LESSON
21.1
Name
Probability and Set
Theory
Date
21.1 Probability and Set Theory
Essential Question: How are sets and their relationships used to calculate probabilities?
Resource
Locker
Common Core Math Standards
Explore
The student is expected to:
COMMON
CORE
Class
Working with Sets
A set is a collection of distinct objects. Each object in a set is called an element of the set. A set
is often denoted by writing the elements in braces.
S-CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using
characteristics ... of the outcomes, or as unions, intersections, or
complements of other events (“or,”“and,”“not”).
The set with no elements is the empty set, denoted by ⌀ or { }.
Mathematical Practices
Identifying the number of elements in a set is important for calculating probabilities.
COMMON
CORE
The set of all elements under consideration is the universal set, denoted by U.
A
MP.6 Precision
Use set notation to identify each set described in the table and identify the number of
elements in each set.
Language Objective
Set
Explain to a partner how to find the probability of rolling a certain
number on a number cube and how to find its complement.
Essential Question: How are sets and
their relationships used to calculate
probabilities?
To calculate the probability of an event, you need to
know the number of items in the set of outcomes
for that event, as well as the number of items in the
set of all possible outcomes. The theoretical
probability of the event is the ratio of the two
numbers.
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
Number of
Elements in
the Set
Set Notation
Set A is the set of prime
numbers less than 10.
⎧
⎫
A = ⎨2, 3, 5 , 7⎬
⎩
⎭
n(A) = 4
Set B is the set of even natural
numbers less than 10.
⎧
⎫
B=⎨ 2 , 4 , 6 , 8 ⎬
⎩
⎭
n(B) = 4
Set C is the set of natural
numbers less than 10 that are
multiples of 4.
The universal set is all natural
numbers less than 10.
⎧
⎫
⎩
⎭
C = ⎨4, 8 ⎬
n(C) = 2
⎫
⎭
⎧
⎩
U = ⎨1, 2, 3, 4, 5, 6, 7, 8, 9⎬
n
( U )=9
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the
photograph. Ask students to describe math problems
that could be illustrated by the two dogs. Then
preview the Lesson Performance Task.
Module 21
be
ges must
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DO NOT Key=NL-B;CA-B
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Lesson 1
1083
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
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HARDCOVER PAGES 905914
Turn to these pages to
find this lesson in the
hardcover student
edition.
of
Number in
Elements
the Set
ion
Set Notat
Set
set of even
Set B is the than 10.
less
numbers
n(A) = 4
⎫
⎧
5 , 7⎬
⎭
A = ⎨2, 3,
⎩
set of prime
Set A is the than 10.
less
numbers
⎫
⎬
⎧
6 , 8 ⎭
2 , 4 ,
B = ⎨⎩
natural
n(B) =
y
g Compan
Publishin
Harcour t
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© Houghto
all natura
sal set is
The univer than 10.
less
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⎧
U = ⎨⎩1,
2,
4
n(C) =
⎫
⎧
8 ⎬
⎭
C = ⎨⎩4,
l
set of natura
Set C is the than 10 that are
less
numbers
of 4.
multiples
⎫
7, 8, 9⎬⎭
3, 4, 5, 6,
n
2
( U )=9
Lesson 1
1083
Module 21
1083
Lesson 21.1
1L1 1083
01_U8M2
ESE3858
GE_MNL
07/11/14
10:09 AM
16/06/15 2:35 PM
The following table identifies terms used to describe relationships among sets. Use sets A, B,
C, and U from the previous table. You will supply the missing Venn diagrams in the Example
column, including the referenced elements of the sets, as you complete steps B–I following.
Term
Notation
C⊂B
Set C is a subset of
set B if every element
of C is also an element
of B.
B
C
Example
4,8
2,6
U
A
B
Working with Sets
1,3,5,7,9
U
A⋂B
The intersection of sets
A and B is the set of all
elements that are in
both A and B.
EXPLORE
Venn Diagram
INTEGRATE TECHNOLOGY
Students have the option of doing the Explore activity
either in the book or online.
1,9
3,5,7
2
4,6,8
A ⋂ B is the double-shaded region.
A⋃B
The union of sets A
and B is the set of all
elements that are in
A or B.
U
A
B
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Discuss how the Venn diagrams provide
1,9
3,5,7
2
4,6,8
A ⋃ B is the entire shaded region.
A or ∼A
C
The complement of
set A is the set of all
elements in the universal
set U that are not in A.
U
A
pictures of set relationships to help students
understand the terminology. Encourage students to
practice drawing Venn diagrams to use when
investigating set theory. For example, discuss how a
Venn diagram can make it easier to identify the
complement of an intersection.
1,4,6,8,9
2,3,5,7
A c is the shaded region.
B
Since C is a subset of B, every element of set C, which consists of the
numbers 4 and 8 , is located not only in oval C, but also within oval B.
Set B includes the elements of C as well as the additional elements 2 and 6 ,
which are located in oval B outside of oval C. The universal set includes the
elements of sets B and C as well as the additional elements 1 , 3 , 5 , 7 ,
and 9 , which are located in region U outside of ovals B and C.
