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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
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7th Grade Math
Probability
2013-05-31
www.njctl.org
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PROBABILITY
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Sampling
Click on a topic to
go to that section.
Comparing Two Populations
Introduction to Probability
Experimental and Theoretical
Word Problems
Fundamental Counting Principle
Permutations and Combinations
Probability of Compound Events
Probabilities of Mutually Exclusive and Overlapping Events
Complementary Events
Common Core: 7.SP.1-8
Sampling
Return to table
of contents
Your task is to count the number of whales in the
ocean or the number of squirrels in a park.
How could you do this?
What problems might you face?
A sample is used to make a prediction about an event
or gain information about a population.
A whole group is called a POPULATION.
A part of a group is called a SAMPLE.
A sample is considered random (or unbiased) when every
possible sample of the same size has an equal chance of
being selected. If a sample is biased, then information
obtained from it may not be reliable.
Example: To find out how many people in New York feel
about mass transit, people at a train station are asked
their opinion. Is this situation representative of the general
population?
No. The sample only includes people who take the train
and does not include people who may walk, drive, or bike.
Determine whether the situation would produce a random
sample.
You want to find out about music preferences of people
living in your area. You and your friends survey every tenth
person who enters the mall nearest you.
1
Food services at your school wants to
increase the number of students who eat
hot lunch in the cafeteria. They conduct a
survey by asking the first 20 students that
enter the cafeteria to determine the
students' preferences for hot lunch. Is this
survey reliable? Explain your answer.
A Yes
B No
2
The guidance counselors want to organize a
career day. They will survey all students
whose ID numbers end in a 7 about their
grades and career counseling needs. Would
this situation produce a random sample?
Explain your answer.
A Yes
B No
3
The local newspaper wants to run an
article about reading habits in your town.
They conduct a survey by asking people in
the town library about the number of
magazines to which they subscribe. Would
this produce a random sample? Explain
your answer.
A Yes
B No
How would you estimate the size of a crowd?
What methods would you use?
Could you use the same methods to estimate the
number of wolves on a mountain?
One way to estimate the number of wolves on a
mountain is to use the CAPTURE - RECAPTURE
METHOD.
Suppose this represents all the wolves on the mountain.
Wildlife biologists first find some wolves and tag them.
Then they release them back onto the mountain.
They wait until all the wolves have mixed together.
Then they find a second group of wolves and count
how many are tagged.
Biologists use a proportion to estimate the total number
of wolves on the mountain:
tagged wolves on mountain
=
total wolves on mountain
tagged wolves in second group
total wolves in second group
For accuracy, they will often conduct more than one
recapture.
8 = 2
w
9
2w = 72
w = 36
There are 36 wolves on the mountain
Try This:
Biologists are trying to determine how many fish are in
the Rancocas Creek. They capture 27 fish, tag them and
release them back into the Creek. 3 weeks later, they
catch 45 fish. 7 of them are tagged. How many fish are
in the creek?
There are 174 fish in the river
27
7
=
f
45
27(45) = 7f
1215 = 7f
173.57 = f
A whole group is called a POPULATION.
A part of a group is called a SAMPLE.
When biologists study a group of wolves, they are
choosing a sample. The population is all the wolves on
the mountain.
Population
Sample
Try This:
315 out of 600 people surveyed voted for Candidate A.
How many votes can Candidate A expect in a town with
a population of 1500?
4
860 out of 4,000 people surveyed watched
Dancing with the Stars. How many people
in the US watched if there are 93.1 million
people?
5
Six out of 150 tires need to be realigned.
How many out of 12,000 are going to need
to be realigned?
6
You are an inspector. You find 3 faulty
bulbs out of 50. Estimate the number of
faulty bulbs in a lot of 2,000.
7
You survey 83 people leaving a voting site.
15 of them voted for Candidate A. If 3,000
people live in town, how many votes
should Candidate A expect?
8
The chart shows the number of people
wearing different types of shoes in Mr.
Thomas' English class. Suppose that there
are 300 students in the cafeteria. Predict
how many would be wearing high-top
sneakers. Explain your reasoning.
Shoes
Number of Students
Low-top sneakers
12
High-top sneakers
Sandals
Boots
7
3
6
Multiple Samples
The student council wanted to determine which lunch was
the most popular among their students. They conducted
surveys on two random samples of 100 students. Make at
least two inferences based on the results.
Student Sample Hamburgers Tacos Pizza
#1
#2
12
12
14
11
74
77
Total
100
100
• Most students prefer pizza.
• More people prefer pizza than hamburgers and tacos
combined.
Try This!
The NJ DOT (Department of Transportation) used two
random samples to collect information about NJ drivers.
The table below shows what type of vehicles were being
driven. Make at least two inferences based on the results
of the data.
Driver Sample
#1
#2
Cars
37
33
SUVs
43
46
Mini Vans
12
11
Motorcycles
8
10
Total
100
100
The student council would like to sell potato chips at the
next basketball game to raise money. They surveyed
some students to figure out how many packages of each
type of potato chip they would need to buy. For home
games, the expected attendance is approximately 250
spectators. Use the chart to answer the following
questions.
Student
Sample
#1
#2
Regular
BBQ
Cheddar
8
8
10
11
7
6
9
How many students participated in each
survey?
10 According to the two random samples, which
flavor potato chip should the student council
purchase the most of?
A
B
C
Regular
BBQ
Cheddar
11
Use the first random sample to evaluate the
number of packages of cheddar potato
chips the student council should purchase.
