MAT 141 ‐ Chapter 7
Finite Mathematics
MAT 141: Chapter 7 Notes
Sets
Set Theory and Probability
David J. Gisch
Definition
• A set is a well-defined collection of objects.
▫ We denote sets with capital letters
▫ We write sets with brackets as follows
3, 4, 5
▫ This is referred to as roster form of a set.
• Any item belonging to a set is called an element or
member of that set.
▫ We denote elements of a set as follows
3 ∈ 3, 4, 5
7 ∉ 3, 4, 5
Why well-defined?
Give me the set of people in this room who are nice.
Definition
• Repetitions of elements do not matter. Whether it is
listed once or twice it is still a member of the set and that
is all that matters.
• Order also does not matter in sets, unless it is used to
establish a pattern.
3, 4, 5
4, 3, 5
3, 3, 3, 4, 5
5, 5, 3, 4, 4, 4
• A set can also contain no elements. We call this the
empty set.
▫ We denote the empty set as ∅
▫ For example, the set of months starting with the letter z.
∅ all have different meanings.
▫ Note that ∅, 0, 0 ,
1
MAT 141 ‐ Chapter 7
Definition
Sets
• The set of all things being discussed is referred to as the
universal set. We denote the universal set as set .
Example 7.1.1: Explain the difference for each of the
following.
• For example, if we were discussing arithmetic in third
grade we might use the universal set of whole numbers.
In college algebra the universal set would be all real
numbers.
∅
0
0
∅
Sets
Example 7.1.2: Write out each of the following sets in roster
form.
(a) The set of all numbers between 2 and 7.
Definition
• Sets can also be written in set-builder notation.
| ∈ 2
5
The set
Of things
(b) The set days of the week that begin with the letter S.
Such that
They are integers and between
2 and 5.
In roster form
3, 4
Obviously above it doesn’t help but what about
| (c) The set of planets in our solar system that begin with the
letter C.
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MAT 141 ‐ Chapter 7
Set-Builder Notation
Example 7.1.3: Write each of the following sets in setbuilder notation.
Set-Builder Notation
Example 7.1.4: Write each of the following sets roster form.
(a)
| (b)
| (a) 2, 4, 6, 8, 10, …
(b)
3, 2, 0, 1, 2, 3, 4
(c) 15, 16, 17, 18, … .
Definition
• Sometimes every element of set is also the element of
another set.
Subsets
Example 7.1.5: List all of the subsets of the set 3, 4
▫
| | ▫
▫ Here, every element of set A is also an element of set B.
• Think of a proper subset as being strictly smaller. When
in doubt do not write ⊂ , write ⊆ .
3
MAT 141 ‐ Chapter 7
Subsets
Example 7.1.5: List all of the subsets of the set
Number of Subsets
, ,
Number of Subsets
Example 7.1.6: How many subsets does the following set
have?
, , , 3, 2
Example 7.1.5: List all of the subsets of the set
, ,
Set Relations
• Sets can be
▫ Proper subsets (one is contained in the other).
| 
| 
▫ Have some overlap (called the intersection).
| 
| 
▫ Have no overlap (called disjoint sets).

| | 
4
MAT 141 ‐ Chapter 7
Set Relations
Set Relations
Set Relations
Set Relations
5
MAT 141 ‐ Chapter 7
Sets, Union, Intersection, Complement
Example 7.1.7: Use the given sets to state each of the
following in roster form.
, , , , , , ,
, ,
, ,
,
Sets, Union, Intersection, Complement
Example 7.1.8: Use the given sets to state each of the
following in roster form.
(a)
∪
(a)
∪
(b)
∩
(b)
∩
(c)
∪∅
(c) B′
(d)
′
(d)
′∩ ′
(e)
′∩
(e)
∪
(f)
∩
′
′
Venn Diagram (2 Sets)
• 2 sets split the diagram up into 3-4 regions.
Applications of Venn Diagrams
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MAT 141 ‐ Chapter 7
Venn Diagrams
Example 7.2.1: Write each shaded region using set notation.