In the first row of the table, draw the corresponding Venn diagram that includes the
elements of B, C, and U. See first row of table.
D
To determine the intersection of A and B, first define the elements of set A and set B
separately, then identify all the elements found in both sets A and B.
© Houghton Mifflin Harcourt Publishing Company
C
AVOID COMMON ERRORS
⎧
⎫
⎬
A=⎨ 2
, 3 , 5 , 7 ⎭
⎩
Students may assume that a set that contains only 0 is
the same as the empty set. Contrast the empty set, { },
with the set that contains only 0, {0}.
QUESTIONING STRATEGIES
What word corresponds to the intersection of
two sets? Is it union? Explain. And means the
elements are in both sets, which corresponds to the
intersection. Or means the elements can be in either
set, which corresponds to the union.
⎧
⎫
⎬
B=⎨ 2
, 4 , 6 , 8 ⎭
⎩
⎧
⎫
A⋂B=⎨ 2 ⎬
⎩
⎭
Module 21
1084
Lesson 1
PROFESSIONAL DEVELOPMENT
GE_MNLESE385801_U8M21L1 1084
Math Background
A German mathematician, Georg Cantor (1845–1918), is considered to be the
father of set theory. Cantor discovered that the rational numbers are countable but
the real numbers are uncountable.
03/09/14 8:22 PM
How is an intersection different from a
subset? The intersection consists of the
elements two sets have in common, while all of the
elements of a subset lie within the set of which it is a
subset.
How do you know when sets overlap? Sets
will overlap when they have some elements
in common.
Two French mathematicians, Blaise Pascal (1623–1662) and Pierre de Fermat
(1601–1665), are considered to be the founders of probability theory. The roots of
probability theory lie in the letters they exchanged analyzing games of chance.
Probability and Set Theory 1084
EXPLAIN 1
E
In the second row of the table, draw the Venn diagram for A ⋂ B that includes the elements
of A, B, and U and the double-shaded intersection region. See second row of table.
Calculating Theoretical Probabilities
F
To determine the union of sets A and B, identify all the elements found in either set A or set B
by combining all the elements of the two sets into the union set.
⎧
⎫
⎬
A⋃B=⎨ 2
, 3 , 4 , 5 , 6 , 7 , 8 ⎭
⎩
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Discuss the set notation used to define
theoretical probability. Connect the notation to a
word description of the probability ratio, such as, the
ratio of favorable outcomes in sample space to total
number of outcomes in sample space.
G
In the third row of the table, draw the Venn diagram for A ⋃ B that includes the elements of
A, B, and U and the shaded union region. See third row of table.
H
To determine the complement of set A, first identify the elements of set A and universal
set U separately, then identify all the elements in the universal set that are not in set A.
⎧
⎫
⎬
A=⎨ 2
, 3 , 5 , 7 ⎭
⎩
⎧
⎫
U=⎨ 1
2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ⎬
,
⎩
⎭
⎧
⎫
⎬
AC = ⎨ 1
, 4 , 6 , 8 , 9 ⎭
⎩
AVOID COMMON ERRORS
Students may have difficulty identifying an event
based on a union or intersection. Suggest that
students draw Venn diagrams to model the
experiment. They can begin by defining each set and
then create the Venn diagram to show where the sets
overlap.
I
In the fourth row of the table, draw the Venn diagram for A c that includes the elements
of A and U and the shaded region that represents the complement of A. See fourth row of table.
Reflect
1.
Draw Conclusions Do sets always have an intersection that is not the empty set? Provide an
example to support your conclusion.
No. Using the example sets above, A ⋂ C = ⌀ because they do not have any elements
© Houghton Mifflin Harcourt Publishing Company
in common.
Explain 1
Calculating Theoretical Probabilities
A probability experiment is an activity involving chance. Each repetition of the experiment
is called a trial and each possible result of the experiment is termed an outcome. A set of
outcomes is known as an event, and the set of all possible outcomes is called the sample space.
Probability measures how likely an event is to occur. An event that is impossible has a
probability of 0, while an event that is certain has a probability of 1. All other events have a
probability between 0 and 1. When all the outcomes of a probability experiment are equally
likely, the theoretical probability of an event A in the sample space S is given by
n(A)
number of outcomes in the event
= _.
P(A) = ____
number of outcomes in the sample space
n(S)
Module 21
1085
Lesson 1
COLLABORATIVE LEARNING
GE_MNLESE385801_U8M21L1 1085
Small Group Activity
Ask each group to draw a spinner with 6 or 8 equal parts. Then ask them to use
letters, colors, or numbers to distinguish each section of the spinner. Have them
define two events based on the spinners. For example, if letters are used, the set of
vowels and the set of letters in the word math. Ask students to find the probability
of each event, their complements, their union, and their intersection. Have
students share their work. Review which events have a probability of 1, which have
a probability of 0, and why.
1085
Lesson 21.1
03/09/14 8:22 PM
Example 1

Calculate P(A), P(A ⋃ B), P(A ⋂ B), and P(A C) for each situation.
