Chapter 4 Introduction to Probability

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© 2007 Thomson South-Western. All Rights Reserved© 2007 Thomson South-Western. All Rights Reserved

Chapter 4Chapter 4 Introduction to Probability Introduction to Probability

Experiments, Counting Rules, and Assigning Probabilities

Events and Their Probability

Some Basic Relationships of Probability

Conditional Probability

Bayes’ Theorem

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Probability as a Numerical MeasureProbability as a Numerical Measureof the Likelihood of Occurrenceof the Likelihood of Occurrence

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Increasing Likelihood of OccurrenceIncreasing Likelihood of Occurrence

ProbabilitProbability:y:

The eventThe eventis veryis veryunlikelyunlikelyto occur.to occur.

The eventThe eventis veryis veryunlikelyunlikelyto occur.to occur.

The occurrenceThe occurrenceof the event isof the event is

just as likely asjust as likely asit is unlikely.it is unlikely.

The occurrenceThe occurrenceof the event isof the event is

just as likely asjust as likely asit is unlikely.it is unlikely.

The eventThe eventis almostis almostcertaincertain

to occur.to occur.

The eventThe eventis almostis almostcertaincertain

to occur.to occur.

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An Experiment and Its Sample SpaceAn Experiment and Its Sample Space

An An experimentexperiment is any process that generatesis any process that generates well-defined outcomes (e.g. throwing a dice).well-defined outcomes (e.g. throwing a dice). An An experimentexperiment is any process that generatesis any process that generates well-defined outcomes (e.g. throwing a dice).well-defined outcomes (e.g. throwing a dice).

The The sample spacesample space for an experiment is the set of for an experiment is the set of all experimental outcomes (in the experiment of all experimental outcomes (in the experiment of throwing a dice, the sample space is {1,2,3,4,5,6} ).throwing a dice, the sample space is {1,2,3,4,5,6} ).

The The sample spacesample space for an experiment is the set of for an experiment is the set of all experimental outcomes (in the experiment of all experimental outcomes (in the experiment of throwing a dice, the sample space is {1,2,3,4,5,6} ).throwing a dice, the sample space is {1,2,3,4,5,6} ).

An experimental outcome is also called a An experimental outcome is also called a samplesample pointpoint (e.g. a sample point when you throw a dice (e.g. a sample point when you throw a diceis to get a 5).is to get a 5).

An experimental outcome is also called a An experimental outcome is also called a samplesample pointpoint (e.g. a sample point when you throw a dice (e.g. a sample point when you throw a diceis to get a 5).is to get a 5).

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A Counting Rule for A Counting Rule for Multiple-Step ExperimentsMultiple-Step Experiments

If an experiment consists of a sequence of If an experiment consists of a sequence of kk steps steps in which there are in which there are nn11 possible results for the first step, possible results for the first step,

nn22 possible results for the second step, and so on, possible results for the second step, and so on,

then the total number of experimental outcomes isthen the total number of experimental outcomes is given by (given by (nn11)()(nn22) . . . () . . . (nnkk).).

A helpful graphical representation of a multiple-stepA helpful graphical representation of a multiple-step

experiment is a experiment is a tree diagramtree diagram..

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Assigning ProbabilitiesAssigning Probabilities

Classical MethodClassical MethodClassical MethodClassical Method

Relative Frequency MethodRelative Frequency MethodRelative Frequency MethodRelative Frequency Method

Subjective MethodSubjective MethodSubjective MethodSubjective Method

Assigning probabilities based on the assumptionAssigning probabilities based on the assumption of of equally likely outcomesequally likely outcomes

Assigning probabilities based on Assigning probabilities based on experimentationexperimentation or historical dataor historical data

Assigning probabilities based on Assigning probabilities based on judgmentjudgment

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Remember that…Remember that…

The probability of any outcome in an The probability of any outcome in an experiment must be between 0 and 1experiment must be between 0 and 1

The sum of the probabilities of all possible The sum of the probabilities of all possible outcomes in an experiment equals 1.outcomes in an experiment equals 1.

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Classical MethodClassical Method

If an experiment has If an experiment has nn possible outcomes, this method possible outcomes, this method

would assign a probability of 1/would assign a probability of 1/nn to each outcome. to each outcome.

Experiment: Rolling a dieExperiment: Rolling a die

Sample Space: Sample Space: SS = {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

Probabilities: Each sample point has aProbabilities: Each sample point has a 1/6 chance of occurring1/6 chance of occurring

ExampleExample

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Relative Frequency MethodRelative Frequency Method

Number ofNumber ofPolishers RentedPolishers Rented

NumberNumberof Daysof Days

0011223344

44 6618181010 22

Lucas Tool Rental would like to assignprobabilities to the number of car polishersit rents each day. Office records show the followingfrequencies of daily rentals for the last 40 days.