Comparing Two Populations
Return to table
of contents
Measure of Center - Vocabulary Review
Mean (Average) - The sum of the data values divided by the
number of items
Median - The middle data value when the values are written in
numerical order
Mode - The data value that occurs the most often
Measures of Variation - Vocabulary Review
Range - The difference between the greatest data value and
the least data value
Quartiles - are the values that divide the data in four equal
parts.
Lower (1st) Quartile (Q1) - The median of the lower half of the
data.
Upper (3rd) Quartile (Q3) - The median of the upper half of
the data.
Interquartile Range - The difference of the upper quartile and
the lower quartile. (Q3 - Q1)
Mean absolute deviation - the average distance between
each data value and the mean.
Example:
Victor wants to compare the mean height of the players on his favorite
basketball and soccer teams. He thinks the mean height of the players
on the basketball team will be greater but does not know how much
greater. He also wonders if the variability of heights of the athletes is
related to the sport they play. He thinks that there will be a greater
variability in the heights of soccer players as compare to basketball
players. He uses the rosters and player statistics from the team
websites to generate the following lists.
Height of Soccer Players (inches)
73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74,
73, 67, 70, 72, 69, 78, 73, 76, 69
Height of Basketball Players (inches)
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84
from http://katm.org/wp/wp-content/uploads/flipbooks/7th_FlipBookEdited21.pdf
Victor creates two dot plots on the same scale.
x
65
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
80
75
70
85
Height of Soccer Players (inches)
x
65
70
x
75
x
x x x
x x x
80
Height of Basketball Players (inches)
x
x
x
x
x
x
x
85
Victor notices that although generally the basketball players
are taller, there is an overlap between the two data sets.
Both teams have players that are between 73 and 78 inches
tall.
x
65
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
70
x
x
x
x
x
x
x
75
x
80
85
Height of Soccer Players (inches)
x
65
70
x x
75
x x x
x x x
x
x
80
Height of Basketball Players (inches)
x
x
x
x
x
85
Since the teams have different numbers of players, Victor wants to
determine the mean height for each team.
mh1 =
Sum of Soccer Player's Height
Total # of Soccer Players
= 2090 in = 72.07 in
29
The mean height of a soccer
player is about 72 inches.
mh2 =
Sum of Basketball Player's Height
Total # of Basketball Players
= 1276 in = 79.75 in
16
The mean height of a basketball
player is about 80 inches.
The difference between the means (mh2 - mh1) is about 8 inches.
The mean absolute deviation can tell us more about the
variability of data within a set. Victor decides to calculate that
next.
mad = sum of absolute deviation values*
# of players
* absolute deviation value = distance between mean height and individual height value
Example with Soccer players (1st data value):
65 in - 72 in = 7 in
mad1 = sum of absolute deviation values
mad2 = sum of absolute deviation values
# of soccer players
= 62 in = 2.14 in
29
The mean absolute deviation is
2.14 inches for a soccer player.
This means that the average
soccer player varies 2.14 inches
in height from 72 inches.
# of basketball players
= 40 in = 2.5 in
16
The mean absolute deviation is
2.5 inches for a soccer player.
This means that the average
basketball player varies 2.5
inches in height from 80 inches.
The mean absolute deviations for both teams are very close.
This means the variabilities are similar.
To express the difference between centers of two data
sets as a multiple of a measure of variability, first find
the difference between the centers.
*Recall: The difference between the means is
79.75 – 72.07 = 7.68.
Divide the difference by the mean absolute deviations
of each data set.
7.68 ÷ 2.14 = 3.59
7.68 2.5 = 3.07
The difference of the means (7.68) is approximately 3
times the mean absolute deviations.
Use the following data to answer the next set of questions.
Pages per Chapter in Hunger Games
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
10
x
x
x
x
x
x
x
x
x
20
15
25
30
Pages per Chapter in Twilight
x
10
x
x
x
x
15
x
x
x
x
x
x
x
x
20
x
x
x
x
25
x
x
x
x
x
30
12 What is the mean number of pages per
chapter in the Hunger Games?
13 What is the mean number of pages per
chapter in Twilight?
14 What is the difference of the means?
15 What is the mean absolute deviation of the
data set for Hunger Games?
(Hint: Round mean to the nearest ones.)
16 What is the mean absolute deviation of the
data set for Twilight?
(Hint: Round mean to the nearest ones.)
17 Which book has more variability in the number
of pages per chapter?
A Hunger Games
B Twilight
18 The difference of the means between the two
data sets is approximately ______ times the
mean absolute deviation for Twilight?
(Round your answers to the nearest tenths.)
Introduction to
Probability
Return to table
of contents
Probability
One way to express probability is to use a fraction.
Probability
of an event
Number of
favorable outcomes
=
Total number of
possible outcomes
Probability
Example: What is the probability of flipping a nickel and
the nickel landing on heads?
Step 1: What are the possible outcomes?
click
Step 2: What is the number of favorable outcomes?
click
Step 3: Put it all together to answer the question.
The probability of flipping a nickel and landing on
heads is: 1 .
2
Probability can be expressed in many forms. For
example, the probability of flipping a head can be
expressed as:
1
2
or 50% or 1:2 or .5
The probability of randomly selecting a blue marble can
be expressed as:
1 or 1:6 or 16.7% or .167
6
When there is no chance of an event occurring, the
probability of the event is zero (0).
When it is certain that an event will occur, the probability of
the event is one (1).
0
1
4
1
2
3
4
1
The less likely it is for an event to occur, the probability
is closer to 0 (i.e. smaller fraction).