QUESTIONING STRATEGIES
S
You roll a number cube. Event A is rolling a prime number.
Event B is rolling an even number.
A
⎧
⎫
⎧
⎫
S = ⎨1, 2, 3, 4, 5, 6⎬, so n(S) = 6. A = ⎨2, 3, 5⎬, so n(A) = 3.
⎩
⎭
⎩
⎭
When you calculate the theoretical
probabilities of events based on the same
probability experiment, can the term in the
denominator of the probability ratio change?
Explain. No, the term in the denominator
corresponds to the sample space, which does
not change.
B
n(A)
3 =_
1.
So, P(A) = _ = _
2
6
n(S)
n(A ⋃ B)
⎫
⎧
5.
A ⋃ B = ⎨2, 3, 4, 5, 6⎬, so n(A ⋃ B) = 5. So, P(A ⋃ B) = _ = _
⎩
⎭
6
n(S)
n(A ⋂ B)
⎧ ⎫
1.
A ⋂ B = ⎨2⎬, so n(A ⋂ B) = 1. So, P(A ⋂ B) = _ = _
⎩ ⎭
6
n(S)
If a set has no elements, what is the
probability of the event represented by
the set? Explain. 0, because there are no outcomes
in the sample space that correspond to the event
n(A )
⎧
⎫
3 =_
1.
A C = ⎨1, 4, 6⎬, so n(A C) = 3. So, P(A C) = _ = _
⎩
⎭
2
6
n(S)
C

Your grocery basket contains one bag of each of the following items:
oranges, green apples, green grapes, green broccoli, white cauliflower,
orange carrots, and green spinach. You are getting ready to transfer
your items from your cart to the conveyer belt for check-out.
Event A is picking a bag containing a vegetable first. Event B is picking
a bag containing a green food first. All bags have an equal chance of
being picked first.
Order of objects in sets may vary.
⎧
⎫
S = ⎨orange, apple, grape, broccoli, cauliflower, carrot, spinach⎬, so n(S) = 7 .
⎩
⎭
If the elements of a set are the same as the
elements of the sample space, what is the
probability of the event represented by the set?
Explain. 1, because the number of elements in the
set is the same as the number of elements in the
sample space
⎧
A ⋃ B = ⎨broccoli, cauliflower ,
⎩
(
)
carrot ,
spinach ,
apple , grape⎬⎫, so n(A ⋃ B) = 6 .
⎭
n
A ∪ B
6
P(A ⋃ B) = __ = _
n
(S)
7
⎧
⎫
A ⋂ B = ⎨ broccoli , spinach ⎬, so n(A ⋂ B) = 2
⎩
⎭
(
)
n
A ∩ B
2
P(A ⋂ B) = __ = _
7
n
S
( )
n ( A )
3
P( A ) = _ = _
7
n ( S )
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ted
Morrison/Bon Appetit/Alamy
( )
( )
4
A
n
⎫
⎧
A = ⎨broccoli, cauliflower, carrot , spinach ⎬, so n(A) = 4 . So P(A) = _ = _.
⎩
⎭
7
S
n
c
c
Module 21
1086
Lesson 1
DIFFERENTIATE INSTRUCTION
GE_MNLESE385801_U8M21L1.indd 1086
Manipulatives
9/2/14 3:43 PM
Encourage students to design their own experiments to illustrate what they have
learned about probability, such as calculating the complement of rolling a number
with a number cube and then attempting to conform the calculation
experimentally. Invite students to demonstrate their experiments before the class.
Probability and Set Theory 1086
Reflect
EXPLAIN 2
2.
Using the Complement of an Event
Discussion In Example 1B, which is greater, P(A ⋃ B) or P(A ⋂ B)? Do you think this result is true
in general? Explain.
6
2
Since P(A ∪ B) = _
is greater than P(A ∩ B) = _
, the union is more likely than the
7
7
intersection. Yes, this is generally true since the union includes all the elements from both
events, whereas the intersection contains only elements present in both sets. However, if
AVOID COMMON ERRORS
A = B, then the probability of the union and intersection will be the same.
Students may have difficulty understanding when to
use the complement to find a probability. Point out
that students may be able to find the probability
directly but that the complement may provide a
shortcut. Continue to remind students to create Venn
diagrams to help them recognize relationships
between sets.
Your Turn
The numbers 1 through 30 are written on slips of paper that are then placed in a hat.
Students draw a slip to determine the order in which they will give an oral report.
Event A is being one of the first 10 students to give their report. Event B is picking a
multiple of 6. If you pick first, calculate each of the indicated probabilities.
3.
P(A)
n(A)
10
1
P(A) = ____ = __
=_
30
3
n(S)
4.
QUESTIONING STRATEGIES
5.
Why are there different equations that relate
the probability of an event and its
complement? The three equations state the same
relationship in different ways.
P(A ∪ B)
⎫
⎧
⎩
A ∪ B = ⎨1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 24, 30⎬⎭; P(A ∪ B) =
P(A ∩ B)
⎧ ⎫
A ∩ B = ⎨6⎬⎭; P(A ∩ B) =
⎩
6.
