is based on what actually
occurred during such an experiment.


This is the probability you wrote down based on what you
expected to roll.
This is the probability you wrote down based on what you
actually rolled.
The theoretical probability and the experimental
probability of an event may or
may not be the same. As the
number of times an experiment is
conducted increases, the theoretical
probability and the experimental
probability should become closer
together.
8-3-A: Theoretical and Experimental Probability

Experimental Probability

When two number cubes are rolled together 75 times, a
sum of 9 is rolled 10 times. What is the experimental
probability of rolling a sum of 9?



The experimental probability of rolling a sum of 9 is 2/15.
You try!

In the above experiment, what is the experimental
probability of rolling a sum that is not 9?


13/15
A spinner is spun 100 times, and it lands on green 32 times.
What is the experimental probability of spinning green?

8/25
8-3-A: Theoretical and Experimental Probability

Experimental and Theoretical Probability


The graph shows the results of an
experiment in which a spinner with 3
equal sections is spun sixty times. Find
the experimental probability of spinning
red for this experiment.

The graph indicated that the spinner
landed on red 24 times, blue 15 times,
and green 21 times.

The experimental probability is 2/5.
Compare the experimental probability you found above to its
theoretical probability.


The spinner has three equal sections: red, blue, and green. So, the
theoretical probability of spinning red is 1/3.
Since 2/5 is close to 1/3, then the experimental probability is close to
the theoretical probability.
8-3-A: Theoretical and Experimental Probability

You try!

Refer to the table again. If the spinner
was spun three more times and
landed on green each time, find the
experimental probability of spinning
green for this experiment.


Compare the experimental probability
you found above to its theoretical
probability.


8/21
8/21 is close to 1/3
The graph shows the results of an
experiment in which a number cube is
rolled 50 times. Find the experimental
probability of rolling a 4.


9/50
Compare the experimental probability
you found above to its theoretical
probability.

1/6 is close to 9/50.
8-3-A: Theoretical and Experimental Probability


Theoretical and experimental probability can be used to
Last year’s DVD sales at the local video store are shown. What
is the probability that a
customer bought a comedy DVD last
year?


Suppose a medial buyer expects to sell 5,000
DVDs this year. How many comedy DVDs


She should buy 1,450 comedy DVDs.
8-3-A: Theoretical and Experimental Probability

You try!
 Mrs.
about what they are going to choose for lunch this
week. Thirty students choose pizza, 17 students
choose macaroni and cheese, 12 students choose
hamburgers, and 5 students choose chicken
fingers. What is the experimental probability of
someone choosing hamburgers?
 3/16
 Suppose
Mrs. Ramirez surveys all 1,200 students in
the school. How many students can she expect to
choose hamburgers for lunch?
 225 students
Self Assessment: On your own, complete # 1 – 2 on page 460.
Chapter 8: Section 3: Part E
Use Data to Predict
Book pages 468 - 471
8-3-E: Use Data to Predict

The circle graph shows the results of a
survey in which children ages 8 to 12
were asked whether they have a
television in their bedroom.



Can you tell how many were surveyed?
Explain.
Describe how you could use the graph to
predict how many students in your school
have a television in their bedroom?
A survey is designed to collect data about a specific
group of people, call the population. A smaller group of
people called a sample must be chosen. A sample is
used to represent a population.

If a survey is conducted at random, or without preference, you
can assume that the survey represents the population. In this
lesson, you will use the results of randomly conducted surveys to
8-3-E: Use Data to Predict

Refer to the circle graph again.
Predict how many out of 1,725
students would not have a
television in their bedroom.

You can use the percent equation
and the survey results to predict what
part p of the 1,725 students have no
TV in their bedroom.



About 932 students do not have a television in their bedroom.
Refer to the same graph. Predict how many out of
1,370 students have a television in the bedroom?