The more likely it is for an event to occur, the
probability is closer to 1 (i.e. larger fraction).
Without counting, can you determine if the probability of
picking a red marble is lesser or greater than 1/2?
It is very likely you will pick a red marble, so the
Click to Reveal
probability is greater than 1/2 (or 50% or 0.5)
What is the probability of picking a red marble?
5
6
Click
to
Reveal
Add the probabilities of both events. What is the sum?
1 + 5 =1
6Click to6Reveal
Note:
The sum of all possible outcomes is always equal to 1.
There are three choices of jelly beans - grape, cherry and
orange. If the probability of getting a grape is 3/10 and the
probability of getting cherry is 1/5, what is the probability
of getting orange?
3 + 1 + ? =1
10 5
?
5 + ? = 1
10 ?
The probability of getting an orange jelly bean is 5 .
19 Arthur wrote each letter of his name on a
separate card and put the cards in a bag.
What is the probability of drawing an A from
the bag?
A
B
C
D
A
0
1/6
1/2
1
Probability =
R
T
Need
a hint?
Number
of favorable outcomes
Click
the box.of possible outcomes
Total
number
H
U
R
20 Arthur wrote each letter of his name on a
separate card and put the cards in a bag.
What is the probability of drawing an R from
the bag?
A 0
Probability = Need
Number
of favorable outcomes
a hint?
B 1/6
Total
Clicknumber
the box.of possible outcomes
C 1/3
D 1
A
R
T
H
U
R
21 Matt's teacher puts 5 red, 10 black, and 5
green markers in a bag. What is the
probability of Matt drawing a red marker?
A 0
B 1/4
C 1/10
D 10/20
Probability =
Need
hint?
Number
ofafavorable
outcomes
Total number
of possible
outcomes
Click the
box.
22 What is the probability of rolling a 5 on a fair
number cube?
23 What is the probability of rolling a composite
number on a fair number cube?
24 What is the probability of rolling a 7 on a fair
number cube?
25 You have black, blue, and white t-shirts in
your closet. If the probability of picking a
black t-shirt is 1/3 and the probability of
picking a blue t-shirt is 1/2, what is the
probability of picking a white t-shirt?
26 If you enter an online contest 4 times and at
the time of drawing its announced there were
100 total entries, what are your chances of
winning?
27 Mary chooses an integer at random from 1 to
6. What is the probability that the integer she
chooses is a prime number?
A
B
C
D
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,
June, 2011.
28
Each of the hats shown below has colored marbles placed
inside. Hat A contains five green marbles and four red marbles.
Hat B contains six blue marbles and five red marbles. Hat C
contains five green marbles and five blue marbles.
If a student were to randomly pick one marble from each of
these three hats, determine from which hat the student would
most likely pick a green marble. Justify your answer.
Hat A
Hat B
Hat C
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,
June, 2011
Determine the fewest number of marbles, if any, and the color of these
marbles that could be added to each hat so that the probability of
picking a green marble will be one-half in each of the three hats.
Hat A contains five green marbles and four red marbles.
Hat B contains six blue marbles and five red marbles.
Hat C contains five green marbles and five blue marbles.
Hat A
Hat B
Hat C
From the New York State Education Department. Office of Assessment Policy, Development
and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra;
accessed 17, June, 2011
Experimental
&
Theoretical
Probability
Return to table
of contents
What are the possible outcomes of each of these?
Experimental Probability
Probability
of an event
number of times the outcome happened
number of times experiment was repeated
Flip a coin 5
times and
determine the
experimental
probability of
heads.
Heads
Tails
Experimental Probability
Example 1 - Golf
A golf course offers a free game to golfers who make a
hole-in-one on the last hole. Last week, 24 out of 124
golfers achieved this. Find the experimental probability
that a golfer makes a hole-in-one on the last hole.
P(hole-in-one) =
# of successes
# of trials
= 24 = 6
124
31
Out of 31 golfers, you could expect 6 to make a
hole-in-one on the last hole. Or there is a 19% chance of a
golfer making a hole-in-one on the last hole.
Experimental Probability
Example 2 - Surveys
Of the first 40 visitors through the turnstiles at an
amusement park, 8 visitors agreed to participate in a
survey being conducted by park employees. Find
the experimental probability that an amusement park
visitor will participate in the survey.
P(participation) =
# of successes
# of trials
= 8 = 1
40
5
You could expect 1 out of every 5 people to
participate in the survey. Or there is a 20% chance
of a visitor participating in the survey.
Sally rolled a die 10 times and the results are shown below.
Use this information to answer the following questions.
# on Die
Picture of Roll
Results
1
1 one
2
3 twos
3
1 three
4
0 fours
5
4 fives
6
1 six
29 What is the experimental probability of
rolling a 5?
A 1/2
B 5/4
C 4/5
D 2/5
# on Die
Picture of Roll
Results
1
1 one
2
3 twos
3
1 three
4
0 fours
5
4 fives
6
1 six
These are the results after 10 rolls of the die
30 What is the experimental probability of
rolling a 4?
# on Die
A 1/2
B 5/4
C 4/4
D 0
Picture of Roll
Results
1
1 one
2
3 twos
3
1 three
4
0 fours
5
4 fives
6
1 six
These are the results after 10 rolls of the die
31
Based on the experimental probability you
found, if you rolled the die 100 times, how
many sixes would you expect to get?
# on Die
Picture of Roll
Results
A 6 sixes
B 10 sixes
1
1 one
2
3 twos
3
1 three
4
0 fours
5
4 fives
6
1 six
C 12 sixes
D 60 sixes
These are the results after 10 rolls of the die
32 Mike flipped a coin 15 times and it landed on
tails 11 times. What is the experimental
probability of landing on heads?
