Sets and Probability

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Sets and Probability

• Chapter 7

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Ch. 7 Sets and Probabilities

• 7.1 Sets
• 7.2 Applications of Venn Diagrams
• 7.3 Introduction to Probability
• 7.4 Basic Concepts of Probability
• 7.5 Conditional Probability Independent Events
• 7.6 Bayes Theorem

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7.1 Sets

• Set A well defined collection of objects.
• The set consisting of all students in the class.
• The set consisting of all the coins produced by
  the U.S. government.
• The set consisting of the numbers 3, 4, and 5
• 3, 4, 5
• 4 is an element of the set 3, 4, 5
• 4?3, 4, 5
• 8?3, 4, 5
• n(A) the number of elements in finite set A
• A 3, 4, 5, n(A) 3

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7.1 Sets

• Two sets are equal if they contain the same
  elements
• 3, 4, 5 5, 4, 3 4, 3, 5 3, 3, 4, 5,
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• Subsets
• Set A is a subset of set B (A ? B) if every
  element of A is also an element of B.
• Set A is a proper subset (A ? B) if A ? B and A ?
  B.
• A (3, 4, 5, 6, B 2, 3, 4, 5, 6, 7, 8)
• A ? B
• The Universal Set (U)
• The set that contains all objects under
  consideration.

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Relationships Among Sets Venn Diagrams

• (A ? B)

U
B
A
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Relationships Among Sets Venn Diagrams

• Complement of a set
• Set A contains the elements of U that are not in
  A

U
A'
A
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Relationships Among Sets Venn Diagrams

• Intersection of Two Sets
• A ? B

U
B
A
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Relationships Among Sets Venn Diagrams

• Union of Two Sets
• A ? B

U
B
A
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Relationships Among Sets Venn Diagrams

• Disjoint Sets
• A ? B 0

U
B
A
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7.2 Applications of Venn Diagrams

• One set leads to 2 regions

U
2
A
1
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7.2 Applications of Venn Diagrams

• One set leads to 2 regions
• Region 1 A
• Region 2 A

U
2
A
1
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Applications of Venn Diagrams

• Two sets leads to 3 regions

U
A
B
2
3
1
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Applications of Venn Diagrams

• Two sets leads to 3 regions
• Or...

U
A
B
2
3
1
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Applications of Venn Diagrams

• Two sets leads to 3 regions
• Or...

U
A
3
3
2
B
1
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Applications of Venn Diagrams

• Two sets leads to 4 regions

U
A
B
3
4
2
1
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Applications of Venn Diagrams

• 3 sets, 8 regions

U
A
3
B
2
8
4
5
7
1
6
C
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Example

• 140 people in a suburban shopping center were
  interviewed to discover some of their cooking
  habbits.
• 58 use microwave ovens
• 63 use electric ranges
• 58 use gas ranges
• 19 use microwave ovens and electric ranges
• 17 use microwave ovens gas ranges
• 4 use both gas and electric ranges
• 1 uses all three
• 2 use none of the three

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Example
U
M
18
E
23
41
1
16
3
2
38
G
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7.3 Introduction to Probability

• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probability values are always assigned on a scale
  from 0 to 1.
• A probability near 0 indicates an event is very
  unlikely to occur.
• A probability near 1 indicates an event is almost
  certain to occur.
• A probability of 0.5 indicates the occurrence of
  the event is just as likely as it is unlikely.

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An Experiment and Its Sample Space

• An experiment is an activity or occurrence with
  an observable result.
• Each repetition of an experiment is called a
  trial.
• The sample space for an experiment is the set of
  all experimental outcomes.
• An event is a subset of the sample space.

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An Experiment and Its Sample Space

• An experiment consists of studying the number of
  boys and girls in families with exactly 3
  children.
• Let b represent boy and g represent girl.
• Event B the family has only boys. B b, b, b
• Event H the family has exactly two girls

Sample Space
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Set Operations for Events

• Let E and F be events for a sample space S.
• E ? F occurs when both E and F occur
• E ? F occurs when E or F or both occur
• E occurs when E does not occur.
• Mutually Exclusive Events
• E and F are mutually exclusive if E ? F Ø.

