Sets and Probability

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Sets and Probability

• Chapter 8

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Ch. 8 Sets and Probabilities

• 8.3 Introduction to Probability
• 8.4 Basic Concepts of Probability
• 8.5 Conditional Probability Independent Events
• 8.6 Bayes Theorem

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8.3 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 any process that generates
  well-defined outcomes.
• The sample space for an experiment is the set of
  all experimental outcomes.
• A sample point is an element of the sample space,
  any one particular experimental outcome.

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Example Bradley Investments

• Bradley has invested in two stocks, Markley Oil
  and
• Collins Mining. Bradley has determined that the
• possible outcomes of these investments three
  months
• from now are as follows.
• Investment Gain or Loss
• in 3 Months (in 000)
• Markley Oil Collins Mining
• 10 8
• 5 -2
• 0
• -20

Sample Point
Sample Space
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Assigning Probabilities

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

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Classical Method

• If an experiment has n possible outcomes, this
  method
• would assign a probability of 1/n to each
  outcome.
• Example
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance
• of occurring.

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Relative Frequency Method

• Example Lucas Tool Rental
• Lucas would like to assign probabilities to the
• number of floor polishers it rents per day.
  Office
• records show the following frequencies of daily
  rentals
• for the last 40 days.
• Number of Number
• Polishers Rented of Days
• 0 4
• 1 6
• 2 18
• 3 10
• 4 2

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Relative Frequency Method

• Example Lucas Tool Rental
• The probability assignments are given by
  dividing
• the number-of-days frequencies by the total
  frequency
• (total number of days).
• Number of Number
• Polishers Rented of Days Probability
• 0 4 .10 4/40
• 1 6 .15 6/40
• 2 18 .45 etc.
• 3 10 .25
• 4 2 .05
• 40 1.00

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Subjective Method

• When economic conditions and a companys
  circumstances change rapidly it might be
  inappropriate to assign probabilities based
  solely on historical data.
• We can use any data available as well as our
  experience and intuition, but ultimately a
  probability value should express our degree of
  belief that the experimental outcome will occur.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

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Example Bradley Investments

• Applying the subjective method an analyst
• made the following probability assignments.
• Exper. Outcome (Markley, Collins)
  Net Gain/Loss Probability
• ( 10, 8) 18,000 Gain
  .20
• ( 10, -2) 8,000 Gain
  .08
• ( 5, 8) 13,000 Gain
  .16
• ( 5, -2) 3,000 Gain
  .26
• ( 0, 8) 8,000 Gain
  .10
• ( 0, -2) 2,000 Loss
  .12
• (-20, 8) 12,000 Loss
  .02
• (-20, -2) 22,000 Loss
  .06

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Events and Their Probability

• An event is a collection of sample points.
• The probability of any event is equal to the sum
  of the probabilities of the sample points in the
  event.

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Example Bradley Investments

• Events and Their Probabilities
• Event M Markley Oil Profitable
• M (10, 8), (10, -2), (5, 8), (5,
  -2)
• P(M) P(10, 8) P(10, -2) P(5, 8)
  P(5, -2)
• .2 .08 .16 .26
• .70
• Event C Collins Mining Profitable
• C (10, 8), (5, 8), (0, 8), (-20,
  8)
• P(C) .48 (found using the same logic)

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8.4 Basic Concepts of Probability

• Complement of an Event
• Union of Two Events
• Intersection of Two Events
• Mutually Exclusive Events

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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 Ac.
• The Venn diagram below illustrates the concept of
  a complement.

Sample Space S
Event A
Ac
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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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Example Bradley Investments

• Union of Two Events
• Event M Markley Oil Profitable
• Event C Collins Mining Profitable
• M ??C Markley Oil Profitable
• or Collins Mining Profitable
• M ??C (10, 8), (10, -2), (5, 8), (5, -2),
  (0, 8), (-20, 8)
• P(M ??C) P(10, 8) P(10, -2) P(5, 8) P(5,
  -2)
• P(0, 8) P(-20, 8)
• .20 .08 .16 .26 .10 .02
• .82

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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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Example Bradley Investments

• Intersection of Two Events
• Event M Markley Oil Profitable
• Event C Collins Mining Profitable
• M ??C Markley Oil Profitable
• and Collins Mining Profitable
• M ??C (10, 8), (5, 8)
• P(M ??C) P(10, 8) P(5, 8)
• .20 .16
• .36

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Addition Law

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

Event A
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Addition Law

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

Event B
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Addition Law

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

Event A
Event B
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Example Bradley Investments

• Addition Law
• Markley Oil or Collins Mining Profitable
• We know P(M) .70, P(C) .48, P(M ??C)
  .36
• Thus P(M ? C) P(M) P(C) - P(M ? C)
• .70 .48 - .36
• .82
• This result is the same as that obtained
  earlier using
• the definition of the probability of an event.

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Addition Law 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 Law for Mutually Exclusive Events
• P(A ??B) P(A) P(B)

Sample Space S
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
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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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8.5 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 Bradley Investments

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

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Example Bradley Investments

• Conditional Probability
• Collins Mining Profitable given Markley Oil
  Profitable

P(C ? M) ? .36 P(M) .70
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Multiplication Law

• The multiplication law 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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Example Bradley Investments

• Multiplication Law
• Markley Oil and Collins Mining Profitable
• We know P(M) .70, P(CM) .51
• Thus P(M ? C) P(M) P(CM)
• (.70)(.51)
• .36
• This result is the same as that obtained
  earlier using
• the definition of the probability of an event.

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Multiplication Law 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 multiplication law also can be used as a test
  to see if two events are independent.

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Example Bradley Investments

• Multiplication Law for Independent Events
• Are M and C independent?
• ????????? Does?P(M ? C) P(M)P(C) ?
• We know P(M ? C) .36, P(M) .70,
  P(C) .48
• But P(M)P(C) (.70)(.48) .34
• .34???????so?M and C are not independent.
• Also,
• P(C) .48,
• P(CM) .51

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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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8.6 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 8