P(A c)
⎧
⎩
(A ∩ B)
n
______
n(S)
⎫
A c = ⎨11, 12,…, 30⎬⎭; P(A c) =
© Houghton Mifflin Harcourt Publishing Company
Explain 2
(A ∪ B)
n
______
n(S)
14
= __
30
1
= __
30
n(A )
20
2
____
= __
=_
c
n(S)
30
3
Using the Complement of an Event
You may have noticed in the previous examples that the probability of an event occurring and
the probability of the event not occurring (i.e., the probability of the complement of the event)
have a sum of 1. This relationship can be useful when it is more convenient to calculate the
probability of the complement of an event than it is to calculate the probability of the event.
Probabilities of an Event and Its Complement
P(A) + P(A c) = 1
The sum of the probability of an event and the
probability of its complement is 1.
P(A) = 1 - P(A c)
The probability of an event is 1 minus the
probability of its complement.
P(A c) = 1 - P(A)
The probability of the complement of an event
is 1 minus the probability of the event.
Module 21
1087
Lesson 1
LANGUAGE SUPPORT
GE_MNLESE385801_U8M21L1.indd 1087
Connect Vocabulary
Have students create a set of cards with diagrams to help them become familiar
with the vocabulary introduced in this lesson. Help students connect the
vocabulary to the notation used to represent a set, an element, the universal set, a
subset, union, intersection, and complement. Have students use different colors to
highlight and distinguish each relationship.
1087
Lesson 21.1
9/2/14 3:43 PM

Use the complement to calculate the indicated probabilities.
You roll a blue number cube and a white number cube at the
same time. What is the probability that you do not roll doubles?
Step 1 Define the events. Let A be that you do not roll
doubles and A c that you do roll doubles.
Step 2 Make a diagram. A two-way table is one helpful
way to identify all the possible outcomes in the
sample space.
Blue Number Cube
White Number Cube
Example 2
1
2
3
4
5
6
1
1, 1 1, 2 1, 3 1, 4 1, 5 1, 6
2
2, 1 2, 2 2, 3 2, 4 2, 5 2, 6
3
3, 1 3, 2 3, 3 3, 4 3, 5 3, 6
4
4, 1 4, 2 4, 3 4, 4 4, 5 4, 6
5
5, 1 5, 2 5, 3 5, 4 5, 5 5, 6
6
6, 1 6, 2 6, 3 6, 4 6, 5 6, 6
Step 3 Determine P(A ). Since there are fewer outcomes for rolling doubles, it is more
convenient to determine the probability of rolling doubles, which is P(A c). To determine
n(A c), draw a loop around the outcomes in the table that correspond to A c and then
calculate P(A c).
n(A c)
6 =_
1
P(A c) = _ = _
36
6
n(S)
c
Step 4 Determine P(A). Use the relationship between the probability of an event and its
complement to determine P(A).
5
1 =_
P(A) = 1 - P(A c) = 1 - _
6
6
So, the probability of not rolling doubles is __56 .

One pile of cards contains the numbers 2 through 6 in red hearts. A second pile of cards
contains the numbers 4 through 8 in black spades. Each pile of cards has been randomly
shuffled. If one card from each pile is chosen at the same time, what is the probability that
the sum will be less than 12?
Step 1 Define the events. Let A be the event that the sum is less than 12 and A c be the
the sum is not less than 12 .
event that
Module 21
GE_MNLESE385801_U8M21L1 1088
1088
Black Spades
Step 3 Determine P(A c). Circle the outcomes in the table that
correspond to A c, then determine P(A c).
⎛
⎞
n ⎜ Ac ⎟
6
⎝
⎠
P(A c) = _ = _
⎛
⎞
25
n ⎜ S ⎟
⎝
⎠
Red Hearts
2 3 4 5 6
4 4+2 4+3 4+4 4+5 4+6
5 5+2 5+3 5+4 5+5 5+6
6 6+2 6+3 6+4 6+5 6+6
7 7+2 7+3 7+4 7+5 7+6
8 8+2 8+3 8+4 8+5 8+6
© Houghton Mifflin Harcourt Publishing Company
Step 2 Make a diagram. Complete the table to show all the
outcomes in the sample space.
Lesson 1
03/09/14 8:22 PM
Probability and Set Theory 1088
Step 4 Determine P(A). Use the relationship between the probability of an event and its
complement to determine P(A c).
ELABORATE
P(A) = 1 - P
AVOID COMMON ERRORS
( A )=
c
6
19
1 -_=_
25
25
19
So, the probability that the sum of the two cards is less than 12 is _.
25
Students may have trouble identifying some
outcomes associated with an event. Encourage
students to carefully identify all outcomes by using
tables, lists, or diagrams. They can circle the
outcomes of interest (often called the favorable
outcomes) in the sample space.
Reflect
7.
Describe a different way to calculate the probability that the sum of the two cards will be less than 12.
Use the table to count the number of outcomes in event A instead of A c, which is 19, then
19
divide that by the total number of outcomes to get __
.