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Basic Probability Principle

• For sample spaces with equally likely outcomes,
  the probability of an event is defined as follows.

BASIC PROBABILITY PRINCIPLE Let S be a sample
space of equally likely outcomes, and let Event E
be a subset of S. Then the probability that E
occurs is
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Basic Probability Principle
If a single playing card is drawn at random from
a standard 52-card deck, find the probability of
each event

• Drawing an ace

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Basic Probability Principle
If a single playing card is drawn at random from
a standard 52-card deck, find the probability of
each event

• Drawing a spade
• Drawing a spade or a heart

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7.4 Basic Concepts of Probability

• Union (Addition) Rule for Probability
• Union Rule for Mutually Exclusive Events
• Complement Rule
• Odds

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Union of Two Events

• The union of events A and B is the event
  containing all sample points that are in A or B
  or both.
• The union is denoted by A ??B?
• The union of A and B is illustrated below.
• P(A ??B) The probability of the occurrence of
  Event A or Event B.

Sample Space S
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Addition Rule

• The addition rule provides a way to compute the
  probability of event A, or B, or both A and B
  occurring.
• The rule is written as
• P(A ??B) P(A) P(B) - P(A ? B?

Event A
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Addition Rule

• The addition rule provides a way to compute the
  probability of event A, or B, or both A and B
  occurring.
• The rule is written as
• P(A ??B) P(A) P(B) - P(A ? B?

Event B
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Addition Rule

• The addition rule provides a way to compute the
  probability of event A, or B, or both A and B
  occurring.
• The rule is written as
• P(A ??B) P(A) P(B) - P(A ? B?

Event A
Event B
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Union of Two Events

• ADDITION RULE FOR PROBABILITY
• For any events E and F from sample space S

Consider an assembly plant with 50 employees.
Each worker is expected to complete work
assignments on time and in such a way that the
assembled product will pass a final inspection.
On occasion, some of the workers fail to meet the
performance standards by completing work late
and/or assembling a defective product. Recently,
the manager found that 5 workers had completed
work late, 6 had assembled a defective product,
and 2 had both completed work late and assembled
a defective product.
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Union of Two Events

• Let
• L the event that the work is completed late
• D the event that the assembled product is
  defective

The manager decided to assign a poor performance
rating to any employee whose work was either late
or defective. What is the probability an employee
was given a poor performance rating?
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Addition Rule forMutually Exclusive Events

• Two events are said to be mutually exclusive if
  the events have no sample points in common. That
  is, two events are mutually exclusive if, when
  one event occurs, the other cannot occur.
• Addition Rule for Mutually Exclusive Events
• P(A ??B) P(A) P(B)

Event A
Event B
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Roll the Dice

• If you roll 2 dice, whats the probability of
  rolling a 7 or 11?

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Die 1
Die 2
P(7) 6/36 .167
P(11) 2/36 .056
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Roll the Dice

• If you roll 2 dice, whats the probability of
  rolling a 7 or 11?

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Complement of an Event

• The complement of event A is defined to be the
  event consisting of all sample points that are
  not in A.
• The complement of A is denoted by A.

Event A
A
COMPLEMENT RULE P(E) 1 P(E) and
P(E) 1 P(E)
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7.5 Conditional Probability

• The probability of an event given that another
  event has occurred is called a conditional
  probability.
• The conditional probability of A given B is
  denoted by P(AB).

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Conditional Probability
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Joint Probability Table
P(A ??M)
P(M)
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Joint Probability Table
Joint probabilities
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Joint Probability Table
Marginal probabilities
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Joint Probability Table
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Conditional Probability

• The probability of an event given that another
  event has occurred is called a conditional
  probability.
• The conditional probability of A given B is
  denoted by P(AB).
• A conditional probability is computed as follows
• If P(AB) 0, then event A and event B are
  mutually exclusive.

