3 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Probability Chapter 3.

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© 2000 Prentice-Hall, Inc.© 2000 Prentice-Hall, Inc.

Statistics for Business Statistics for Business and Economicsand Economics

ProbabilityProbabilityChapter 3Chapter 3

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Learning ObjectivesLearning Objectives

1.1. Define Experiment, Outcome, Event, Define Experiment, Outcome, Event, Sample Space, & ProbabilitySample Space, & Probability

2.2. Explain How to Assign ProbabilitiesExplain How to Assign Probabilities

3.3. Use a Contingency Table, Venn Use a Contingency Table, Venn Diagram, or Tree to Find ProbabilitiesDiagram, or Tree to Find Probabilities

4.4. Describe & Use Probability RulesDescribe & Use Probability Rules

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Thinking ChallengeThinking Challenge

What’s the probability of What’s the probability of getting a getting a headhead on the on the toss of a single fair toss of a single fair coin? Use a scale from coin? Use a scale from 00 ((nono wayway) to ) to 11 ( (sure sure thingthing).).

So toss a coin twiceSo toss a coin twice. . Do it! Did you get one Do it! Did you get one head & one tail? head & one tail? What’s it all mean?What’s it all mean?

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Many Repetitions!*Many Repetitions!*

Number of TossesNumber of Tosses

Total Heads /Total Heads /Number of TossesNumber of Tosses

0.000.00

0.250.25

0.500.50

0.750.75

1.001.00

00 2525 5050 7575 100100 125125

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Experiments, Experiments, Outcomes, & EventsOutcomes, & Events

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Experiments & Experiments & OutcomesOutcomes

1.1. ExperimentExperiment Process of Obtaining an Observation, Process of Obtaining an Observation,

Outcome or Simple EventOutcome or Simple Event

2.2. Sample PointSample Point Most Basic Outcome of Most Basic Outcome of

an Experimentan Experiment

3.3. Sample Space (S) Sample Space (S) Collection of Collection of AllAll Possible Outcomes Possible Outcomes

Sample Space Sample Space Depends on Depends on Experimenter!Experimenter!

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Outcome ExamplesOutcome Examples

Toss a Coin, Note FaceToss a Coin, Note Face Head, TailHead, Tail

Toss 2 Coins, Note FacesToss 2 Coins, Note Faces HH, HT, TH, TTHH, HT, TH, TT

Select 1 Card, Note Kind Select 1 Card, Note Kind 22, 2, 2, ..., A, ..., A (52) (52)

Select 1 Card, Note ColorSelect 1 Card, Note Color Red, Black Red, Black

Play a Football GamePlay a Football Game Win, Lose, TieWin, Lose, Tie

Inspect a Part, Note QualityInspect a Part, Note Quality Defective, OKDefective, OK

Observe GenderObserve Gender Male, FemaleMale, Female

ExperimentExperiment Sample SpaceSample Space

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Outcome PropertiesOutcome Properties

1.1. Mutually ExclusiveMutually Exclusive 2 Outcomes Can Not 2 Outcomes Can Not

Occur at the Same TimeOccur at the Same Time BothBoth Male & Female in Male & Female in

Same PersonSame Person2.2. Collectively ExhaustiveCollectively Exhaustive

1 Outcome in Sample Space Must 1 Outcome in Sample Space Must OccurOccur

Male or FemaleMale or Female

Experiment: Observe Experiment: Observe GenderGender

© 1984-1994 T/Maker Co.

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EventsEvents

1.1. Any Collection of Sample PointsAny Collection of Sample Points

2.2. Simple EventSimple Event Outcome With 1 CharacteristicOutcome With 1 Characteristic

3.3. Compound Event Compound Event Collection of Outcomes or Simple EventsCollection of Outcomes or Simple Events 2 or More Characteristics2 or More Characteristics Joint Event Is a Special CaseJoint Event Is a Special Case

2 Events Occurring Simultaneously2 Events Occurring Simultaneously

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Event ExamplesEvent Examples

Sample SpaceSample Space HH, HT, TH, TTHH, HT, TH, TT

1 Head & 1 Tail1 Head & 1 Tail HT, THHT, TH

Heads on 1st CoinHeads on 1st Coin HH, HTHH, HT

At Least 1 HeadAt Least 1 Head HH, HT, THHH, HT, TH

Heads on BothHeads on Both HHHH

Experiment: Toss 2 Coins. Note Faces.Experiment: Toss 2 Coins. Note Faces.