25
Your Turn
One bag of marbles contains two red, one yellow, one green, and one blue marble.
Another bag contains one marble of each of the same four colors. One marble from
each bag is chosen at the same time. Use the complement to calculate the indicated
probabilities.
QUESTIONING STRATEGIES
How does listing the elements in a set help you
find the probability of an event associated
with the set? The probability is based on the
number of elements in the set, so you can just count
the elements for the numerator of the probability
ratio.
8.
9.
5
1
1
3
(G, G), (B, B); P(A c) = __
=_
, so P(A) = 1 - _
=_
.
4
4
4
20
Probability of not selecting a yellow marble A c is selecting at least one yellow marble: (Y, R), (Y, Y),
8
3
2
2
(Y, G), (Y, B), (R 1, Y), (R 2, Y), (G, Y), (B, Y); P(A c) = __
=_
, so P(A) = 1 - _
=_
.
5
5
5
20
Elaborate
10. Can a subset of A contain elements of A C? Why or why not?
No. The elements of a subset are contained completely within the parent set A, whereas
SUMMARIZE THE LESSON
none of the elements of the complement of a set A are in set A by definition, and thus they
© Houghton Mifflin Harcourt Publishing Company
How can you use set theory to help you
calculate theoretical probabilities? You can
use the number of elements in a set to define
theoretical probability: the theoretical probability
n(A)
that an event A will occur is given by P(A) = ___,
n(S)
where S is the sample space.
Probability of selecting two different colors A c is selecting the same color: (R 1, R), (R 2, R), (Y, Y),
cannot be in a subset of A.
11. For any set A, what does A ∩ Ø equal? What does A ⋃ Ø equal? Explain.
The intersection of set A and the empty set is the empty set, A ∩ Ø = Ø, since the two
sets do not have any elements in common. The union of set A and the empty set is set A,
A ∪ Ø = A, since the elements of the union are the elements in set A or the empty set.
12. Essential Question Check-In How do the terms set, element, and universal set correlate to the terms
used to calculate theoretical probability?
Possible answer: To calculate probability, you need to know the number of possible
outcomes in the sample space, which is the number of elements in the universal set.
You also need to know the number of possible outcomes in the defined event, which
is the number of elements in the defined set.
Module 21
GE_MNLESE385801_U8M21L1 1089
1089
Lesson 21.1
1089
Lesson 1
03/09/14 8:22 PM
Evaluate: Homework and Practice
EVALUATE
Set A is the set of factors of 12, set B is the set of even natural
numbers less than 13, set C is the set of odd natural numbers less than
13, and set D is the set of even natural numbers less than 7. The universal
set for these questions is the set of natural numbers less than 13.
⎧
⎩
⎫
⎭
⎧
⎩
• Online Homework
• Hints and Help
• Extra Practice
⎫
⎭
So, A = ⎨1, 2, 3, 4, 6, 12⎬, B = ⎨2, 4, 6, 8, 10, 12⎬,
⎧
⎫
⎧
⎫
⎩
⎭
⎩
⎭
⎧
⎫
U = ⎨1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12⎬. Answer each question.
⎩
⎭
C = ⎨1, 3, 5, 7, 9, 11⎬, D = ⎨2, 4, 6⎬, and
1.
Is D ⊂ A? Explain why or why not.
2.
Yes, because every element of D is also
an element of A.
3.
5.
7.
What is A ⋂ B ?
⎧
⎫
⎨2, 4, 6, 12⎬
⎩
⎭
What is A ⋃ B ?
⎧
⎫
⎨1, 2, 3, 4, 6, 8, 10, 12⎬
⎩
⎭
What is A C?
⎧
⎫
⎨5, 7, 9, 10, 11⎬
⎩
⎭
ASSIGNMENT GUIDE
Is B ⊂ A? Explain why or why not.
No, because there is at least one element of
B that is not an element of A. For example,
8 is an element of B that is not an element
of A.
4.
6.
8.
What is A ⋂ C ?
⎧
⎫
⎨1, 3⎬
⎩
⎭
What is A ⋃ C ?
⎧
⎫
⎨1, 2, 3, 4, 5, 6, 7, 9, 11, 12⎬
⎩
⎭
What is B C?