Venn Diagram (3 Sets)
• 3 sets split the diagram up into at most 8 regions.
Venn Diagrams
Example 7.2.2: Write each shaded region using set notation.
Venn Diagrams
Example 7.2.3: Write each shaded region using set notation.
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MAT 141 ‐ Chapter 7
Venn Diagrams
Venn Diagrams
Example 7.2.4: Assume that A: Set of athletes, B: Set of
honors students, and C: Set of Band students. Describe each
of the following regions in words.
Example 7.2.5: Write each indicated region using set
notation.
II
II
IV and V
IV and V
I and II and III
I and II and III
Sets, Union, Intersection, Complement
Example 7.2.6: Use the given sets to state each of the
following in roster form.
(a)
∪
∪
(b)
∩
∩
, , , , , , , , ,
, , , ,
, , , ,
,
Making Venn Diagrams
• Peel your way out!!!!
▫ Start with the inner-most region first.
▫ Go to the intersections and subtract off what you already have.
▫ Go to the remainder of the sets and subtract off what you already
have.
▫ Always check if any amount is unused.
• For example: Let’s say there are 20 total elements and A
has 12 elements, B has 10 elements and A intersect B has
8 elements.
(c)
∪
∩
(d)
∪
∩
∪
(e)
∩ ′ ∩
∪
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MAT 141 ‐ Chapter 7
The Number of Elements
Venn Diagrams
Example 7.2.7: Eight hundred students were surveyed and the results of
the campus blood drive survey indicated that 490 students were willing
to donate blood,340 students were willing to help serve a free breakfast
to blood donors, and 120 students were willing to donate blood and
serve breakfast.
(a) How many students were willing to donate blood or serve
breakfast?
(b) How many were willing to do neither?
Venn Diagrams
Venn Diagrams
Example 7.2.8: A survey of 120 college students was taken at
registration. Of those surveyed, 75 students registered for a math
course, 65 for an English course, and 40 for both math and English. Of
those surveyed,
Example 7.2.9: Use the Venn Diagram below to answer the following.
(a) How many registered only for a math course?
(b) How many read Business Week and Fortune but not the Journal?
(b) How many registered only for an English course?
(a) How many students read none of the publications?
(c) How many read Business Week or the Journal?
(c) How many registered for a math course or an English course?
(d) How many did not register for either a math course or an English
course?
(d) How many read all three?
(e) How man do not read the Journal?
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MAT 141 ‐ Chapter 7
Venn Diagrams
Example 7.2.10: A survey of 180 college men was taken to determine
participation in various campus activities. Forty-three students were in
fraternities, 52 participated in campus sports, and 35 participated in
various campus tutorial programs. Thirteen students participated in
fraternities and sports, 14 in sports and tutorial programs, and 12 in
fraternities and tutorial programs. Five students participated in all
three activities. Create the Venn Diagram for this scenario.
Introduction to Probability
Definitions
Theoretical Method for Equally Likely Outcomes
• An experiment is an activity or occurrence with an
observable result.
Step 1:
Count the total number of possible
outcomes.
• Outcomes are the most basic possible results of
observations or experiments.
Step 2:
Among all the possible outcomes, count
the number of ways the event of interest,
E, can occur.
• The set of all possible outcomes of an experiment is
called the sample space.
Step 3:
Determine the probability,
• An event consists of one or more outcomes that share a
property of interest. Or think of an event as a subset of
the sample space.
.
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MAT 141 ‐ Chapter 7
Outcomes and Events
Expressing Probability
The probability of an event,
expressed as
, is
always between 0 and 1
(inclusive).
0
Certain
1
Likely
1
0.5
A probability of 0 means the
event is impossible and a
probability of 1 means the
event is certain.
50-50 Chance
Unlikely
0
Impossible
Listing Outcomes (Sample Space)
• To help list all of the outcomes use charts and tables.
▫ In the last example we used a chart.
▫ You could have also used a tree diagram as shown below.
Of the 16 possible
outcomes, 6 have the event
two girls and two boys.
