Finite Math B: Chapter 7 Notes
Sets and Probability
1
Chapter 7: Sets and Probability
Part 1: Sets
7.1 Sets
What is a set?
A set is a __________________________ collection of objects.
We should always be able to answer the question: “Is object X in this set or not?”
Examples of “sets”
{Mrs. Leahy’s Semester 2 Discrete Math Class}
{Coins minted by the US Treasury in 2011}
{even integers}
Generally in this chapter we will be talking about sets of numbers.
Set Notation:
Use set braces: { } to enclose the numbers in set.
Example 1:
1,3,5, 7,9
The numbers 1, 3, 5, 7, 9 are called the __________________ or _________________ of the set.
Naming Sets:
Use the symbol: 
3 1,3,5, 7,9
SAY: “3 is an element of the set”
Use the symbol: 
4 1,3,5,7,9
SAY: “4 is NOT an element of the set”
Sets are often named with letters:
Example 2:
True or False?
A  10, 20,30
A  10, 20,30
11 A
An empty set is a set with __________________________.
25  A
10  A
Symbol: _____
Can you think of a “collection” or group that would have no members?
The universal set is a set that contains all the objects being discussed. (integers, people, whole numbers, etc.)
Finite Math B: Chapter 7 Notes
Sets and Probability
0
= the number zero

= an empty set, a set with no elements
0
= a set that has one element – the number zero

= a set that has one element – the empty set
CAUTION:
Example 3:
2
How many elements are in each set?
Symbol: n( A) = the number of elements in a finite set A.
a. A  5, 6, 7
b. B  {positive integers less than 5}
c.
C  0
Two sets are equal if they contain ______________ the ____________ elements.
Example 4:
True or False?
NOTE:
a. {1,10,100}  {100,1,10}
If Set A equals Set B, we say
b. {32,33,34}  {32,33,34,35}
c.
positive even numbers
d.
3,5, 7,9  5, 7,9,3
Set-Builder Notation:
 8  {0, 2, 4,6,8}
A B
If Set A does not equal Set B,
we say:
A B
Useful when we are looking for objects that share a common property
x x has property P
SAY: “The set of all elements x such that x has property P”
Example 5:
x x is a whole number between 7 and 10
Elements: _________________________
Finite Math B: Chapter 7 Notes
Sets and Probability
3
*****
Subsets
Consider the two sets A and B:
A  {3, 4,5,6}
B  {2,3, 4,5,6,7,8}
Notice that EVERY element is A is also an element in B.
We say that A is a subset of B.
A  {1, 2,3, 4}
B  {1, 2,3, 4,5}
Example 6:
True or False:
Example 7:
A  {5, 6, 7,8}
B  {7, 6,5,8}
True or False:
A B
A B
A B
A B
B A
B A
B A
B A
A  {1, 2,3, 4, 7}
B  {1, 2,3, 4,5,12}
Example 8:
True or False:
A B
A B
The empty set is a subset of every set.
A set is always a subset of itself.
B
A
Finite Math B: Chapter 7 Notes
Sets and Probability
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Counting Subsets
Example 9: How many subsets are possible? What are they?
(Tree diagrams can sometimes be useful)
a.
5, 6
b.
 x, y, z
Basically, there are two different possibilities for each element = yes or no
SO:
Example 10: Find the number of subsets for each set.
a.
{9,8,7,6}
b.
x x is a day of the week
c. 
Finite Math B: Chapter 7 Notes
Sets and Probability
5
Venn Diagrams:
Helps to show elements of sets & subsets.
U is the universal set
A is a subset of B.
B is a subset of U.
Set Operators:
Often we will form new sets by combining or manipulating one or more existing sets.
Complement:
The elements in the universal set NOT in
your set.
A (white) is a set.
A’ (pink) --- say “A prime” is everything
else and is called the complement of A.
For Example:
Let U = the students in a class
A = the set of all female students in the
class.
A’ = the set of all male students in the
class.
Example 11: Let U  {1, 2,3, 4,5,6,7,8,9,10} , A  1, 2,3, 4 , and B  {1,3,5,7,9}
Find each set.
a.
A'
c.  '
b.
B'
d.
 A ' '
Finite Math B: Chapter 7 Notes
Sets and Probability
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Intersection:
The intersection of two sets is the set of all elements
belonging to BOTH A and B.
A B
Symbol:
Example:
Let
U = the students in a class
A = the set of all male students
B = the set of all students with “B”
averages.
Then
A  B = the set of all male students with “B” averages.
Example 11: Let A  1, 2,3 , B  {1,3,5,7} , and the universal set U  {1, 2,3, 4,5,6,7,8}
Find each set:
a.
A B
Disjoint Sets:
Disjoint sets have no elements in common.