⎧
⎫
⎨1, 3, 5, 7, 9, 11⎬
⎩
⎭
_ _ _
11. P (A ∪ B)
12. P (A ∩ B)
⎧
⎫
The sample space S = ⎨⎩1, 2, 3, 4, 5, 6, 7, 8, 9, 10⎬⎭; The sample space S = ⎧⎨1, 2, 3, 4, 5, 6, 7, 8, 9, 10⎫⎬;
⎩
⎭
⎧
⎫
⎧
⎫
A ∪ B = ⎨⎩1, 2, 3, 4, 5, 6, 7, 9⎬⎭
A ∩ B = ⎨⎩1, 3, 5⎬⎭
n(A ∪ B)
8
4
n(A ∩ B)
3
=
=
P(A ∪ B) =
=
P (A ∩ B ) =
5
10
n(S)
10
n(S)
14. P(B C)
13. P(A C)
⎧
⎫
⎧
⎫
The sample space S = ⎨⎩1, 2, 3, 4, 5, 6, 7, 8, 9, 10⎬⎭; The sample space S = ⎨⎩1, 2, 3, 4, 5, 6, 7, 8, 9, 10⎬⎭;
⎧
⎫
⎧
⎫
A C = ⎨⎩7, 8, 9, 10⎬⎭
B C = ⎨⎩2, 4, 6, 8, 10⎬⎭
C
C
n(B )
n (A )
5
4
= 2
= 1
=
=
P(A C) =
P(B C) =
10
10
5
2
n(S)
n(S)
_ _
_ _ _
Module 21
Exercise
GE_MNLESE385801_U8M21L1.indd 1090
Exercises 1–8
Example 1
Calculating Theoretical Probabilities
Exercises 9–14,
22, 26, 29
Example 2
Using the Complement of an Event
Exercises 15–21,
23–25, 27–28
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Discuss when a Venn diagram might be
useful in solving a probability problem, and when
another method might be easier.
_ _ _
Lesson 1
1090
Depth of Knowledge (D.O.K.)
Explore
Working with Sets
Discuss the importance of understanding the sample
space. Encourage students to always list the members
of the sample space before they find a probability.
Discuss why this can help avoid errors, such as
finding the probability of rolling a 2 with a number
cube as __15 .
© Houghton Mifflin Harcourt Publishing Company
9. P(A)
10. P(B)
⎧
⎫
⎧
⎫
The sample space S = ⎨⎩1, 2, 3, 4, 5, 6, 7, 8, 9, 10⎬⎭; The sample space S = ⎨⎩1, 2, 3, 4, 5, 6, 7, 8, 9, 10⎬⎭;
⎧
⎫
⎧
⎫
B = ⎩⎨1, 3, 5, 7, 9⎬⎭
A = ⎨⎩1, 2, 3, 4, 5, 6⎬⎭
n(B)
n(A)
5
1
3
6
=
=
P(B) =
=
=
P(A) =
10
2
5
10
n(S)
n(S)
_ _ _
Practice
COMMUNICATING MATH
You have a set of 10 cards numbered 1 to 10. You choose a card
at random. Event A is choosing a number less than 7. Event B is
choosing an odd number. Calculate the probability.
_ _ _
Concepts and Skills
COMMON
CORE
Mathematical Practices
1–8
1 Recal
MP.4 Modeling
9–14
1 Recal
MP.2 Reasoning
15–20
2 Skills/Concepts
MP.2 Reasoning
21
3 Strategic Thinking
MP.6 Precision
22
3 Strategic Thinking
MP.4 Modeling
23–26
3 Strategic Thinking
MP.3 Logic
27
3 Strategic Thinking
MP.3 Logic
9/2/14 3:43 PM
Probability and Set Theory 1090
Use the complement of the event to find the probability.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Review the connection between likelihood
15. You roll a 6-sided number cube. What is the probability that you do not roll a 2?
1
The probability of rolling a 2, P(2), is .
6
5
1
The probability of not rolling a 2 is 1 - P(2) = 1 - = .
6
6
16. You choose a card at random from a standard deck of cards. What
is the probability that you do not choose a red king?
_
and probability with students. Discuss how this can
be useful when solving problems. When students
calculate the probability of an event, be sure they
understand what this means in the context of the
original problem. For example, students should
recognize that an event with a probability of 0.9 is
very likely to occur, while an event with a probability
of 0.1 is unlikely to occur.
The probability of drawing a red
1
2
=
.
king, P(red king), is
52
26
The probability of not drawing a red king
25
1
is 1 - P(red king) = 1 =
.
26
26
17. You spin the spinner shown. The spinner is divided into 12 equal sectors.
What is the probability of not spinning a 2?
_ _
_ _
11
AVOID COMMON ERRORS
12 1
2
10
3
4
9
8
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Oleg
Golovnev/Shutterstock
Students may not consider the sample space when
finding probabilities. Suggest that they summarize
the probability ratio using words before they compute
the probability.
_ _
5
7
6
The probability of spinning a 2, P(2), is
1
_
.
12
11
1
The probability of not spinning a 2 is 1 - P(2) = 1 =
.
12
12
_ _
18. A bag contains 2 red, 5 blue, and 3 green balls. A ball is chosen at random.
What is the probability of not choosing a red ball?
1
2
= .
The probability of choosing a red ball, P(red ball), is
5
10
4
1
The probability of not choosing a red ball is 1 - P(red ball) = 1 - = .
5
5
19. Cards numbered 1–12 are placed in a bag. A card is chosen at random. What is the
probability of not choosing a number less than 5?
1
4
The probability of choosing a number less than 5, P(less than 5), is
= .
12
3
2
1
The probability of not choosing a number less than 5 is 1 - P(less than 5) = 1 - = .
3
3
20. Slips of paper numbered 1–20 are folded and placed into a hat, and then a slip of
paper is drawn at random. What is the probability the slip drawn has a number which
is not a multiple of 4 or 5?