Example: Lucas Tool Rental

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Subjective MethodSubjective Method

When economic conditions and a company’sWhen economic conditions and a company’s circumstances change rapidly it might becircumstances change rapidly it might be inappropriate to assign probabilities based solely oninappropriate to assign probabilities based solely on historical data.historical data. We can use any data available as well as ourWe can use any data available as well as our experience and intuition, but ultimately a probabilityexperience and intuition, but ultimately a probability value should express our value should express our degree of beliefdegree of belief that the that the experimental outcome will occur.experimental outcome will occur.

The best probability estimates often are obtained byThe best probability estimates often are obtained by combining the estimates from the classical or relativecombining the estimates from the classical or relative frequency approach with the subjective estimate.frequency approach with the subjective estimate.

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An An eventevent is a collection of sample points .is a collection of sample points . An An eventevent is a collection of sample points .is a collection of sample points .

The The probability of any eventprobability of any event is equal to the sum of is equal to the sum of the probabilities of the sample points in the event.the probabilities of the sample points in the event. The The probability of any eventprobability of any event is equal to the sum of is equal to the sum of the probabilities of the sample points in the event.the probabilities of the sample points in the event.

If we can identify all the sample points of anIf we can identify all the sample points of an experiment and assign a probability to each, weexperiment and assign a probability to each, we can compute the probability of an event.can compute the probability of an event.

If we can identify all the sample points of anIf we can identify all the sample points of an experiment and assign a probability to each, weexperiment and assign a probability to each, we can compute the probability of an event.can compute the probability of an event.

Events and Their ProbabilitiesEvents and Their Probabilities

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Some Basic Relationships of ProbabilitySome Basic Relationships of Probability

There are some There are some basic probability relationshipsbasic probability relationships that thatcan be used to compute the probability of an eventcan be used to compute the probability of an eventwithout knowledge of all the sample point probabilities.without knowledge of all the sample point probabilities.

Complement of an EventComplement of an Event Complement of an EventComplement of an Event

Intersection of Two EventsIntersection of Two Events Intersection of Two EventsIntersection of Two Events

Mutually Exclusive EventsMutually Exclusive Events Mutually Exclusive EventsMutually Exclusive Events

Union of Two EventsUnion of Two EventsUnion of Two EventsUnion of Two Events

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The complement of The complement of AA is denoted by is denoted by AAcc.. The complement of The complement of AA is denoted by is denoted by AAcc..

The The complementcomplement of event of event A A is defined to be the eventis defined to be the event consisting of all sample points that are not in consisting of all sample points that are not in A.A. The The complementcomplement of event of event A A is defined to be the eventis defined to be the event consisting of all sample points that are not in consisting of all sample points that are not in A.A.

Complement of an EventComplement of an Event

Event Event AA AAccSampleSpace SSampleSpace S

VennVennDiagraDiagra

mm

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The union of events The union of events AA and and BB is denoted by is denoted by AA BB The union of events The union of events AA and and BB is denoted by is denoted by AA BB

The The unionunion of events of events AA and and BB is the event containing is the event containing all sample points that are in all sample points that are in A A oror B B or both.or both. The The unionunion of events of events AA and and BB is the event containing is the event containing all sample points that are in all sample points that are in A A oror B B or both.or both.

Union of Two EventsUnion of Two Events

SampleSpace SSampleSpace SEvent Event AA Event Event BB

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The intersection of events The intersection of events AA and and BB is denoted by is denoted by AA The intersection of events The intersection of events AA and and BB is denoted by is denoted by AA

The The intersectionintersection of events of events AA and and BB is the set of all is the set of all sample points that are in bothsample points that are in both A A and and BB.. The The intersectionintersection of events of events AA and and BB is the set of all is the set of all sample points that are in bothsample points that are in both A A and and BB..


Intersection of Two EventsIntersection of Two Events

Intersection of A and BIntersection of A and B

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The The addition lawaddition law provides a way to compute the provides a way to compute the probability of event probability of event A,A, or or B,B, or both or both AA and and B B occurring.occurring. The The addition lawaddition law provides a way to compute the provides a way to compute the probability of event probability of event A,A, or or B,B, or both or both AA and and B B occurring.occurring.

Addition LawAddition Law

The law is written as:The law is written as: The law is written as:The law is written as:

PP((AA BB) = ) = PP((AA) + ) + PP((BB) ) PP((AA BBPP((AA BB) = ) = PP((AA) + ) + PP((BB) ) PP((AA BB

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Mutually Exclusive EventsMutually Exclusive Events

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

Two events are mutually exclusive if, when one eventTwo events are mutually exclusive if, when one event occurs, the other cannot occur.occurs, the other cannot occur. Two events are mutually exclusive if, when one eventTwo events are mutually exclusive if, when one event occurs, the other cannot occur.occurs, the other cannot occur.