The intersection of disjoint sets is the empty set.
Example:
{1, 2,3, 4} {5,6,7,8}  
b.
A ' B
Finite Math B: Chapter 7 Notes
Sets and Probability
7
Union:
The set of all elements belonging to set A, to
set B, or to both sets.
Symbol: A  B
Example:
Let U = students in a class
A = set of all female students
B = set of all students with B averages
A  B = any student who is either
female OR has a “B” average
Example 12:
Let A  {1, 2,3, 4,5} , B  {1,3,5,7,9} , and C  {1, 2,5,6,9,10}
for the universal set U  {x x is an integer such that 1  x  12}
Find each set:
a.
A B
b.
AC '
c.
 A  B  C
d.
( A ' B)  C '
Finite Math B: Chapter 7 Notes
Sets and Probability
8
*****
Applications of Sets
Example:
Pg 348
The following table give the 52-week high and
low prices, the closing price, and the change
from the previous day for six stocks in the
Standard & Poor’s 100 on April 11, 2006.
Let the universal set U consist of the six stocks
listed in the table. Let A contain all stocks with
a high price greater than $34. Let B contain all
stocks with a closing price between $26 and
$30. Let C contain all stocks with a positive
price change.
State the elements in each set:
A=
A’ =
B=
AC =
C=
A B =
Example:
A department store classifies credit applicants by gender, marital status, and employment status. Let the
universal set be the set of all applicants, M be the set of all male applicants, S be the set of single applicants, and
E be the set of employed applicants. Describe each set in words.
a.
M E
b.
M ' S
c.
M ' S '
d.
M E'
Finite Math B: Chapter 7 Notes
Sets and Probability
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7.2 Applications of Venn Diagrams
pg 353 “The responses to a survey of 100 household show that 21 have a DVD player, 56 have a videocassette
recorder, and 12 have both. How many have neither a DVD player nor a videocassette recorder?”
Venn Diagrams are very useful for “sorting out” this information.
Things that might happen:
1 set leads to 2 regions
2 sets lead to 4 regions
2 sets lead to 3 regions
2 sets lead to 3 regions
3 sets lead to 8 regions
Example 1: Shade the following Venn Diagrams
a.
A B
b.
A B
A
C
c.
A ' B '
A
B
C
A
B
C
B
Finite Math B: Chapter 7 Notes
d.
A ' C '
Sets and Probability
A ' ( B  C ')
e.
f.
B  AC
A
A
C
10
B
C
A
B
C
B
Example 2a:
“The responses to a survey of 100 household show that
21 have a DVD player, 56 have a videocassette recorder,
and 12 have both. How many have neither a DVD player
nor a videocassette recorder?”
In general: These questions are often referred to as Both/Neither Questions and usually have information
about two types of elements: Type A and Type B.
Total = Type A + Type B + Neither – Both
Example 2b:
Mrs. Leahy surveys the 35 students in her period 2 finite math class. She notes that 15 students said they like
donuts, 25 students say they like bagels, and 5 students were absent the day of the survey were able to
respond. Assuming that everyone who was present answered, how many students like BOTH bagels and
donuts?
Finite Math B: Chapter 7 Notes
Sets and Probability
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Example 3: A survey of 77 freshman business students at a large university produced the following results.
25 read Business Week
19 read The Wall Street Journal
27 do not read Fortune
11 read Business Week but not The Wall Street Journal
11 read The Wall Street Journal and Fortune
13 read Business Week and Fortune
9 read all three
Questions to answer:
How many read none of the publications?
How many read only Fortune?
How many read Business Week and The
Wall Street Journal but not Fortune?
The Union Rule For Sets:
Essentially:
x  y  z  n(A) + n(B)  y
In general, if you feel like you are “missing” information in a survey
problem, you need probably need to use the union rule in some
way.
Example 4: A group of 10 students are all majoring in either
accounting or economics or both. Five of the students are
economics majors and 7 are majors in accounting. How many
major in both subjects?
Finite Math B: Chapter 7 Notes
Sets and Probability
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Example 5:
The table gives the number
of threatened and
endangered animal species
in the world as of April
2006. (pg 358) Using the
letters given in the table to
denote each set, find the
number of species in each
of the following sets.
a.
EB
b.
EB
c.
(F  M )  T '
Example 6: Suppose that a group of 150 Students have joined at least one of three chat rooms: one on autoracing, one on bicycling, and one for college students. For simplicity, we will call these rooms A, B, and C. In
addition,
90 students joined room A
50 students joined room B
70 students joined room C
15 students joined rooms A and C
12 students joined rooms B and C
10 students joined all three rooms
How many students joined both A and B?
