1 1 Slide © 2016 Cengage Learning. All Rights Reserved. Probability is a numerical measure of the likelihood Probability is a numerical measure of the

11 Slide Slide

© 2016 Cengage Learning. All Rights Reserved. © 2016 Cengage Learning. All Rights Reserved.

ProbabilityProbability is a numerical measure of the likelihood is a numerical measure of the likelihood that an event will occur.that an event will occur. ProbabilityProbability is a numerical measure of the likelihood is a numerical measure of the likelihood that an event will occur.that an event will occur.

Probability values are always assigned on a scaleProbability values are always assigned on a scale from 0 to 1.from 0 to 1. Probability values are always assigned on a scaleProbability values are always assigned on a scale from 0 to 1.from 0 to 1.

A probability near zero indicates an event is quiteA probability near zero indicates an event is quite unlikely to occur.unlikely to occur. A probability near zero indicates an event is quiteA probability near zero indicates an event is quite unlikely to occur.unlikely to occur.

A probability near one indicates an event is almostA probability near one indicates an event is almost certain to occur.certain to occur. A probability near one indicates an event is almostA probability near one indicates an event is almost certain to occur.certain to occur.

Chapter 4 - Introduction to ProbabilityChapter 4 - Introduction to Probability

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

00 11..55

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.

33 Slide Slide


Statistical ExperimentsStatistical Experiments

In statistics, the notion of an experiment differsIn statistics, the notion of an experiment differs somewhat from that of an experiment in thesomewhat from that of an experiment in the physical sciences.physical sciences.

In statistics, the notion of an experiment differsIn statistics, the notion of an experiment differs somewhat from that of an experiment in thesomewhat from that of an experiment in the physical sciences.physical sciences.

In statistical experiments, probability determinesIn statistical experiments, probability determines outcomes.outcomes. In statistical experiments, probability determinesIn statistical experiments, probability determines outcomes.outcomes.

Even though the experiment is repeated in exactlyEven though the experiment is repeated in exactly the same way, an entirely different outcome maythe same way, an entirely different outcome may occur.occur.

Even though the experiment is repeated in exactlyEven though the experiment is repeated in exactly the same way, an entirely different outcome maythe same way, an entirely different outcome may occur.occur.

For this reason, statistical experiments are some-For this reason, statistical experiments are some- times called times called random experimentsrandom experiments.. For this reason, statistical experiments are some-For this reason, statistical experiments are some- times called times called random experimentsrandom experiments..

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An Experiment and Its Sample SpaceAn Experiment and Its Sample Space

An An experimentexperiment is any process that generates well-is any process that generates well- defined outcomes.defined outcomes. An An experimentexperiment is any process that generates well-is any process that generates well- defined outcomes.defined outcomes.

The The sample spacesample space for an experiment is the set of for an experiment is the set of all experimental outcomes.all experimental outcomes. The The sample spacesample space for an experiment is the set of for an experiment is the set of all experimental outcomes.all experimental outcomes.

An experimental outcome is also called a An experimental outcome is also called a samplesample pointpoint.. An experimental outcome is also called a An experimental outcome is also called a samplesample pointpoint..

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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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ExperimentExperiment

Toss a coinToss a coinInspection of a Inspection of a partpartConduct a sales Conduct a sales callcallRoll a dieRoll a diePlay a football Play a football gamegame

Experiment OutcomesExperiment Outcomes

Head, tailHead, tailDefective, non-Defective, non-defectivedefectivePurchase, no purchasePurchase, no purchase1, 2, 3, 4, 5, 61, 2, 3, 4, 5, 6Win, lose, tieWin, lose, tie

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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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Classical MethodClassical Method

If an experiment has If an experiment has nn possible outcomes, the possible outcomes, theclassical method would assign a probability of 1/classical method would assign a probability of 1/nnto each outcome.to each outcome.