_ _
_ _
_ _
_ _
Multiples of 4 up to 20: 4, 8, 16, 20
Multiples of 5 up to 20: 5, 10, 15, 20
⎧
⎫
The set of multiples of 4 or 5 is ⎨⎩4, 5, 8, 10, 15, 16, 20⎬⎭.
7
P(multiple of 4 or 5) =
20
The probability of not selecting a card that is a multiple of 4 or 5 is
13
7
=
.
1 - P(multiple of 4 or 5) = 1 20
20
_
_ _
Module 21
Exercise
GE_MNLESE385801_U8M21L1.indd 1091
1091
Lesson 21.1
Lesson 1
1091
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
28
3 Strategic Thinking
MP.3 Logic
29
3 Strategic Thinking
MP.3 Logic
2/26/16 11:06 PM
21. You are going to roll two number cubes, a white number cube and a red
number cube, and find the sum of the two numbers that come up.
VISUAL CUES
When students create Venn diagrams to model a
sample space and sets, caution them to be sure that
an element is not used more than once on the
diagram. For example, have students check that a
number does not appear both in Set A and in its
intersection with Set B.
a. What is the probability that the sum will be 6?
There are 36 possible outcomes. There are 5 ways to get a sum
of 6, where the first addend is from the white cube and the second
addend is from the red cube: 5 + 1, 4 + 2, 3 + 3, 2 + 4, and 1 + 5.
5
So the probability of getting a sum of 6, P(6), is
.
36
_
b. What is the probability that the sum will not be 6?
5
31
= __
.
The probability that the sum will not be 6 is P(not 6) = 1 - P(6) = 1 - __
36
36
22. You have cards with the letters A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P. Event U is
choosing the cards A, B, C or D. Event V is choosing a vowel. Event W is choosing a
letter in the word “APPLE”. Find P(U ⋂ V ⋂ W).
1
U ⋂ V ⋂ W = ⎨A⎬⎭ ; P(U ∩ V ∩ W) = __
16
⎩
⎧ ⎫
A standard deck of cards has 13 cards (2, 3, 4, 5, 6, 7, 8, 9, 10,
jack, queen, king, ace) in each of 4 suits (hearts, clubs, diamonds,
spades). The hearts and diamonds cards are red. The clubs and
spades cards are black. Answer each question.
23. You choose a card from a standard deck of cards at random. What is the probability
that you do not choose an ace? Explain.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Sung-Il
Kim/Corbis
12
4
1
__
; there are 4 aces in the 52-card deck, so P(ace) = __
= __
. This means
13
52
13
1
12
__
__
P(not ace) = 1 - = .
13
13
24. You choose a card from a standard deck of cards at random. What is the probability
that you do not choose a club? Explain.
13
1
_3; there are 13 clubs in the 52-card deck, so P(club) = __
=_
. This means
4
4
52
3
1
P(not club) = 1 - _ = _.
4
4
25. You choose a card from a standard deck of cards at random. Event A is choosing a
red card. Event B is choosing an even number. Event C is choosing a black card. Find
P(A ∩ B ∩ C). Explain.
A ∩ B ∩ C = Ø because you can never draw a card that is both red and black.
Therefore, P(A ∩ B ∩ C) = 0.
Module 21
GE_MNLESE385801_U8M21L1 1092
1092
Lesson 1
03/09/14 8:22 PM
Probability and Set Theory 1092
26. You are selecting a card at random from a standard deck of cards. Match each event
with the correct probability. Indicate a match by writing the letter of the event on the
line in front of the corresponding probability.
1
B
_
A. Picking a card that is both red and a heart.
52
1
A
_
B. Picking a card that is both a heart and an ace.
4
51
C
_
C. Picking a card that is not both a heart and an ace.
52
JOURNAL
Have students write and solve their own probability
problems. Remind students to use set notation in
their solutions to the problems.
n(red ∩ heart)
13
1
P(red ∩ heart) = __________ = __
=_
4
52
n(deck)
(heart ∩ ace) __
n
1
__________
= 52
P(heart ∩ ace) =
n(deck)
51
1
= __
; the only card
P(not (heart ∩ ace)) = 1 - P(heart ∩ ace) = 1 - __
52
52
that is both a heart and an ace is the ace of hearts, so there are 51 cards in
the event not (heart ∩ ace).
H.O.T. Focus on Higher Order Thinking
27. Critique Reasoning A bag contains white tiles, black tiles, and gray tiles. Someone
is going to choose a tile at random. P(W), the probability of choosing a white tile,
is __14 . A student claims that the probability of choosing a black tile, P(B), is __34 since
P(B) = 1 - P(W) = 1 - __14 = __34 . Do you agree? Explain.
No; choosing a black tile is not the complement of choosing a
white tile since the bag also contains gray tiles. It is not possible to
calculate P(B) from the given information.
© Houghton Mifflin Harcourt Publishing Company
28. Communicate Mathematical Ideas A bag contains 5 red marbles and 10 blue
marbles. You are going to choose a marble at random. Event A is choosing a red
marble. Event B is choosing a blue marble. What is P(A ∩ B)? Explain.