Theoretical Probability
Theoretical Probability
Theoretical Probability
What is the theoretical probability
of spinning green?
Theoretical Probability
Probability
of an event
number of favorable outcomes
total number of possible outcomes
Theoretical Probability
Example 1 - Marbles
Find the probability of randomly choosing a white
marble from the marbles shown.
P(white) =
# of favorable outcomes
# of possible outcomes
= 4 =2
10 5
There is a 2 in 5 chance of picking a white marble or a
40% possibility.
Theoretical Probability
Example 2 - Marbles
Suppose you randomly choose a gray marble. Find
the probability of this event.
P(gray) = # of favorable outcomes
# of possible outcomes
= 3
10
There is a 3 in 10 chance of picking a gray marble or a
30% possibility.
Theoretical Probability
Example 3 - Coins
Find the probability of getting tails when you flip a
coin.
P(tails) = # of favorable outcomes
# of possible outcomes
= 1
2
There is a 1 in 2 chance of getting tails when you
flip a coin or a 50% possibility.
33 What is the theoretical probability of picking
a green marble?
A
1/8
B
7/8
C
1/7
D 1
34 What is the theoretical probability of picking
a black marble?
A
1/8
B
7/8
C
1/7
D
0
35 What is the theoretical probability of picking
a white marble?
A
1/8
B
7/8
C
1/4
D
1
36 What is the theoretical probability of not
picking a white marble?
A
3/4
B
7/8
C
1/7
D
1
37 What is the theoretical probability of rolling
a three?
A
B
C
D
1/2
3
1/6
1
38 What is the theoretical probability of rolling an
odd number?
A
B
C
D
1/2
3
1/6
5/6
39 What is the theoretical probability of rolling a
number less than 5?
A
B
C
D
2/3
4
1/6
5/6
40 What is the theoretical probability of not
rolling a 2?
A
B
C
D
2/3
2
1/6
5/6
41 Seth tossed a fair coin five times and got five
heads. The probability that the next toss will
be a tail is
A
B
0
C
D
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
42 Which inequality represents the probability, x,
of any event happening?
A
x≥0
B
C
D
0<x<1
x<1
0≤x≤1
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011
Class Activity
Each student flips a coin 10 times and records the
number of heads and the number of tail outcomes.
Each student calculates the experimental probability of
flipping a tail and flipping a head.
Use the experimental probabilities determined by each
student to calculate the entire class's experimental
probability for flipping a head and flipping a tail.
Answer the following:
What is the theoretical probability for flipping a tail? A
head?
Compare the experimental probability to the theoretical
probability for 10 experiments.
Compare the experimental probability to the theoretical
probability when the experiments for all of the students
are considered?
Word Problems
Return to table
of contents
The Marvelous Marble Company produces batches of
marbles of 1000 per batch. Each batch contains 317
blue marbles, 576 red marbles, and 107 green marbles.
Determine the theoretical probability of selecting each
color marble if 1 color is selected by a robotic arm.
Number of Outcomes in the Event Total Number of Possible Outcomes
107
317
576
107+317+576=1000
Theoretical Probability
1000
107/1000=0.107
0.107  100 =
10.7%
1000
317/1000=0.317
0.317  100 =
31.7%
click
1000
1000/1000
576/1000=0.576
0.576  100 =
57.6%
click
1000/1000 = 1  100 =
click
100%
Bob, the manager of the Marvelous Marble Company tells
Pete that it is time to add a yellow marble to the batch. In
addition, Bob tells Pete to start making the batches in equal
proportion so the customer can receive an equal amount of
colors in a batch. He tells Pete he needs this taken care of
right away.
If you were Pete, how would you use theoretical probability to solve
this problem? Assume 1000 marbles per batch (red, green, blue and
yellow colored marbles)
• Start with 1000 marbles
• Divide 1000 into 4 equal parts (equal
colors)
• Each part is equal to 250 marbles
• Reduce to lowest terms
Do you have an explanation of the
probability for Bob? Click on black circle
to find answer.
The customer
has a 1 in 4 or
25%
Click chance
Here of
picking any
color!
Erica loves soccer! The ladies' coach tells Erica that she scored 19% of
her attempts on goal last season. This season, the coach predicts the
same percentage for Erica. Erica reports she attempted approximately
1,100 shots on goal last season. Her coach suggests they estimate the
number of goals using experimental probability.
What do you know about percentages to
figure out the relationship of goals scored
to goals attempted?
19 shots made
click to
100 shots attempted
reveal
= 19%
Experimental Probability =number of times the outcome happened
number of times experiment was repeated
Move toofReveal
number
goals
Erica's Experimental
=
number of attempts
Probability
Move to Reveal
Please continue on next slide...
Let's estimate the number of goals Erica scored.
Erica makes 19% of her
shots on goal.
Erica takes 1,100 shots on
goal.
About what percent would
be a good estimate to use?
About how many attempts
did Erica take?
click
19
20
100
100
so she makes about 20%
of her shots on goal.
is very close to
click1,100
is very close to
1,000. So we will
estimate that Erica has
about 1,000 attempts
Erica wants to find 20% of 1,000. Her math looks like
this:
Erica figures she made about 200 of her shots on
goal.
Challenge
Can you find the actual values
that will give you 19%?