Experiment: Rolling a die

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

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

Example: Rolling a Die

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Relative Frequency MethodRelative Frequency Method

• Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day and office records show the following frequencies of daily rentals for the last 40 days.

• Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days).

4/40

ProbabilityNumber ofPolishers Rented

Numberof Days

01234

4 61810 240

.10 .15 .45 .25 .051.00

Example: Lucas Tool Rental

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Subjective MethodSubjective Method

When economic conditions or a company’s 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 estimate.

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An analyst made the following probability estimates.An analyst made the following probability estimates.

Exper. Outcome Net Gain or Loss 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

1.00

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A second useful counting rule enables us to countA second useful counting rule enables us to countthe number of experimental outcomes when the number of experimental outcomes when n n objectsobjectsare to be selected from a set of are to be selected from a set of NN objects. objects.

Counting Rule for CombinationsCounting Rule for Combinations

CN

nN

n N nnN

!

!( )!C

N

nN

n N nnN

!

!( )!

Number of Number of CombinationsCombinations of of NN Objects Objects Taken Taken nn at a Time at a Time

where: where: NN! = ! = NN((NN 1)( 1)(NN 2) . . . (2)(1) 2) . . . (2)(1) nn! = ! = nn((nn 1)( 1)(nn 2) . . . (2)(1) 2) . . . (2)(1) 0! = 10! = 1

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Number of Number of PermutationsPermutations of of NN Objects Objects Taken Taken nn at a Time at a Time

where: where: NN! = ! = NN((NN 1)( 1)(NN 2) . . . (2)(1) 2) . . . (2)(1) nn! = ! = nn((nn 1)( 1)(nn 2) . . . (2)(1) 2) . . . (2)(1) 0! = 10! = 1

!!

( )!Nn

N NP n

n N n

!!

( )!Nn

N NP n

n N n

Counting Rule for PermutationsCounting Rule for Permutations

A third useful counting rule enables us to A third useful counting rule enables us to countcount

the number of experimental outcomes when the number of experimental outcomes when n n objects are to be selected from a set of objects are to be selected from a set of NN

objects,objects,where the order of selection is important.where the order of selection is important.

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Some Basic Relationships of Some Basic Relationships of ProbabilityProbability

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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Bradley has invested in two stocks, Markley Oil Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that theand Collins Mining. Bradley has determined that thepossible outcomes of these investments three monthspossible outcomes of these investments three monthsfrom now are as follows.from now are as follows.

Investment Gain or LossInvestment Gain or Loss in 3 Months (in $000)in 3 Months (in $000)

Markley OilMarkley Oil Collins MiningCollins Mining

1010 55 002020

8822

Example: Bradley InvestmentsExample: Bradley Investments


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Tree DiagramTree Diagram

Gain 5Gain 5

Gain 8Gain 8

Gain 8Gain 8

Gain 10Gain 10

Gain 8Gain 8

Gain 8Gain 8

Lose 20Lose 20

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

EvenEven

Markley OilMarkley Oil(Stage 1)(Stage 1)

Collins MiningCollins Mining(Stage 2)(Stage 2)

ExperimentalExperimentalOutcomesOutcomes

(10, 8) (10, 8) Gain $18,000 Gain $18,000

(10, -2) (10, -2) Gain $8,000 Gain $8,000

(5, 8) (5, 8) Gain $13,000 Gain $13,000

(5, -2) (5, -2) Gain $3,000 Gain $3,000

(0, 8) (0, 8) Gain $8,000 Gain $8,000

(0, -2) (0, -2) Lose Lose $2,000$2,000

(-20, 8) (-20, 8) Lose Lose $12,000$12,000

(-20, -2)(-20, -2) Lose Lose $22,000$22,000


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An analyst made the following probability estimates.An analyst made the following probability estimates.