Finite Math B: Chapter 7 Notes
Sets and Probability
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Example 7: A survey was conducted of 150 High School students who were asked about which international city
they would like to visit from Athens, Dublin, and Hong Kong. The results were tallied as follows:
60 students wanted to visit Athens
70 students wanted to visit Dublin
80 students wanted to visit Hong Kong
10 students did not respond to the survey
25 wanted to visit both Athens and Dublin
15 wanted to visit both Athens and Hong Kong
35 wanted to visit both Dublin and Hong Kong
How many students responded that they
wanted to visit all three cities?
Finite Math B: Chapter 7 Notes
Sets and Probability
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7.3 Intro to Probability
Vocabulary:
Experiment: an activity or occurrence with an observable result
Trial: Each repetition of the experiment
Outcome: A possible result of a trial
Sample Space: The set of all possible outcomes of that experiment
Event: a subset of the sample space
Example: Experiment – Tossing a coin
Sample space:
Example: Give the sample space for each experiment
a) Rolling a die
b) Drawing a card from a deck containing only the 13 spades
c) Measuring the weight of a person to the nearest half-pound (the scale will not
measure more than 300 lb)
d) Tossing a coin 3 times
Finite Math B: Chapter 7 Notes
Sets and Probability
Example: For each sample space, write the set the represents the given event
a) Sample Space: Rolling a die
Event 1: The die is even
Event 2: The die shows a multiple of 3
Event 3: The die shows a 1
Event 4: The die shows a number less than 4
b) Sample space: A family with three children
Event 1: The family has exactly two girls
Event 2: The family has three children of the same sex
Event 3: The family has at least one boy
Set Operators: Remember
E AND F both occur:
E OR F or both occur:
E does not occur:
E and F are DISJOINT (Mutually Exclusive):
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Finite Math B: Chapter 7 Notes
Sets and Probability
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Probability
Probability is the likelihood of an event:
0
1
Let S be a sample space of equally likely outcomes and let event E be a subset of S,
Then the probability that the event E occurs is:
P( E ) 
n( E )
n( S )
Example: Find the Following Probabilities
1. You roll two fair dice
a.
P(a sum of 5)
b.
P(a sum less than 5 OR a sum greater than 10)
c.
P (a sum greater than or equal to 10 AND a sum that is even)
2. You draw a card from a standard deck of 52 playing cards
a. P (a heart)
b) A red queen
c) a face card
d) P (a heart OR a queen)
e) P(a heart AND a queen)
Finite Math B: Chapter 7 Notes
Sets and Probability
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7.4 Basic Concepts of Probability
Reminder:
intersection:
union:
A B
When dealing with probability you often have to be
very careful about overlap. We will use the union
rule to help us.
A
F)
B
B
Union Rule for Probability
For any events E and F from a sample space S:
F )  P( E)  P( F )  P( E
A
A
A
P( E
A B
B
A
C
B
C
Basically:
The probability of E or F happening is equal to the
probability of E + the probability of F – the probability
of the overlap of the two events.
C
Example 1: You have a box with several geometric shaped tiles that are lettered A, B, or C. (See picture above)
You draw out a single tile from the box. Find the following probabilities.
a) P(triangle or square)
b) P(triangle or letter B)
c) P(square or letter C)
d) P(square or letter A)
e) P(circle and letter C)
f) P(triangle and letter C)
Example 2: You draw a card from a standard deck of 52 playing cards. Let R represent the event “draw a red
card” , let F represent the event “draw a face card”, and let J represent the event “draw a Jack.” Find the
following probabilities.
a) P( R
J)
b) P(J F)
c) P(F R)
Finite Math B: Chapter 7 Notes
Sets and Probability
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Complements
Often in probability it is easier to find the complement of an event than the event itself.
Complement Rule for Probabilities
P( E )  1  P( E)
Example: Two fair dice are rolled and their sum is calculated. Find the following probabilities.
a) P(sum is not 12)
b) P(sum is larger than 3)
Odds vs. Probability
Sometimes probability is given in terms of odds.
Example: You have a 2 out of 5 chance that you will win a door
prize. What are the odds in favor of you winning the prize?
Odds in favor of an Event E
The ratio of P( E ) : P( E)
P( E )
where P( E ')  0
P( E ')
You can also turns odds back into probability by determining n( S )
If the odds in favor of an event E are m to n, then
P( E ) 
m
mn
and P( E ') 
n
mn
Example: You read that the odds of contracting the flu
this year are 7 to 50. What is the probability that you will
get the flu?
Example: The following table lists the probability that a dollar spent by US
Advertisers is spent on a particular medium.
a) P(Newspapers)
b) P(Newspapers or Broadcast TV)
c) P(Not Magazines nor Yellow Pages)
Direct Mail
Newspapers
Broadcast TV
Cable TV
Radio
Yellow Pages
Magazines
Other
0.1979
0.1767
0.1754
0.0816
0.0742
0.0531
0.0464
0.1946
Finite Math B: Chapter 7 Notes
Sets and Probability
Properties of Probability
You are told the responses to a survey were as follows:
30% yes
40% no
35% undecided
Problem?