Experimental Probability
Example 3 - Gardening
Last year, Lexi planted 12 tulip bulbs, but only 10
of them bloomed. This year she intends to plant 60 tulip
bulbs. Use experimental probability to predict how
many bulbs will bloom.
10 bloom = x bloom
12 total
60 total
Solve this proportion by looking at it
times 5
10 bloom = 50 bloom
12 total
60 total
Based on her experience last year, Lexi can expect
50 out of 60 tulips to bloom.
Experimental Probability
Example 4 - Basketball
Today you attempted 50 free throws and made 32 of
them. Use experimental probability to predict how many
free throws you will make tomorrow if you attempt 75
free throws.
32 made
= x made
50 attempts
75 attempts
Solve this proportion using cross products
32  75 = 50  x
2400 = 50x
48 = x
Based on your performance yesterday, you can
expect to make 48 free throws out of 75 attempts.
Now, its your turn. Calculate the experimental
probability for the number of goals.
Number of
attempts
Number of goals
100
30
1000
600
500
150
2000
1600
Experimental
Probability
43 Tom was at bat 50 times and hit the ball 10
times. What is the experimental probability for
hitting the ball?
44 Tom was at bat 50 times and hit the ball 10
times. Estimate the number of balls Tom hit if
he was at bat 250 times.
45 What is the theoretical probability of
randomly selecting a jack from a deck of
cards?
46 Mark rolled a 3 on a die for 7 out of 20 rolls.
What is the experimental probability for
rolling a 3?
47 What is the theoretical probability for rolling
a 3 on a die?
48 Some books are laid on a desk. Two are
English, three are mathematics, one is
French, and four are social studies. Theresa
selects an English book and Isabelle then
selects a social studies book. Both girls take
their selections to the library to read. If
Truman then selects a book at random, what
is the probability that he selects an English
book?
49 What is the probability of drawing a king or an
ace from a standard deck of cards?
A
B
C
D
2/52
4/52
2/13
8/52
50 What is the probability of drawing a five or a
diamond from a standard deck of cards?
A
B
C
D
4/13
13/52
2/13
16/52
Fundamental
Counting
Principle
Return to table
of contents
What should I wear today?
Buddy has 2 shirts and 3 pairs of pants to
choose from. How many different outfits
can he make?
Let's find out how many outfits Buddy can make
using a tree diagram.
Or we could use multiplication to find out how many
outfits Buddy could make.
3
pants
x
2
shirts
= 6
outfits
How many different meals can we create using the
following menu?
Side
Soup
Salad
French Fries
Entree
Lasagna
Chicken Fajita
Burrito
Pizza
Hamburger
Dessert
Ice Cream
Cake
Create a tree diagram by dragging the items.
Side
Soup
Salad
French Fries
Entree
Lasagna
Chicken Fajita
Pizza
Burrito
Hamburger
Lasagna
Soup
Ice Cream
Cake
Dessert
Ice Cream
Cake
Now try to solve the same problem using multiplication.
Side
Soup
Salad
French Fries
Entree
Chicken Fajita
Lasagna
Pizza
Burrito
Hamburger
x
Sides
x
Entrees
Dessert
Ice Cream
Cake
=
Desserts
Meals
If you were to pick 4 digits to be your identification
number, how many choices are there?
Before we begin we must consider if once a number is
chosen if it can be repeated.
If a digit can repeat its called replacement,
because once it chosen it placed back on the list.
If a digit cannot repeat it is said to be without
replacement, because the number does not back on to
the list of choices.
If you were to pick 4 digits to be your identification
number, how many choices are there if there is no
replacement?
_______ ________ _________ __________
First consider how choices there are for a digit:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
So 10 choices for the first digit.
For the second digit there will be only 9 choices left.
For the third digit there are only 8 choices left.
For the fourth digit there are only 7 choices.
Students are given a lock for their gym lockers. Each code
requires you to enter 4 single digit numbers. If the
numbers cannot be repeated, how many different codes are
possible?
x
x
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
=
1
2
3
4
5
6
7
Total
Possibilities
5,040
combinations
Move to Reveal Answer
If you were to pick 4 digits to be your identification
number, how many choices are there if there is
replacement?
__________ _________ _________ __________
First consider how choices there are for a digit:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
So 10 choices for the first digit.
For the second digit there will be only 10 choices because
with replacement there can be repeats.
For the third digit there are only 10 choices left.
For the fourth digit there are only 10 choices.
Using the Counting Principle: (10)(10)(10)(10)= 10,000
combos
Students are given a lock for their gym lockers. Each code
requires you to enter 4 single digit numbers. If the
numbers can be repeated, but zero cannot be the first
number how many different codes are possible?
x
x
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
0
x
1
2
3
4
5
6
7
8
9
0
=
1
2
3
4
5
6
7
8
9
0
Total
Possibilities
9,000
combinations
Move to Reveal Answer
7,893,600
combinations
This cryptex has a map to
treasure buried somewhere in
New Jersey inside of it! Each
of the 5 columns lists every
letter in the alphabet once.
What are the total number of
codes that can be created if the
letters cannot be repeated?
Click on lock to reveal answers
26___
11,881,376
This cryptex has a map to
treasure buried somewhere in
New Jersey inside of it! Each
of the 5 columns lists every
letter in the alphabet once.
What is the probability of the
codes containing the letters
MATH (in that order) as the first
4 letters in the code? (Last
letter can be a repeat)
Click on lock to reveal answers
Challenge
Version
51 Robin has 8 blouses, 6 skirts, and 5 scarves.
Which expression can be used to calculate
the number of different outfits she can
choose, if an outfit consists of a blouse, a
skirt, and a scarf?