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Example Conditional Probability

• Conditional Probability
• Officer promoted given the officer is a man
• Officer promoted given the officer is a woman

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Intersection of Two Events

• The intersection of events A and B is the set of
  all sample points that are in both A and B.
• The intersection is denoted by A ????
• The intersection of A and B is the area of
  overlap in the illustration below.
• P(A ???) The probability of the occurrence of
  Event A and Event B.

Sample Space S
Intersection
Event A
Event B
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Product Rule

• The product rule provides a way to compute the
  probability of an intersection of two events.
• The law is written as

Event A
Event B
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Product Rule for Independent Events

• Events A and B are independent if P(AB) P(A).
• Multiplication Law for Independent Events
• P(A ? B) P(A)P(B)
• The product rule also can be used as a test to
  see if two events are independent.

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You are given the following information on Events
A, B,
C, and D.
È
P(A) .4
P(A
D) .6
ô
P(B) .2
P(A
B) .3
Ç
P(C) .1
P(A
C) .04
Ç
P(A
D) .03
b. Compute P(A ? B)
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7.5 Bayes Theorem

• The probability of an event A B is generally
  different from the probability of B A.
• However, there is a definite relationship between
  the two.
• Bayes' theorem is the statement of that
  relationship.
• Medical researchers know that the probability of
  getting lung cancer if a person smokes is .34.
• The probability that a nonsmoker will get lung
  cancer is .03.
• With Bayes theorem, we can calculate the
  probability that a person with lung cancer is (or
  was) a smoker.

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Bayes Theorem

• Prior Probabilities
• Let

S Person is a smoker N Person is a
non-smoker
According to the Center for Disease Control and
Prevention, approximately 22 of the population
18 years or older smokes tobacco products
regularly.
P(S) .22, P(N) .78
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Bayes Theorem

• Conditional Probabilities
• Let

C Person has (or will have) lung cancer H
Person will not have lung cancer
Based upon medical research
P(CS) .34
P(CN) .03
P(HS) .66
P(HN) .97
Hence
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Bayes Theorem

• We can illustrate the different possible outcomes
  with a tree diagram (2-step experiment).

Step 2 Health
Step 1 Smoker or non-smoker
Experimental Outcomes
C
(S, C)
S
(S, H)
H
C
(N, C)
N
(N, H)
H
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Bayes Theorem

• Now we can fill in the probabilities

Step 2 Health
Step 1 Smoker or non-smoker
Experimental Outcomes
P(CS) .34
P(S ? C) P(S)P(CS) .07
P(S) .22
P(S ? H) .15
P(HS) .66
P(CN) .03
P(N ? C) .02
P(N) .78
P(N ? H) .76
P(HN) .97
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Bayes Theorem

• Now suppose we want to determine the probability
  that a person who has been diagnosed with lung
  cancer is a smoker. In other words,
• From the law of conditional probabilities, we
  know that
• From the probability tree, we know that
• Event C can occur in only two ways (S ? C) and
  (N ? C)

Posterior probability
Equation 1
Equation 2
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Bayes Theorem
Bayes Theorem (2 events)
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Bayes Theorem

• To find the posterior probability that event
  Ai will
• occur given that event B has occurred, we
  apply
• Bayes theorem.

• Bayes theorem is applicable when the events
  for
• which we want to compute posterior
  probabilities
• are mutually exclusive and their union is
  the entire
• sample space.

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Bayes Theorem, example

• A local bank reviewed its credit card policy with
  the intention of recalling some of its credit
  cards. In the past, approximately 5 of
  cardholders defaulted, leaving the bank unable to
  collect the outstanding balance. Hence,
  management established a prior probability of .05
  that any particular cardholder will default. The
  bank also found that the probability of missing a
  monthly payment is .20 for customers who do not
  default. Of course, the probability of missing a
  monthly payment for those who default is 1.
• Given that a customer missed a monthly payment,
  compute the posterior probability that the
  customer will default.

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Bayes Theorem

• M missed payment
• D1 customer defaults
• D2 customer does not default
• P(D1) .05 P(D2) .95 P(MD2) .2
  P(MD1) 1

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End of Chapter 4