The probability of any event must be between _____ and ______ . No exceptions.
The SUM of all the probabilities of events in a sample space must equal ______ or _______.
(rounding could cause issues with this in the tiny decimals, so be careful)
Example: Are these possible outcomes of an experiment? Why or why not?
a)
Outcome
S1 S2 S3 S4 S5
probability .3 .3 .3 .1 .1
b)
Outcome
S1 S2
S3
S4 S5
probability 1.5 -0.5 -0.5 0.3 0.2
c)
Outcome
S1 S2 S3 S4 S5
probability .3 .1 .1 .4 .1
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Finite Math B: Chapter 7 Notes
Sets and Probability
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7.5 Conditional Probability
A manager at a brokerage firm notices that
some of his stockbrokers follow the company’s
suggestions based on research while others
tend to operate on more of a “gut feeling” and
make their own calls.
Using the data in the chart, find the following probabilities.
1. P(a stockbroker picked stocks that went up)
2. P(a stockbroker did not pick stocks that went up)
3. P(a stockbroker used research)
4. P(a stockbroker didn’t use research)
What if the manager wants to compare the success of his brokers who are using research to those that are not?
5. P(a stockbroker who used research
picked stocks that went up)
6. P (a stockbroker who didn’t use research
picked stocks that went up)
When you reduce your _____________________ you are using a concept called “Conditional Probability.”
So for our example:
5. P(Stocks went up|used research)= P(A|B)
6. P(Stocks went up|did not use research)= P(A|B′)
Finite Math B: Chapter 7 Notes
Sets and Probability
21
Example 1: Given the following Venn Diagram
a) 𝑃(𝐸) =
b) 𝑃(𝐹) =
c) 𝑃(𝐸 ∩ 𝐹) =
d) 𝑃(𝐸|𝐹) =
e) 𝑃(𝐹|𝐸) =
Example 2: Suppose you ask 100 people their
age and what they prefer to drink at lunch and
record the results in the table to the right.
Find the following probabilities.
a) P(soda)
b) P(40-49)
c) P(30 or older)
d) P(soda or coffee)
e) P(coffee or 20-29)
f) P(coffee and 20-29)
g) P(𝑠𝑜𝑑𝑎|30 − 39)
h) P(50 − 59|𝑡𝑒𝑎)
i) 𝑃(𝑤𝑎𝑡𝑒𝑟|𝑢𝑛𝑑𝑒𝑟 40)
Finite Math B: Chapter 7 Notes
Sets and Probability
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Example 3: A single card is drawn from a deck of 52.
a) P(A Queen given the card is black)
b) P(a face card given the card is a heart)
Example 4: A fair die is rolled.
a) P( a 2, given the number is even)
b) P(a 1, given the number was less than 4)
Product Rule of Probability:
Example 3: You know that 54% of your high school is female and 46% of your high school is male. A
survey in the school newspaper states that 32% of female students are planning to go to prom. 45% of
the male students are planning on attending prom. If you randomly choose a name from the school
roster, what is the probability….
P(female attending prom) =
P(male not attending prom) =
Example 4: You draw one card from a standard deck of cards. Use the Product rule to find the
probability that the card is a red queen.
Finite Math B: Chapter 7 Notes
Sets and Probability
Conditional probability may need to be considered with
independent/dependent events. For example:
You draw a card from a deck and, without replacing it, you draw a second card.
1st draw:
2nd draw:
You flip a coin. You flip the coin again.
1st flip:
2nd flip:
Events are considered to
be independent if:
Are the following events independent?
Drawing a card from a deck, without replacing it, drawing another card.
Drawing a card from a deck, replacing it, drawing another card.
Flipping a coin twice.
Flipping a coin. Rolling a die.
Choosing a name off the course roster to be the president. Choosing another name to be the vice
president.
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Finite Math B: Chapter 7 Notes
Sets and Probability
24
Example 5: You draw a card from a standard deck and then without replacing it, you draw again. Find the
following probabilities.
a) the second card is red given the first card was red
b) the second card is red given the first card was black
c) the second card is an ace given the first card is a queen
d) the second card is a heart given the first card is a heart
Example 6: You draw a card from a standard deck and then REPLACE IT before drawing a second card. Find the
probability.
a) the second card is an ace given the first card is a queen
b) the second card is a heart given the first card is a heart
Example 7: You draw a card from a standard deck and without replacing it, you draw a second card. Find the
probability.
a) Both cards are hearts.
b) The first card is a king and the second card is a queen.