A
B
C
D
8+6+5
8•6•5
8! 6! 5!
19C3
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
52 In a school building, there are 10 doors that
can be used to enter the building and 8
stairways to the second floor. How many
different routes are there from outside the
building to a class on the second floor?
A 1
B 10
C 18
D 80
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 20
53 Joe has 4 different hats, 3 different shirts,
and 2 pairs of pants. How many different
outfits can Joe make?
A
9 outfits
B
14 outfits
C
24 outfits
D
12 outfits
54 Stacy is trying to find out how many different
combinations of license plates there. She
lives in New Jersey where there are 3 letters
followed by 3 numbers. How many different
combinations of license plates are there?
A 17,576,000 license plates
B 12,812,904 license plates
C 729 license plates
D 17,576 license plates
55 If you wanted to maximize the amount of
available license plates and could add an
additional letter or number to the existing
combination of 3 letters and 3 numbers, would
you add a letter or a number?
A
B
letter
number
56 Becky and Andy are going on their first date
to the movies. Andy wants to buy Becky a
snack and drink, but she is taking forever to
make a decision. Becky says that there are
too many combinations to choose from. If
there are 6 different types of drinks and 15
different snacks, how many options does
Becky actually have?
A 45 choices
B 90 choices
C 21 choices
D 42 choices
57 Ali is making bracelets for her and her
friends out of beads. She figured that each
bracelet should be about 10 beads. If she
only has blue and green beads, how many
different bracelets can she possibly make?
A
1,024 bracelets
B
1,000 bracelets
C
100 bracelets
D 20 bracelets
58 5 styles of bikes come in 4 colors each, how
many different bikes choices are available?
59 If the book store has four levels of algebra
books, each level is available in soft back or
hardcover, and each comes in three different
typefaces, how many options of algebra
books are available?
60 How many ways can 3 students be named
president, vice president, and secretary if
each holds only 1 office?
61 How many ways can a 8-question multiple
choice quiz be answered if the there are 4
choices per question?
62 A locker combination system uses three
digits from 0 to 9. How many different threedigit combinations with no digit repeated are
possible?
A
B
C
D
30
504
720
1000
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
63 How many different five-digit numbers can
be formed from the digits 1, 2, 3, 4, and 5 if
each digit is used only once?
A
120
B
C
D
60
24
20
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
64 All seven-digit telephone numbers in a town
begin with 245. How many telephone
numbers may be assigned in the town if the
last four digits do not begin or end in a zero?
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
65 The telephone company has run out of
seven-digit telephone numbers for an area
code. To fix this problem, the telephone
company will introduce a new area code. Find
the number of new seven-digit telephone
numbers that will be generated for the new
area code if both of the following conditions
must be met:
• The first digit cannot be a zero or a one.
• The first three digits cannot be the
emergency number (911) or the number used
for information (411).
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Permutations
and
Combinations
Return to table
of contents
How many ways can the following animals be arranged?
There are two methods to solve this problem:
Method 1: List all the possible groupings
Method 2: Use the permutation.
Method 1: List all possible groupings.
There are 24 arrangements of 4 animals in 4 positions.
Method 2: Use the permutation.
A permutation is an arrangement of n objects in which
order is important.
There are 4 choices for the first position.
There are 3 choices for the second position.
There are 2 choices for the third position.
There is 1 choice for the fourth position.
4  3  2  1 = 24
There are 24 arrangements of 4 animals in 4 positions.
The expression 4  3  2  1 can be written as 4!, which
is read as "4 factorial."
66
What is the value of 5! ?
67
How many ways can the letters in FROG be
arranged?
68
In how many ways can a police officer,
fireman and a first aid responder enter a
room single file?
A
B
C
D
E
3
3!
6
6!
1
69
In how many ways can four race cars finish a
race that has no ties?
A
B
C
D
E
4
4!
24
24!
12
70 How many ways can the letters the word
HOUSE be arranged?
71 How many ways can 6 books be arranged
on a shelf?
How many ways can the letters in the word DEER be
rearranged?
There are 2 E's! So DEER and DEER are consider to be
the same combo. Since there are 2 repeated letters
calculate the combos using the Counting Principle and
the divide by 2.
(4)(3)(2)(1) = 12 ways
2
72
In how many ways can the letters in JERSEY
be arranged?
73 How many different three-letter arrangements
can be formed using the letters in the word
ABSOLUTE if each letter is used only once?
A
B
C
D
56
112
168
336
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Permutation Formula
Key concept: an arrangement of n objects in which order is
important is a permutation.
A race is an example of a situation where order is important.
Can you name other examples where order is important?
____________________________________________________
The number of permutations of n objects taken r at a time
can be written as nPr, where
nPr =
If 5 cars were in a race and prizes were awarded
for first, second and third, this is the number of
possible ways for the prizes to be awarded.
5P3
=
Always remember that:
Note
0! = 1
1! = 1
If 5 cars were in a race and prizes were awarded for each
racer, the number of possible ways for the prizes to be
awarded would be
5P5 =
5! = 5! = 120 = 120
(5-5)! 0! 1
74
Find the value of 6P2
75
Find the value of
4P1
76
Find the value of
6P6
Twenty young ladies entered a beauty contest. Prizes will
be awarded for first, second and third place. How many
different ways can the first, second and third place prizes
be awarded?
20P3 =
= 20!
17!
= 20  19  18  17!
= 6840
17!
Find the number of permutations of 4 objects taken 3 at a
time.