0; A ∩ B = Ø since a marble cannot be both red and blue. So P(A ∩ B) = 0.
29. Critical Thinking Jeffery states that for a sample space S where all outcomes are
equally likely, 0 ≤ P(A) ≤ 1 for any subset A of S. Create an argument that will justify
his statement or state a counterexample.
Assume A is a subset of S. Then 0 ≤ n(A) ≤ n(S). For example, if S has
10 elements, the number of elements of A is greater than or equal to 0 and less
than or equal to 10. No subset of S can have fewer than 0 elements or more
than 10 elements. So 0 ≤
n(A)
n(S)
n(A)
___
≤ 1. When all the outcomes are equally likely,
n(S)
P(A) = ___. Therefore 0 ≤ P(A) ≤ 1.
Module 21
GE_MNLESE385801_U8M21L1.indd 1093
1093
Lesson 21.1
1093
Lesson 1
9/2/14 3:43 PM
Lesson Performance Task
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Call attention to the point in the Lesson
For the sets you’ve worked with in this lesson, membership in a set is binary: Either
something belongs to the set or it doesn’t. For instance, 5 is an element of the set of odd
numbers, but 6 isn’t.
Performance Task graph where the green mediumsized dog line and the blue big-dog line intersect. Ask
students to give as much information as they can
about that point. Sample answer: The point
represents a weight of around 100 pounds and a
degree of membership of around 0.3. The point
represents the highest degree of membership that a
dog of around 100 pounds can obtain
simultaneously in both the medium-sized and
big-weight categories.
In 1965, Lofti Zadeh developed the idea of “fuzzy” sets to deal with sets for which
membership is not binary. He defined a degree of membership that can vary from 0 to 1.
For instance, a membership function mL(w) for the set L of large dogs where the degree of
membership m is determined by the weight w of a dog might be defined as follows:
• A dog is a full member of the set L if it weighs 80 pounds or more. This can be
written as mL(w) = 1 for w ≥ 80.
• A dog is not a member of the set L if it weighs 60 pounds or less. This can be
written as mL(w) = 0 for w ≤ 60.
1.
Degree of Membership
• A dog is a partial member of the set L if it weighs between 60 and 80 pounds.
This can be written as 0 < mL(w) < 1 for 60 < w < 80.
Small Dogs
The “large dogs” portion of the graph shown displays the
m
membership criteria listed above. Note that the graph shows
1
only values of m(w) that are positive.
Using the graph, give the approximate weights for which
a dog is considered a full member, a partial member, and
not a member of the set S of small dogs.
0 pounds to 20 pounds; between 20 pounds and
30 pounds; more than 30 pounds
2.
0
Medium-Sized
Dogs
Large Dogs
0
20
40
60
80 100 120 140 160
Weight (lb)
The intersection of A and B is given by the membership rule
mA⋂B(x) = minimum(mA(x), mB(x)). So, for a dog of a given size, the degree of its
membership in the set of dogs that are both small and medium-sized (S ⋂ M) is
the lesser of its degree of membership in the set of small dogs and its degree of
membership in the set of medium-sized dogs.
Using the graph above and letting S be the set of small dogs, M be the set of
medium-sized dogs, and L be the set of large dogs, draw the graph of each set.
S∪ M
Degree of Membership
Degree of Membership
1
0
b. M ⋂ L
m
w
0
20
40
60
80 100 120 140 160
Weight (lb)
Module 21
1094
m
1
0
x and a medium-sized dog degree of membership of
y. Is x > y, x < y, or does the relationship between x
and y depend on Arf ’s weight? Explain. The
relationship depends on Arf’s weight. The red and
green graphs intersect at about 40 pounds. If Arf
weighs less than 40 pounds, x > y. If Arf weighs
more than 40 pounds, x < y. If Arf weighs 40
pounds, x = y.
© Houghton Mifflin Harcourt Publishing Company
The union of two “fuzzy” sets A and B is given by the membership rule
mA∪B(x) = maximum(mA(x), mB(x)). So, for a dog of a given size, the degree of its
membership in the set of small or medium-sized dogs (S ∪ M) is the greater of its
degree of membership in the set of small dogs and its degree of membership in the set
of medium-sized dogs.
a.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Arf has a small-dog degree of membership of
w
w
0
20
40
60
80 100 120 140 160
Weight (lb)
Lesson 1
EXTENSION ACTIVITY
GE_MNLESE385801_U8M21L1.indd 1094
Have students draw graphs showing fuzzy sets ranging from cold to hot. (Three
sets could show cold, warm, and hot. Four sets could show cold, cool, warm, and
hot. However, leave the choice of adjectives and the number of sets to students.)
The vertical axis should record degrees of membership from 0 to 1. The horizontal
axis should show either Fahrenheit or Celsius temperatures. Encourage students
to be creative with their graphs, for example, by using colors to distinguish sets
from one another. Students should write and answer at least three questions
involving unions, intersections, and complements of the sets they have graphed.
2/26/16 11:06 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Probability and Set Theory 1094