8-3-E: Use Data to Predict

Use this information to predict how many of
the 2,450 students at Washington Middle
School use emoticons on their instant
messengers.
 You need to predict how many of the 2,450
students use emoticons.
 n = 0.85 ∙ 2,450
n = 2,082.5
8-3-E: Use Data to Predict

You try!

This same survey found that 59% of people use sound on their
instant messengers. Predict how many of the 2,450 students use
sound on their instant messengers.


The table shows the results of a
survey in which people were asked
whether their house pets watch
television.
There are 540 students at McCloskey
Middle School who owns pets. Predict
how many of them would say their
pets watch TV.


According to one survey, 25% of high school students reported
that they would not get summer jobs. Predict how many of the
948 students at Mohawk High School will not get summer jobs.

Self Assessment: On your own, complete # 1 – 2 on page 469.
Chapter 8: Section 3: Part F
Unbiased and Biased Samples
Book pages 472 - 476
8-3-F: Unbiased and Biased Samples

The manager of a News Channel 5 wants to
conduct a survey to determine which sport
people consider their favorite to watch.
1.
2.
3.
Suppose she surveys a group of 100 people at a
basketball game. Do the results represent all of
the people in the viewing area? Explain.
Suppose she surveys 100 students at your middle
school. Do the results represent all of the people
in the viewing area? Explain.
Suppose she calls every 100th household in the
telephone book. Do the results represent all of
the people in the viewing area? Explain.
8-3-F: Unbiased and Biased Samples

The manager of News Channel 5 cannot survey everyone in the
viewing area. To get valid results, a sample must be chosen very
carefully. An unbiased sample is selected so that it accurately
represents the entire population. Two ways to pick an unbiased
sample are below.
Unbiased Samples
Type
Description
Example
Simple Random
Sample
Each item of 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.
Systematic
Random Sample
The items or people are
selected according to a
specific time or item interval.
Every 20th person is chosen
from an alphabetical list of all
students attending a school.
8-3-F: Unbiased and Biased Samples

In a biased sample, one of more parts of the population
are favored over others. Two ways to pick a biased
sample are listed below.
Biased Samples
Type
Description
Example
Convenience
Sample
A convenience sample
consists of members of a
population that are easily
accessed.
To represent all the students
attending a school, the
principal surveys students in
one math class.
Voluntary
Response
Sample
A voluntary response sample
Students at a school who wish
involves only those who want
to express their opinions
to participate in the
complete an online survey.
sampling.
8-3-F: Unbiased and Biased Samples

Determining the Validity of Conclusions



for one or two candidates in an upcoming election.
Seventy-two percent of the listeners who responded
preferred candidate A, so the radio station announced that
candidate A would win the election. Is the conclusion valid?
Every tenth students who walked into the cafeteria was
surveys to determine his of her favorite lunch. Out of 40
students, 19 students stated that they liked the burgers best.
The cafeteria staff concluded that about 50% of the
students like burgers best. Is the conclusion valid? Justify
Janet surveyed the student athletes on the girls’ field
hockey team to determine which sports teenagers liked
best. Of these, 65% said that they liked field hockey the
best. Janet concluded that over half of teenagers like field
8-3-F: Unbiased and Biased Samples

A valid sampling method uses unbiased samples. If a sampling
method is valid, you can make generalizations about the
population.

A store sells 4 styles of pants: jeans, capris, cargos,
and khakis. The store workers survey 50 customers
at random. The survey responses are indicated at
the right. If 450 pairs of pants are to be ordered,
how many should be jeans?

First, determine whether the sampling method was
valid. The sample is a simple random sample since
customers were randomly selected. Thus the sample method is valid.



25/50 or 50% of the customers prefer jeans. So, find 50% of 450.
0.5 &times; 450 = 225, so about 225 pairs of jeans should be ordered.
You try!

An instructor at a swimming pool asker her students if they would
be interested in an advanced swimming course, and 60% stated
that they would. Is the sample method valid? If so, suppose there
are 870 pool members. How many people can the instructor
expect to take the course?
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