chapter2 introduction to probability

8/12/2019 CHAPTER2 Introduction to Probability

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Slide 2006 Thomson South-Western. All Rights Reserved

Chapter 2Introduction to Probability

Experiments and the Sample Space Assigning Probabilities to

Experimental Outcomes

Events and Their Probability

Some Basic Relationshipsof Probability

Bayes Theorem


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Probability as a Numerical Measureof the Likelihood of Occurrence

0 1.5

Increasing Likelihood of Occurrence

Probability:

The event

is very

unlikelyto occur.

The occurrence

of the event is

just as likely asit is unlikely.

The event

is almost

certainto occur.


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3


Assigning Probabilities

Classical Method

Relative Frequency Method

Subjective Method

Assigning probabilities based on the assumptionof equally likely outcomes

Assigning probabilities based on experimentationor historical data

Assigning probabilities based on judgment


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Events and Their Probabilities

An experimentis any process that generateswell-defined outcomes.

The sample space for an experiment is the set ofall sample points.

An experimental outcome is also called a sample

point.

An event is a collection of particular sample points.


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Classical Method

If an experiment has npossible outcomes, this method

would assign a probability of 1/nto each outcome.

Experiment: Rolling a dieSample Space: S= {1, 2, 3, 4, 5, 6}

Probabilities: Each sample point has a

1/6 chance of occurring

Example


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Example: Lucas Tool Rental


Lucas Tool Rental would like to

assign probabilities to the number of car

polishers it rents each day. Office records show the

following frequencies of daily rentals for the last40 days.

Number ofPolishers Rented

Numberof Days

0

1234

4

618102


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Each probability assignment is given by

dividing the frequency (number of days) bythe total frequency (total number of days).


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Probability

Number of

Polishers Rented

Number

of Days012

34

46

18

10240

.10

.15

.45

.25.051.00



Subjective Method

When economic conditions and a companys

circumstances change rapidly it might beinappropriate to assign probabilities based solely onhistorical data.

We can use any data available as well as our

experience and intuition, but ultimately a probabilityvalue should express our degree of belief that theexperimental outcome will occur.

The best probability estimates often are obtained by

combining the estimates from the classical or relativefrequency approach with the subjective estimate.



Example: Bradley Investments

Bradley has invested in two stocks, Markley Oil and

Collins Mining. Bradley has determined that thepossible outcomes of these investments three months

from now are as follows.

Investment Gain or Lossin 3 Months (in $000)

Markley Oil Collins Mining

10

50

-20

8

-2



Applying the subjective method, an analyst

made the following probability assignments.

Exper. Outcome Net Gain orLoss Probability

(10, 8)

(10, -2)(5, 8)

(5, -2)

(0, 8)

(0, -2)(-20, 8)

(-20, -2)

$18,000 Gain

$8,000 Gain$13,000 Gain

$3,000 Gain

$8,000 Gain

$2,000 Loss$12,000 Loss

$22,000 Loss

.20

.08

.16

.26

.10

.12

.02

.06

Example: Bradley Investments




EventM= Markley Oil ProfitableM= {(10, 8), (10, -2), (5, 8), (5, -2)}

P(M) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2)

= .20 + .08 + .16 + .26

= .70




Event C= Collins Mining ProfitableC= {(10, 8), (5, 8), (0, 8), (-20, 8)}

P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(-20, 8)

= .20 + .16 + .10 + .02

= .48



Some Basic Relationships of Probability

There are some basic probability relationships that

can be used to compute the probability of an eventwithout knowledge of all the sample point probabilities.

Complement of an Event

Intersection of Two Events

Mutually Exclusive Events

Union of Two Events


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14Slide 2006 Thomson South-Western. All Rights Reserved

The complement ofAis denoted byAc.

The complement of eventA is defined to be the eventconsisting of all sample points that are not inA.

Complement of an Event

EventA AcSampleSpace S

VennDiagram


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The union of eventsAand Bis denoted byA B

The union of eventsAand Bis the event containingall sample points that are inA orB or both.

Union of Two Events

SampleSpace SEventA Event B


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Union of Two Events

EventM= Markley Oil ProfitableEvent 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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The intersection of eventsAand Bis denoted byA

The intersection of eventsAand Bis the set of allsample points that are in bothA and B.

SampleSpace SEventA Event B


Intersection ofAand B


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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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The addition law provides a way to compute theprobability of eventA,or B,or bothAand B occurring.