2
6/16
0.357
Example 7.3.1: Assuming equal chance of having a boy or
girl at birth, what is the probability of having two girls and
two boys in a family of four children?
Scenarios Possible
Combinations
All 4 Girls {GGGG}
3 Girls
{GGGB}, {GGBG},
and 1 Boy {GBGG}, {BGGG}
{GGBB}, {GBGB},
2 Girls
{GBBG}, {BGBG},
and 2
{BBGG}, {BGGB}
Boys
1 Girl and {GBBB}, {BGBB},
3 Boys
{BBGB}, {BBBG}
All 4 Boys {BBBB}
Of the 16 possible
outcomes, 6 have the event
two girls and two boys.
2
6/16
0.357
Theoretical Method for Equally Likely Outcomes
Step 1:
Count the total number of possible
outcomes,
.
Step 2:
Among all the possible outcomes, count
the number of ways the event of interest,
E, can occur,
.
Step 3:
Determine the probability,
.
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MAT 141 ‐ Chapter 7
Sets
Sets
Example 7.3.2: Consider the following.
Example 7.3.3: Consider the following.
Experiment: You roll a fair six-sided die.
Experiment: Flip two coins.
Outcomes: The six different numbers on the die.
Outcomes: Heads or tails on each coin.
Sample Space: 1, 2, 3,4,5,
Sample Space:
6 , n S
6
Events:
You roll a 1
,
,
,
,n S
4
Events:
1,
1
You roll an even number
You roll a number great than 2
4
,
One head
2, 4, 6 ,
3
Events and Sets
• Since events are sets, we can use set operations to find
unions, intersections, and complements of events.
1
,
Both tails
3, 4, 5, 6 ,
2
,
,
At least one tail
,
3
,
Sets
Example 7.3.4: Consider the following.
Experiment: You roll a fair six-sided die.
Outcomes: The six different numbers on the die.
Sample Space: 1, 2, 3,4,5,
6 , n S
6
Events:
1,
You roll a 1
1
You roll an even number
You roll a number great than 2
,
2, 4, 6 ,
3, 4, 5, 6 ,
3
4
,
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MAT 141 ‐ Chapter 7
Standard Deck of Cards
• We will use a standard deck of cards for several
examples as it allows to do a number of different types
of questions. So become familiar with a deck of cards if
you are not already.
Probability of an Event
Example 7.3.5: Calculate each of the following
probabilities.
(a) Rolling a die and getting an even.
(b) Drawing a card from a standard deck of cards and getting a face
card.
(c) Drawing a card from a standard deck of cards and not getting a
face card.
Probability of an Event
Example 7.3.6: One jar contains balls numbered 1, 2, 3, and 4. A
second jar contains 3 balls numbered 1, 2, and 3. An experiment
consists of taking one ball form the first jar, and then taking a ball
from the second.
Example 7.3.6 Continued
(c) Write the event that the sum of the numbers on the two balls is
five as a set.
(a) Write out the sample space
(b) Write the event of the number on the first ball being even as a set.
(d) What is the probability that the number on the first ball being
even ?
(e) What is the probability that the sum of the numbers on the two
balls is five?
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MAT 141 ‐ Chapter 7
Probability of an Event
Example 7.3.7: There were 2,447,864 U.S. deaths in 2002. They are
listed according to cause in the following table. If a randomly selected
person died in 2002, use the information to find the following
probabilities.
(a) The probability that the cause of
death was heart disease.
Basic Concepts of Probability
(b) The probability that the probability of death was cancer or heart disease.
Union Rule for Probability
Recall the union rule for sets
∪
Probability of an Event
∩
Example 7.4.1: Calculate each of the following
probabilities.
(a) Rolling a die and getting an even or a 3.
If we divide each side by
∪
Which then becomes
∪
we have
∩
∩
∪3
3
∩3
(b) Drawing a card from a standard deck of cards and getting a face
card or a spade.