How many 4-digit numbers can you make using each of
the digits 1, 2, 3, and 4 exactly once?
4P4
= 4! = 4  3  2  1 = 24
0!
1
77 10 cars are in a race. How many ways can
prizes be awarded for first, second and third
place?
78 How many ways can four out of seven books
be arranged on a shelf?
79 You are taking 7 classes, three before
lunch. How many possible arrangements
are there for morning classes?
80 The teacher is going to select a president
and vice-president from the 24 students in
class. How many possible arrangements are
there for president and vice-president?
Combinations
A combination is a selection of objects when order is not
important.
Example: A combination pizza, since it does not matter in
which order the toppings were placed.
Can you think of other examples when order does not
matter?
81
You must read 5 of the 10 books on the
summer reading list. This is an example of a
_________
A
B
Combination
Permutation
82 You must fit 5 of the 10 books on the shelf.
How many different ways are there to place
them on the shelf? This is an example of a
____________
A
B
Combination
Permutation
83 10 people are in a room. How many different
pairs can be made? This is an example of a
____________
A
B
Combination
Permutation
84 10 people are about to leave a room. How
many different ways can they walk out of the
room? This is an example of a ____________
A
B
Combination
Permutation
85 You have 100 relatives and can only invite 50
to your 16th birthday party. The possibilities
of who can be invited is an example of a
____________
A
Combination
B
Permutation
Combinations
________________________________________________
To find the number of combinations of n objects taken r
at a time, divide the number of permutations of n objects
taken r at a time by r!
nCr = nPr
r!
_________________________________________________
There are 7 pizza toppings and you are choosing
four of them for your pizza. How many different
pizzas are possible to create?
The order in which you choose the toppings is not
important, so this is a combination. To find the
number of different ways to choose 4 toppings from
7, find 7C4.
7C4 = 7P4 =
4!
7  6  5  4 = 35
4321
86
Find the number of combinations.
5C2
87 There are 40 students in the computer club.
Five of these students will be selected to
compete in the ALL STAR competition. How
many different groups of five students can
be chosen?
88 There are 45 flowers in the shop. How many
different arrangements containing 10 flowers
can be created?
89 Eight people enter the chess tournament.
How many different pairings are possible?
90 Mary can select 3 of 5 shirts to pack for the
trip. How many different groupings are
possible?
91 How many different three-member teams can
be selected from a group of seven students?
A
B
C
D
1
35
210
5040
Probability of
Compound
Events
Return to table
of contents
Probability of Compound Events
First - decide if the two events are independent
or dependent.
When the outcome of one event does not affect the
outcome of another event, the two events are
independent.
Use formula:
Probability (A and B) = Probability (A)  Probability (B)
Independent Example
Select a card from a deck of cards, replace it in the deck,
shuffle the deck, and select a second card.
What is the probability that you will pick a 6 and then a
king?
P (6 and a king) = P(6)  P(king)
4  4 = _1_
52 52 169
When the outcome of one event affects the outcome of
another event, the two events are dependent.
Use formula:
Probability (A & B) = Probability(A)  Probability(B given A)
Dependent Example
Select a card from a deck of cards, do not replace it in the
deck, shuffle the deck, and select a second card.
What is the probability that you will pick a 6 and then a king?
P(6 and a king) = P(6)  P(king given a six has been selected)
4  4 = 4
52 51 663
92 The names of 6 boys and 10 girls from your
class are put in a hat. What is the probability
that the first two names chosen will both be
boys?
93 A lottery machine generates numbers
randomly. Two numbers between 1 and 9
are generated. What is the probability
that both numbers are 5?
94 The TV repair person is in a room with 20
broken TVs. Two sets have broken wires
and 5 sets have a faulty computer chip.
What is the probability that the first TV
repaired has both problems?
95
What is the probability that the first
two cards drawn from a full deck are
both hearts?
(without replacement)
96 A spinner containing 5 colors: red, blue,
yellow, white and green is spun and a die,
numbered 1 thru 6, is rolled. What is the
probability of spinning green and rolling a
two?
97 A drawer contains 5 brown socks, 6 black
socks, and 9 navy blue socks. The power is
out. What is the probability that Sam
chooses two socks that are both black?
98 At a school fair, the spinner represented in
the accompanying diagram is spun twice.
R
G
B
What is the probability that it will land in
section G the first time and then in
section B the second time?
C
A
B
D
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
99 A student council has seven officers, of which
five are girls and two are boys. If two officers
are chosen at random to attend a meeting with
the principal, what is the probability that the
first officer chosen is a girl and the second is a
boy?
A
C
B
D
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
100
The probability that it will snow on Sunday
is .
The probability that it will snow on both
Sunday and Monday is .
What is the probability that it will snow on
Monday, if it snowed on Sunday?
A
B
C
2
D
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Probabilities of Mutually
Exclusive & Overlapping
Events
Return to table
of contents
Events are mutually exclusive or disjoint if they have no
outcomes in common.
Example:
Event A: Roll a 3
Event B: Roll an even number
Event A
3
Event B
2 4 6
Overlapping Events are events that have one or
more outcomes in common
Example
Event A: Roll an even number
Event B: Roll a number greater than 3
Event A
2
4
6
Event B
5
101 Are the events mutually exclusive?
Event A: Selecting an Ace
Event B: Selecting a red card
A
B
Yes
No
102
Are the events mutually exclusive?
Event A: Rolling a prime number
Event B: Rolling an even number
A
B
Yes
No
103 Are the events mutually exclusive?
Event A: Rolling a number less than 4
Event B: Rolling an even number
A
B
Yes
No
104
Are the events mutually exclusive?