Exper. OutcomeExper. OutcomeNet Gain Net Gain oror Loss Loss ProbabilityProbability(10, 8)(10, 8)(10, (10, 2)2)(5, 8)(5, 8)(5, (5, 2)2)(0, 8)(0, 8)(0, (0, 2)2)((20, 8)20, 8)((20, 20, 2)2)

$18,000 Gain$18,000 Gain $8,000 Gain$8,000 Gain $13,000 Gain$13,000 Gain $3,000 Gain$3,000 Gain $8,000 Gain$8,000 Gain $2,000 Loss$2,000 Loss $12,000 Loss$12,000 Loss $22,000 Loss$22,000 Loss

.20.20

.08.08

.16.16

.26.26

.10.10

.12.12

.02.02

.06.06


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Events and Their ProbabilitiesEvents and Their Probabilities

Event Event MM = Markley Oil Profitable = Markley Oil Profitable

MM = {(10, 8), (10, = {(10, 8), (10, 2), (5, 8), (5, 2), (5, 8), (5, 2)}2)}

PP((MM) = ) = PP(10, 8) + (10, 8) + PP(10, (10, 2) + 2) + PP(5, 8) + (5, 8) + PP(5, (5, 2)2)

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

= .70= .70


Event Event CC = Collins Mining Profitable = Collins Mining Profitable

CC = {(10, 8), (5, 8), (0, 8), ( = {(10, 8), (5, 8), (0, 8), (20, 8)}20, 8)}

PP((CC) = ) = PP(10, 8) + (10, 8) + PP(5, 8) + (5, 8) + PP(0, 8) + (0, 8) + PP((20, 8)20, 8)

= .20 + .16 + .10 + .02 = 0.48= .20 + .16 + .10 + .02 = 0.48

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Union of Two EventsUnion of Two Events



MM CC = Markley Oil Profitable = Markley Oil Profitable oror Collins Mining Profitable (or both) Collins Mining Profitable (or both)

MM CC = {(10, 8), (10, = {(10, 8), (10, 2), (5, 8), (5, 2), (5, 8), (5, 2), (0, 8), (2), (0, 8), (20, 8)}20, 8)}

PP((MM C)C) = = PP(10, 8) + (10, 8) + PP(10, (10, 2) + 2) + PP(5, 8) + (5, 8) + PP(5, (5, 2)2)

+ + PP(0, 8) + (0, 8) + PP((20, 8)20, 8)

= .20 + .08 + .16 + .26 + .10 + .02= .20 + .08 + .16 + .26 + .10 + .02

= .82= .82


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Intersection of Two EventsIntersection of Two Events



MM CC = Markley Oil Profitable = Markley Oil Profitable andand Collins Mining Profitable Collins Mining Profitable

MM CC = {(10, 8), (5, 8)} = {(10, 8), (5, 8)}

PP((MM C)C) = = PP(10, 8) + (10, 8) + PP(5, 8)(5, 8)

= .20 + .16= .20 + .16

= .36= .36


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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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Event Event MM = Markley Oil Profitable = Markley Oil ProfitableEvent Event CC = Collins Mining Profitable = Collins Mining Profitable

MM CC = Markley Oil Profitable = Markley Oil Profitable oror Collins Mining Profitable Collins Mining Profitable

We know: We know: PP((MM) = .70, ) = .70, PP((CC) = .48, ) = .48, PP((MM CC) = .36) = .36

Thus: Thus: PP((MM C) C) = = PP((MM) + P() + P(CC) ) PP((MM CC))

= .70 + .48 = .70 + .48 .36 .36

= .82= .82

Addition LawAddition Law

(This result is the same as that obtained earlier(This result is the same as that obtained earlierusing the definition of the probability of an event.)using the definition of the probability of an event.)


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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 is no need toThere is no need toinclude “include “ PP((AA BB””There is no need toThere is 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.. 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..