∪
∩
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MAT 141 ‐ Chapter 7
Probability and Venn Diagrams
Probability of an Event
Example 7.4.2: A study is taken where 55% of respondents
were female, 45% of the respondents agree with Bill 145A, and 20% were female who agreed with the bill. Fill in
the appropriate amounts in the Venn diagram.
You can take any Venn diagram and turn the amounts
(values) into probabilities.
(a) What percent were male?
(b) What percent were female and
disagreed?
(c) What percent were male and
agreed?
Probability of an Event
Probability of an Event
Example 7.4.3: A study is taken where 80% of
respondents were football fans, 15% of the respondents
were hockey fans, and 92% liked football or hockey. Fill in
the appropriate amounts in the Venn diagram.
∪
∩
OR
∩
∪
Example 7.4.4: A study is taken where 55% of respondents
were female, and 65% of the respondents agree with
raising taxes. Fill in the appropriate amounts in the Venn
diagram.
It turns out that you cannot
complete the chart, yet.
We will soon learn how to handle
this scenario.
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MAT 141 ‐ Chapter 7
Complement of a Set
• If the probability of something happening is 20%, then
the probability of it not happening is 80%.
▫ In other words, 20%+80%=100%
• Recall that we signify not being in a set as ′.
• So something not happening is signified by ′.
Probability of an Event
Example 7.4.5: A survey of people living in Downton was
taken. Let W represent the set of people who have a White
parent (biologically) and L represent the set of people
with a Latino parent (biologically).
(a) What is the probability that someone has
a white parent but not a Latino parent?
(b) What is the probability that some has a
parent that is not white?
(c) What does the middle region represent?
Odds
Calculating Odds
• Odds are the ratio of the probability that a particular
event will occur to the probability that it will not occur.
▫ Odds for an event E.

, OR
, OR
▫ Note that
′
(a) Rolling a die and the odds of getting an even.
(b) Drawing a card from a standard deck of cards and the odds
against getting a king.
▫ Odds against an event E.

Example 7.4.6: Calculate each of the following odds.
1
(c) Drawing a card from a standard deck of cards and odds of getting
a face card.
16
MAT 141 ‐ Chapter 7
Calculating Odds
Calculating Odds
Example 7.4.7: Calculate each of the following odds.
Example 7.4.8: Calculate each of the following.
(a) The probability of a player making a 3-point shot are 45%. What
are the odds in favor of hitting a 3-pointer?
(a) The odds of losing a game if you play perfectly are 7:1. What is the
probability of winning?
(b) When drawing three cards, the probability that they are all Aces is
0.02%. What are the odds against drawing 3 aces?
(b) A horse at the track is given 9:1 odds of winning. What is the
associated probability of winning?
Probability Distributions
A probability distribution represents the
probabilities of all possible events.
All possible outcomes and a probability distribution for the
sum when two dice are rolled are shown below.
Probability Distributions
• When we collect data it is useful to turn it into a
probability distribution so that we can make predictions.
• In the example below shows the U.S. advertising volume
in millions of dollars by medium in 2004.
These probabilities
should theoretically
and to 1.0000 but
this does not always
occur due to
rounding.
This is acceptable as
we have rounded
out far enough for a
high degree of
accuracy.
Total = $263,766
Total = .9999
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MAT 141 ‐ Chapter 7
Empirical Probability
Example 7.4.9: The following example is from page 386 in
your book.
Conditional Probability; Independent Events
Conditional Probability
• We often have statistics such as
▫ Probability of getting in a car accident during the course of a year.
▫ Probability of completion of college.
▫ Probability of treating cancer.
• These probabilities are good to know but are very broad
in scope. What if we wanted a more detailed analysis?
▫ Probability of getting in a car accident, given you are a teenager.
▫ Probability of completion of college, given you are in a technical
program.
▫ Probability of treating cancer, given it is malignant.
Conditional Probability
• The table below shows the results of 100 surveyed brokers for an
investment firm.
• Of those surveyed the probability that they picked stocks that went up
are
60
.60 60%
100
• Of those surveyed the probability that those who used research picked
stocks that went up are (i.e. picked stock that went up given they used
research)
30
.6667 66.67%
45
• Adding the condition that they used research narrowed the sample
space down to 45 people.