Addition Law

The law is written as:

P(A B) = P(A) + P(B) -P(AB


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M C= Markley Oil Profitable

or Collins Mining Profitable

We know: P(M) = .70, P(C) = .48, P(M C) = .36

Thus: P(MC) = P(M) + P(C) -P(MC)

= .70 + .48 -.36

= .82

Addition Law

(This result is the same as that obtained earlier

using the definition of the probability of an event.)


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Two events are said to be mutually exclusive if theevents have no sample points in common.

Two events are mutually exclusive if, when one eventoccurs, the other cannot occur.

SampleSpace S

EventA Event B


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If eventsAand Bare mutually exclusive, P(AB= 0.

The addition law for mutually exclusive events is:

P(A B) = P(A) + P(B)

There is no need toinclude -P(AB


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The probability of an event given that another eventhas occurred is called a conditional probability.

A conditional probability is computed as follows :

The conditional probability ofAgiven Bis denoted

by P(A|B).

Conditional Probability

( )

( | ) ( )

P A BP A B

P B


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We know: P(M C) = .36, P(M) = .70

Thus:

Conditional Probability

( ) .36( | ) .5143

( ) .70

P C MP C M

P M

= Collins Mining Profitable

given Markley Oil Profitable

( | )P C M


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Multiplication Law

The multiplication law provides a way to compute theprobability of the intersection of two events.


P(A B) = P(B)P(A|B)


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We know: P(M) = .70, P(C|M) = .5143

Multiplication Law

M C = Markley Oil Profitable

and Collins Mining Profitable

Thus: P(MC) = P(M)P(M|C)

= (.70)(.5143)

= .36(This result is the same as that obtained earlier

using the definition of the probability of an event.)


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Independent Events

If the probability of eventAis not changed by theexistence of event B, we would say that eventsAand Bare independent.

Two eventsAand Bare independent if:

P(A|B) = P(A) P(B|A) = P(B)or

l l


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The multiplication law also can be used as a test to seeif two events are independent.


P(A B) = P(A)P(B)

Multiplication Lawfor Independent Events

M l i li i L


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Multiplication Lawfor Independent Events


We know: P(MC) = .36, P(M) = .70, P(C) = .48

But: P(M)P(C) = (.70)(.48) = .34, not .36

Are eventsMand Cindependent?

DoesP(MC) = P(M)P(C) ?

Hence: Mand Care not independent.


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Bayes Theorem

NewInformation

Applicationof BayesTheorem

PosteriorProbabilities

PriorProbabilities

Often we begin probability analysis with initial or

prior probabilities.

Then, from a sample, special report, or a producttest we obtain some additional information.

Given this information, we calculate revised orposterior probabilities.

Bayes theoremprovides the means for revising theprior probabilities.


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A proposed shopping center

will provide strong competitionfor downtown businesses like

L. S. Clothiers. If the shopping

center is built, the owner of

L. S. Clothiers feels it would be bestto relocate to the center.

The shopping center cannot be built unless a

zoning change is approved by the town council. The

planning board must first make a recommendation, foror against the zoning change, to the council.

Example: L. S. Clothiers


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Prior ProbabilitiesLet:

Bayes Theorem

A1= town council approves the zoning change

A2= town council disapproves the change

P(A1) = .7, P(A2) = .3

Using subjective judgment:


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New Information

The planning board has recommended against thezoning change. Let Bdenote the event of a negativerecommendation by the planning board.

Given that Bhas occurred, should L. S. Clothiers

revise the probabilities that the town council willapprove or disapprove the zoning change?

Bayes Theorem


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Conditional Probabilities

Past history with the planning board and thetown council indicates the following:

Bayes Theorem

P(B|A1) = .2 P(B|A2) = .9

P(BC|A1) = .8 P(BC|A2) = .1Hence:


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P(Bc|A1) = .8P(A1) = .7

P(A2) = .3

P(B|A2) = .9

P(Bc|A2) = .1

P(B|A1) = .2 P(A1B) = .14

P(A2B) = .27

P(A2Bc) = .03

P(A1Bc) = .56

Bayes Theorem

Tree Diagram

Town Council Planning Board ExperimentalOutcomes


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Bayes Theorem

1 1 2 2

( ) ( | )( | )

( ) ( | ) ( ) ( | ) ... ( ) ( | )

i i

i

n n

P A P B AP A B

P A P B A P A P B A P A P B A

To find the posterior probability that eventAiwill

occur given that eventB has occurred, we applyBayes theorem.