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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We know:We know: P P((MM CC) = .36, ) = .36, PP((MM) = .70 ) = .70

Thus: Thus:

Conditional ProbabilityConditional Probability

( ) .36( | ) .5143

( ) .70P C M

P C MP M

( ) .36( | ) .5143

( ) .70P C M

P C MP M

= Collins Mining Profitable= Collins Mining Profitable givengiven Markley Oil Profitable Markley Oil Profitable

( | )P C M( | )P C M


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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(A)(A)PP(B|A)(B|A)

or or PP((AA BB) = ) = PP(B)(B)PP(A|B)(A|B)PP((AA BB) = ) = PP(A)(A)PP(B|A)(B|A)

or or PP((AA BB) = ) = PP(B)(B)PP(A|B)(A|B)

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We know:We know: P P((MM) = .70, ) = .70, PP((CC||MM) = .5143) = .5143

Multiplication LawMultiplication Law

MM CC = Markley Oil Profitable = Markley Oil Profitable andand Collins Mining Profitable Collins Mining Profitable

Thus: Thus: PP((MM C) C) = = PP((MM))PP(C(C|M|M))= (.70)(.5143)= (.70)(.5143)

= .36= .36

(This result is the same as that obtained earlier(This result is the same as that obtained earlierusing the definition of the probability of an event.)using the definition of the probability of an event.)


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Joint Probability TableJoint Probability Table

Collins MiningCollins MiningProfitable (C) Not Profitable (CProfitable (C) Not Profitable (Ccc))Markley OilMarkley Oil

Profitable (M)Profitable (M)

Not Profitable (MNot Profitable (Mcc))

Total .48 .52Total .48 .52

TotalTotal

.70.70

.30.30

1.001.00

.36 .34.36 .34

.12 .18.12 .18

Joint ProbabilitiesJoint Probabilities(appear in the (appear in the

bodybodyof the table)of the table)

Marginal Marginal ProbabilitiesProbabilities

(appear in the (appear in the marginsmargins

of the table)of the table)

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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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We know:We know: P P((MM CC) = .36, ) = .36, PP((MM) = .70, ) = .70, PP((CC) = .48) = .48 But: But: PP((M)P(C) M)P(C) = (.70)(.48) = .34, not .36= (.70)(.48) = .34, not .36

Are events Are events MM and and CC independent? independent?DoesDoesPP((MM CC) = ) = PP((M)P(C) M)P(C) ??

Hence:Hence: M M and and CC are are notnot independent. independent.


Multiplication LawMultiplication Lawfor Independent Eventsfor Independent Events

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Do not confuse the notion of mutually exclusiveDo not confuse the notion of mutually exclusive events with that of independent events.events with that of independent events. Do not confuse the notion of mutually exclusiveDo not confuse the notion of mutually exclusive events with that of independent events.events with that of independent events.

Two events with nonzero probabilities cannot beTwo events with nonzero probabilities cannot be both mutually exclusive and independent.both mutually exclusive and independent. Two events with nonzero probabilities cannot beTwo events with nonzero probabilities cannot be both mutually exclusive and independent.both mutually exclusive and independent.

If one mutually exclusive event is known to occur,If one mutually exclusive event is known to occur, the other cannot occur.; thus, the probability of thethe other cannot occur.; thus, the probability of the other event occurring is reduced to zero (and theyother event occurring is reduced to zero (and they are therefore dependent).are therefore dependent).

If one mutually exclusive event is known to occur,If one mutually exclusive event is known to occur, the other cannot occur.; thus, the probability of thethe other cannot occur.; thus, the probability of the other event occurring is reduced to zero (and theyother event occurring is reduced to zero (and they are therefore dependent).are therefore dependent).

Mutual Exclusiveness and Mutual Exclusiveness and IndependenceIndependence

Two events that are not mutually exclusive, mightTwo events that are not mutually exclusive, might or might not be independent.or might not be independent. Two events that are not mutually exclusive, mightTwo events that are not mutually exclusive, might or might not be independent.or might not be independent.

Documents

1 1 Slide © 2016 Cengage Learning. All Rights Reserved. Probability is a numerical measure of the likelihood Probability is a numerical measure of the