• The “given” part is an added condition and is what we
mean when we say conditional probability.
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MAT 141 ‐ Chapter 7
Conditional Probability
Conditional Probability
Example 7.5.1: You flip a coin three times. What is the
probability that you get all heads, given the first toss was
tails?
TTT
TTH
THT
HTT
HHH
HHT
HTH
THH
Conditional Probability
Example 7.5.2: You toss two die. What is the probability
that the sum is a greater than 7, given one die was a 4?
Conditional Probability
Example 7.5.3: Calculate each of the following.
(a)
(b)
′
(c)
′ ′
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MAT 141 ‐ Chapter 7
Conditional Probability
Conditional Probability
Example 7.5.4: At a insurance company 80% of companies
have home owners insurance and 45% own fire insurance.
It is also known that 36% have both. What is the
probability that someone owns fire insurance, given that
they own homeowners insurance?
AND Probability
Example 7.5.5: At a insurance company 80% of companies
have home owners insurance and 45% own fire insurance.
It is also known that 36% have both. What is the
probability that someone owns fire insurance, given that
they do not own homeowners insurance?
Conditional Probability
Example 7.5.6: You flip a coin three times. What is the
probability that you get one Head and the first toss was a
Tails?
TTT
TTH
THT
HTT
HHH
HHT
HTH
THH
∩
∩
|
20
MAT 141 ‐ Chapter 7
Conditional Probability
Example 7.5.7: You toss two die. What is the probability
that the sum is a greater than 7 and one die was a 4?
Conditional Probability
Example 7.5.9: A bag of marbles contains 3 red, 5 blue, 7
yellow, and 5 green marbles. What is the probability of
drawing 3 marbles and the are yellow, green, and blue (in
that order)?
Conditional Probability
Example 7.5.8: What is the probability of drawing two
cards from a standard deck of cards and getting two jacks?
And Probability: Independent Events
• Two events are independent if the outcome of one does
not affect the probability of the other event.
∙
• Two events are dependent if the outcome of one affects
the probability of the other event.
∙
|
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MAT 141 ‐ Chapter 7
AND Probability
Disjoint vs. Independent
• Disjoint (mutually exclusive) and Independent events
are not the same thing.
▫ Mammals and Reptiles are mutually exclusive events (no
intersection)
Example 7.5.10: A zoo has 30 reptiles and 170 mammals.
You randomly select two animals to be on the zoo poster.
What is the probability that the first choice was a mammal
and the second choice was a reptile?
Conditional Probability (Independent)
Example 7.5.11: You flip a coin three times. What is the
probability that you get all tails?
TTT
TTH
THT
HTT
HHH
HHT
HTH
THH
Independent Events
Example 7.5.12: It is found that the probability of a
hurricane hitting Florida any given year is 14% (thus the
probability of not getting hit is 86%). What is the
probability of getting hit by a hurricane two years in a
row?
22
MAT 141 ‐ Chapter 7
Probability of an Event (Independent)
Example 7.5.13: A study is taken where 55% of
respondents were female, and 65% of the respondents
agree with raising taxes. Fill in the appropriate amounts
in the Venn diagram.
Recall that we looked at this
example in Section 4 and could not
tackle it at the time.
Baye’s Theorem
• Suppose we have two events E and F. Then the given
probability can be calculated using Baye’s Theorem.
Bayes' Theorem
Bayes' Theorem
Example 7.6.1: For a fixed length of time, the probability of a worker
error, event , on a certain production line is 0.1, the probability that
an accident, event , will occur when there is a worker error is 0.3,
and the probability that an accident will not occur when there is a
worker error is 0.2. Find the probability of a worker error if there is
an accident.
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MAT 141 ‐ Chapter 7
Bayes' Theorem
Example 7.6.2 Cont.