Bayes theorem is applicable when the events forwhich we want to compute posterior probabilitiesare mutually exclusive and their union is the entiresample space.


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Posterior Probabilities

Given the planning boards recommendation notto approve the zoning change, we revise the priorprobabilities as follows:

1 11

1 1 2 2( ) ( | )( | ) ( ) ( | ) ( ) ( | )

P A P B AP A B

P A P B A P A P B A

(. )(. )

(. )(. ) (. )(. )

7 2

7 2 3 9

Bayes Theorem

= .34


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Conclusion

The planning boards recommendation is goodnews for L. S. Clothiers. The posterior probability ofthe town council approving the zoning change is .34compared to a prior probability of .70.

Bayes Theorem


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Tabular Approach

Step 1

Prepare the following three columns:

Column 1 - The mutually exclusive events for whichposterior probabilities are desired.

Column 2-

The prior probabilities for the events.Column 3 - The conditional probabilities of the newinformationgiveneach event.

b l h


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Tabular Approach

(1) (2) (3) (4) (5)

Events

Ai

Prior

Probabilities

P(Ai)

Conditional

Probabilities

P(B|Ai)

A1

A2

.7

.3

1.0

.2

.9

b l A h


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Tabular Approach

Step 2

Column 4

Compute the joint probabilities for each event andthe new information Bby using the multiplicationlaw.

Multiply the prior probabilities in column 2 by thecorresponding conditional probabilities in column 3.That is, P(AiIB) = P(Ai) P(B|Ai).

T b l A h


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Tabular Approach

(1) (2) (3) (4) (5)

Events

Ai

Prior

Probabilities

P(Ai)

Conditional

Probabilities

P(B|Ai)

A1

A2

.7

.3

1.0

.2

.9

.14

.27

Joint

Probabilities

P(AiIB)

.7 x .2

T b l A h


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Tabular Approach

Step 2 (continued)

We see that there is a .14 probability of the towncouncil approving the zoning change and a negativerecommendation by the planning board.

There is a .27 probability of the town council

disapproving the zoning change and a negativerecommendation by the planning board.

T b l A h


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Tabular Approach

Step 3

Column 4Sum the joint probabilities. The sum is the

probability of the new information, P(B). The sum

.14 + .27 shows an overall probability of .41 of a

negative recommendation by the planning board.

T b l A h


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Tabular Approach

(1) (2) (3) (4) (5)

Events

Ai

Prior

Probabilities

P(Ai)

Conditional

Probabilities

P(B|Ai)

A1

A2

.7

.3

1.0

.2

.9

.14

.27

Joint

Probabilities

P(Ai

IB)

P(B) = .41

T b l A h


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Step 4

Column 5Compute the posterior probabilities using the basic

relationship of conditional probability.

The joint probabilities P(AiIB) are in column 4 and

the probability P(B) is the sum of column 4.

Tabular Approach

)()()|(

BP

BAPBAP i

i

T b l A h


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(1) (2) (3) (4) (5)

Events

Ai

Prior

Probabilities

P(Ai)

Conditional

Probabilities

P(B|Ai)

A1

A2

.7

.3

1.0

.2

.9

.14

.27

Joint

Probabilities

P(Ai

IB)

P(B) = .41

Tabular Approach

.14/.41

Posterior

Probabilities

P(Ai|B)

.3415

.6585

1.0000

Using Excel to Compute


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Formula Worksheet

A B C D E

2 A1 0.7 0.2 =B2*C2 =D2/$D$4

3 A2 0.3 0.9 =B3*C3 =D3/$D$4

4 =SUM(B2:B3) =SUM(D2:D3) =SUM(E2:E3)

5

Posterior

Probabilities1 Events

Prior

Probabilities

Conditional

Probabilities

Joint

Probabilities

Using Excel to ComputePosterior Probabilities

Using Excel to Compute


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Value Worksheet

A B C D E

2 A1 0.7 0.2 0.14 0.3415

3 A2 0.3 0.9 0.27 0.6585

4 1.0 0.41 1.0000

5

Posterior

Probabilities1 Events

Prior

Probabilities

Conditional

Probabilities

Joint

Probabilities

Using Excel to ComputePosterior Probabilities

E d f Ch t 2


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End of Chapter 2

chapter2 introduction to probability

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