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Mutually Exclusive EventsMutually Exclusive Events

If events If events AA and and BB are mutually exclusive, are mutually exclusive, PP((AA BB = 0. = 0. If events If events AA and and BB are mutually exclusive, are mutually exclusive, PP((AA BB = 0. = 0.

The addition law for mutually exclusive events is:The addition law for mutually exclusive events is: The addition law for mutually exclusive events is:The addition law for mutually exclusive events is:

PP((AA BB) = ) = PP((AA) + ) + PP((BB))PP((AA BB) = ) = PP((AA) + ) + PP((BB))

there’s no need tothere’s no need toinclude “include “ PP((AA BB””there’s no need tothere’s no need toinclude “include “ PP((AA BB””

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The probability of an event given that another eventThe probability of an event given that another event has occurred is called a has occurred is called a conditional probabilityconditional probability or or joint probabilityjoint probability..

The probability of an event given that another eventThe probability of an event given that another event has occurred is called a has occurred is called a conditional probabilityconditional probability or or joint probabilityjoint probability..

A conditional probability is computed as follows :A conditional probability is computed as follows : A conditional probability is computed as follows :A conditional probability is computed as follows :

The conditional probability of The conditional probability of AA given given BB is denoted is denoted by by PP((AA||BB).). The conditional probability of The conditional probability of AA given given BB is denoted is denoted by by PP((AA||BB).).

Conditional ProbabilityConditional Probability

( )( | )

( )P A B

P A BP B

( )( | )

( )P A B

P A BP B

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Multiplication LawMultiplication Law

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


PP((AA BB) = ) = PP((BB))PP((AA||BB))PP((AA BB) = ) = PP((BB))PP((AA||BB))

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Retrieving information from joint Retrieving information from joint probabilitiesprobabilities

The following table illustrates the promotion status The following table illustrates the promotion status of police officers over the past two years.of police officers over the past two years.

MenMen WomenWomen TotalTotal

PromotedPromoted 288288 3636 324324

Not Not promotedpromoted

672672 204204 876876

TotalTotal 960960 240240 12001200

Based on the information above, calculate all joint Based on the information above, calculate all joint probabilities.probabilities.

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Independent EventsIndependent Events

If the probability of event If the probability of event AA is not changed by the is not changed by the existence of event existence of event BB, we would say that events , we would say that events AA and and BB are are independentindependent..

If the probability of event If the probability of event AA is not changed by the is not changed by the existence of event existence of event BB, we would say that events , we would say that events AA and and BB are are independentindependent..

Two events Two events AA and and BB are independent if: are independent if: Two events Two events AA and and BB are independent if: are independent if:

PP((AA||BB) = ) = PP((AA))PP((AA||BB) = ) = PP((AA)) PP((BB||AA) = ) = PP((BB))PP((BB||AA) = ) = PP((BB))oror

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


PP((AA BB) = ) = PP((AA))PP((BB))PP((AA BB) = ) = PP((AA))PP((BB))

Multiplication LawMultiplication Lawfor Independent Eventsfor Independent Events

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Bayes’ TheoremBayes’ Theorem

NewNewInformationInformation

NewNewInformationInformation

ApplicationApplicationof Bayes’of Bayes’TheoremTheorem

ApplicationApplicationof Bayes’of Bayes’TheoremTheorem

PosteriorPosteriorProbabilitiesProbabilities

PosteriorPosteriorProbabilitiesProbabilities

PriorPriorProbabilitiesProbabilities

PriorPriorProbabilitiesProbabilities

Often we begin probability analysis with initial orOften we begin probability analysis with initial or prior probabilitiesprior probabilities..

Then, from a sample, special report, or a productThen, from a sample, special report, or a product test we obtain some additional information.test we obtain some additional information. Given this information, we calculate revised orGiven this information, we calculate revised or posterior probabilitiesposterior probabilities..

Bayes’ theoremBayes’ theorem provides the means for revising the provides the means for revising the prior probabilities.prior probabilities.

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Bayes’ TheoremBayes’ Theorem

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

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Bayes Theorem – Two events caseBayes Theorem – Two events case

Two events case (ATwo events case (A11 and A and A22, mutually exclusive), mutually exclusive)

)|()()|()(

)|()()|(

)|()()|()(

)|()()|(

2211

222

2211

111

ABPAPABPAP

ABPAPBAP

ABPAPABPAP

ABPAPBAP

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Bayes’ Theorem – Bayes’ Theorem – n n events case events case

1 1 2 2

( ) ( | )( | )

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

in n

P A P B AP A B

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

1 1 2 2

( ) ( | )( | )

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

in 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 event To find the posterior probability that event AAii will will occur given that eventoccur given that event B B has occurred, we applyhas occurred, we apply Bayes’ theoremBayes’ theorem..

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Chapter 4 Introduction to Probability