Example 7.6.2: For a given math class, the probability of a student
getting over 90% on a test is 0.1, the probability that a student will get
above 90% on homework when they get above 90% on their test is
0.92, and the probability that a student will get above 90% on
homework when they do net get above 90% on their test is 0.15. Find
the probability of a student a getting above 90% on a test given they
did not get above 90% on their homework.
Medical Purpose
• Bayes' theorem is largely used for medical and scientific
studies.
• Tests are not the event. We have a cancer test, separate from
the event of actually having cancer.
• Tests are flawed. Tests detect things that don’t exist (false
positive), and miss things that do exist (false negative).
Bayes' Theorem
Example 7.6.3: Suppose you have cancer screening test which yields
the following results.
• 1% of women have breast cancer (and therefore 99% do not).
• 80% of mammograms detect breast cancer when it is there (and
therefore 20% miss it).
• 9.6% of mammograms detect breast cancer when it’s not there
(and therefore 90.4% correctly return a negative result).
• Put in a table, the probabilities look like this:
• Tests give us test probabilities, not the real probabilities.
People often consider the test results directly, without
considering the errors in the tests.
• Bayes’ theorem gives you the actual probability of an
event given the measured test probabilities.
Suppose you get a positive test result, what is
the probability that you actually have cancer?
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MAT 141 ‐ Chapter 7
Cancer Test
What does this mean?
Let be the event that you have a positive test and let be the event
you have cancer. Then the question, “What is the probability that you
actually have cancer (give you had a positive test)?” becomes
|
| ′
• So, our chance of cancer, given the test was positive, is about
7.8%.
• Interesting — a positive mammogram only means you have a
7.8% chance of cancer, rather than 80% (the supposed
accuracy of the test). It might seem strange at first but it
makes sense: the test gives a false positive 10% of the time, so
there will be a ton of false positives in any given population.
There will be so many false positives, in fact, that most of the
positive test results will be wrong.
• If you take 100 people, only 1 person (1%) will have cancer.
Another 10 (~9.6%) will not have cancer but will get a false
positive result. Getting a positive result means you only have a
roughly 1/11 chance of being the person who really has cancer
(7.8% to be exact).
Other Uses
Bayes' Theorem
One clever application of Bayes’ Theorem is in spam filtering. We have
Event A: The message is spam.
Test X: The message contains certain words (X)
• Plugged into a more readable formula:
|
• We have been looking at two sets and their compliments (e.g.
Cancer/No Cancer and Positive/Not positive). However, we can
analyze more complicated situations with the general form of Baye’s
Theorem.
• Bayesian filtering allows us to predict the chance a message is really spam given
the “test results” (the presence of certain words). Clearly, words like “viagra”
have a higher chance of appearing in spam messages than in normal ones.
• Spam filtering based on a blacklist is flawed — it’s too restrictive and false
positives are too great. But Bayesian filtering gives us a middle ground — we use
probabilities. As we analyze the words in a message, we can compute the chance
it is spam (rather than making a yes/no decision). If a message has a 99.9%
chance of being spam, it probably is. As the filter gets trained with more and
more messages, it updates the probabilities that certain words lead to spam
messages. Advanced Bayesian filters can examine multiple words in a row, as
another data point.
For example, we could ask the probability of a certain Age, given a person
is in a car accident, event . Here “Age” can be broken down into many
categories such as
:
16 20
:
21 30
:
31 65
:
66 99
25
MAT 141 ‐ Chapter 7
Bayes’s Theorem
Example 7.6.4: An auto insurance company insures driers of all ages.
An actuary compiles the following statistics on the companies insured
drivers. A randomly selected driver that the company insures has an
accident, what is the probability that they are 21-30 years old?
Bayes’s Theorem
Example 7.6.5: A bank finds that the relationship between mortgage
defaults and the size of the down payment is given in the following
table.
(a)
If a default occurs, what is the probability that is on a mortgage with a 5% down
payment?
(b) If a default occurs, what is the probability that is on a mortgage with a 10% down
payment?
(c)
If a default occurs, what is the probability that is on a mortgage with a 20% down
payment?
(d) If a default occurs, what is the probability that is on a mortgage with a 25% down
payment?
26