Event A: Selecting a piece of fruit
Event B: Selecting an apple
A
Yes
B
No
105
Are the events mutually exclusive?
Event A: Roll a multiple of 3
Event B: Roll a divisor of 19
A
Yes
B
No
106
Are the events mutually exclusive?
Randomly select a football card
Event A: Select a Philadelphia Eagle
Event B: Select a starting quarterback
A
Yes
B
No
107
Are the events mutually exclusive?
Event A: The Yankees won the World Series
Event B: The Mets won the National League
Pennant
A
Yes
B
No
Formula
probability of two mutually
exclusive events
P(A or B) = P(A) + P(B)
What is the probability of drawing a 5 or an Ace from a
standard deck of cards?
There are 52 outcomes for the standard deck. 4 of these
cards are 5s and 4 are Aces. There is not a card that is
both a 5 and an A. So...
Check your answer by pulling down the screen.
P(5 or A) = P(5) + P(A)
4 + 4 = 8
52 52 52
reduce 2
13
Find the probability if you if you roll a pair of number
cubes and the numbers showing are the same or that the
sum is 11.
P(numbers =) + P(sum is 11)
6 +
8 answer
or 2
Click2to=reveal
36
36 36
9
A bag contains the following candy bars:
3 Snickers
4 Mounds
2 Almond Joy
1 Reese's Peanut Butter Cup
You randomly draw a candy bar from the bag. What is the
probability that you select a Snickers or a Mounds bar?
Are the events mutually exclusive?
Find the probability that you select a Snickers bar
Find the probability that you select a Mounds bar
Find the probability that you select a Snickers or a
Mounds bar
108
In a room of 100 people, 40 like Coke, 30
like Pepsi, 10 like Dr. Pepper, and 20 drink
only water. If a person is randomly selected,
what is the probability that the person likes
Coke or Pepsi?
109
In a school election, Bob received 25% of
the vote, Cara received 40% of the vote,
and Sam received 35% of the vote. If a
person is randomly selected, what is the
probability that the person voted for Bob
or Cara?
110
A die is rolled twice. What is the probability
that a 4 or an odd number is rolled?
111
Sal has a small bag of candy containing
three green candies and two red candies.
While waiting for the bus, he ate two
candies out of the bag, one after another,
without looking. What is the probability that
both candies were the same color?
112
Events A and B are disjoint. Find P(A or B).
P(A) =
P(B) =
113
Events A and B are disjoint. Find P(A or B).
P(A) =
P(B) =
What's the problem with this situation...
What is the probability of selecting a
black card or a 7?
P(black or 7)
If the situation is 2 events CAN
occur at the same time, then
these are NOT mutually
exclusive events.
Formula
Probability of two events which are
NOT mutually exclusive
P(A or B) = P(A) + P(B) - P(A and B)
What is the probability of selecting a black card or a 7?
P(black or 7)
P(black or 7) = P(black) + P(7) - P(black and 7)
P(black or 7) =
26
52
+ 4 - 2 = 28 = 7_
52
52
52
13
Of the 300 students at Jersey Devil Middle School, 121
are girls, 16 students play softball, 29 students are on the
lacrosse team and, 25 are girls on the lacrosse team.
Find the probability that a student chosen at random is a
girl or is on the lacrosse team.
300 total students
girls
lacrosse
Of the 300 students at Jersey Devil Middle
School, 121 are girls, 16 students play
softball, 29 students are on the lacrosse team
and, 25 are girls on the lacrosse team. Find
the probability that a student chosen at
random is a girl or is on the lacrosse team.
P(girl or lacrosse) =
P(girl) + P(lacrosse) - P(girl and lacrosse)
121
29
25
+
click to
300
300
300
reveal
125 = 0.416
click to
300
reveal
114 In a special deck of cards each card has
exactly one different number from 1-19
(inclusive) on it. Which gives the probability
of drawing a card with an odd number or a
multiple of 3 on it?
A P(odd) + P(multiple of 3)
B P(odd) x P (multiple of 3) - P(odd and
multiple of 3)
C P(odd) x P(multiple of 3)
D P(odd) + P (multiple of 3) - P(odd and
multiple of 3)
115
Events A and B are overlapping. Find
P(A or B).
P(A) =
P(B) =
P(A and B) =
116 Events A and B are overlapping. Find
P(A or B).
P(A) =
P(B) =
P(A and B) =
117
What is the probability of rolling a number
less than two or an odd number?
118
What is the probability of rolling a number
that is not even or that is not a multiple
of 3?
Complementary Events
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of contents
Complementary Events
Two events are complementary events if they are
mutually exclusive and one event or the other must occur.
The sum of the probabilities of complementary events is
always 1.
P(A) + P(not A) = 1
Example:
The forecast calls for a 30% chance of rain. What is the
probability that it will not rain?
P(rain) + P(not rain) = 1
.3
+ ? =1
P(not rain) = .7
119 Given P(A), find P(not A).
P(A) = 52%
P(not A) = ______ %
120 Given P(A), find P(not A).
P(A) =
P(not A) = ______
121
The spinner below is divided into eight equal
regions and is spun once. What is the
probability of not getting red?
Green Yellow
Red
Blue
Red
White
Red Purple
A
C
B
D
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
122 The faces of a cube are numbered from 1
to 6. What is the probability of not rolling a
5 on a single toss of this cube?
A
C
B
D
From the New York State Education Department. Office of Assessment
Policy, Development and Administration. Internet. Available from
www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.