EventEvent Outcomes in EventOutcomes in Event

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Sample SpaceSample Space

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Visualizing Visualizing Sample SpaceSample Space

1.1. ListingListing S = {Head, Tail}S = {Head, Tail}

2.2. Venn Diagram Venn Diagram

3.3. Contingency TableContingency Table

4.4. Decision Tree DiagramDecision Tree Diagram

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SS

TailTail

HHHH

TTTT

THTHHTHT

Sample SpaceSample SpaceS = {HH, HT, TH, TT}S = {HH, HT, TH, TT}

Venn DiagramVenn Diagram

OutcomeOutcome


Compound Compound Event Event

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22ndnd CoinCoin11stst CoinCoin HeadHead TailTail TotalTotal

HeadHead HHHH HTHT HH, HTHH, HT

TailTail THTH TTTT TH, TTTH, TT

TotalTotal HH,HH, THTH HT,HT, TTTT SS

Contingency TableContingency Table


S = {HH, HT, TH, TT}S = {HH, HT, TH, TT} Sample SpaceSample Space

Outcome Outcome (Count, (Count, Total % Total % Shown Shown Usually) Usually)

SimpleSimpleEvent Event (Head on(Head on1st Coin)1st Coin)

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Tree DiagramTree Diagram

Outcome Outcome

S = {HH, HT, TH, TT}S = {HH, HT, TH, TT} Sample SpaceSample Space


TT

HH

TT

HH

TT

HHHH

HTHT

THTH

TTTT

HH

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Compound EventsCompound Events

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Forming Forming Compound Events Compound Events

1.1. IntersectionIntersection Outcomes in Both Events A Outcomes in Both Events A andand B B ‘‘ANDAND’ Statement’ Statement Symbol (i.e., A Symbol (i.e., A B) B)

2.2. UnionUnion Outcomes in Either Events A Outcomes in Either Events A oror B or Both B or Both ‘‘OROR’ Statement’ Statement Symbol (i.e., A Symbol (i.e., A B) B)

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SS

BlackBlack

AceAce

Event Intersection: Event Intersection: Venn DiagramVenn Diagram

Joint Event (Ace Joint Event (Ace Black): Black):

AABB, , AABB

Event Event Black: Black:

22BB, ..., , ...,

AABB

Sample Sample Space: Space:

22RR, , 22RR, ,

22BB, ..., , ..., AABB

Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.

Event Ace: Event Ace:

AARR, , AARR, , AABB, , AABB

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ColorColorTypeType RedRed BlackBlack TotalTotal

AceAce Ace &Ace &RedRed

Ace &Ace &BlackBlack

AceAce

Non-AceNon-Ace Non &Non &RedRed

Non &Non &BlackBlack

Non-Non-AceAce

TotalTotal RedRed BlackBlack SS

Event Intersection: Event Intersection: Contingency TableContingency Table

Sample Sample Space (S):Space (S):

22RR,, 22RR,,

22BB, ..., , ..., AABB


Joint Event Joint Event Ace Ace ANDAND Black: Black:

AABB, , AABB

Simple Simple Event Event Ace:Ace:

AARR, ,

AARR, ,

AABB, ,

AABBSimple Event Black: Simple Event Black: 22BB, ..., , ..., AABB

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SS

BlackBlack

AceAce

Event Union : Event Union : Venn DiagramVenn Diagram

Event (Ace Event (Ace Black): Black):

AARR, ..., , ..., AABB, , 22BB, ..., , ..., KKBB

Event Event Black: Black:

22BB, ,

22BB,,..., ...,

AABB


22RR, , 22RR, ,

22BB, ..., , ..., AABB

Event Ace: Event Ace:

AARR, , AARR, , AABB, , AABB


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AceAce Ace &Ace &RedRed

Ace &Ace &BlackBlack

AceAce

Non-AceNon-Ace Non &Non &RedRed

Non &Non &BlackBlack

Non-Non-AceAce

TotalTotal RedRed BlackBlack SS

Event Union : Event Union : Contingency TableContingency Table

Sample Sample Space (S):Space (S):

22RR,, 22RR,,

22BB, ..., , ..., AABBJoint EventJoint EventAce Ace ORORBlack: Black:

AARR, ..., , ..., AABB,,22BB, ..., , ..., KKBB

Simple Simple Event Event Ace:Ace:

AARR, ,

AARR, ,

AABB, ,

AABB

Simple Event Black:Simple Event Black:

22BB, ..., , ..., AABB


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Special EventsSpecial Events

1.1. Null EventNull Event Club & Diamond on Club & Diamond on

1 Card Draw1 Card Draw

2.2. Complement of EventComplement of Event For Event A, All For Event A, All

Events Not In A: AEvents Not In A: A’’

3.3. Mutually Exclusive EventMutually Exclusive Event Events Do Not Occur Events Do Not Occur

SimultaneouslySimultaneously

Null EventNull Event

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SS

BlackBlack

Complement of Complement of Event ExampleEvent Example

Event Black: Event Black:

22BB,, 22BB, ..., , ..., AABB

Complement of Event Black, Complement of Event Black,

Black ’: Black ’: 22RR, , 22RR, ..., , ..., AARR, , AARR


22RR, , 22RR, ,

22BB, ..., , ..., AABB


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SS

Mutually Exclusive Mutually Exclusive Events ExampleEvents Example

Events Events & & Mutually Exclusive Mutually Exclusive

Experiment: Draw 1 Card. Note Kind & Suit.Experiment: Draw 1 Card. Note Kind & Suit.

Outcomes Outcomes in Event in Event Heart: Heart:

22, 3, 3, ,

44, ..., A, ..., A


22, 2, 2, ,

22, ..., A, ..., A

Event Spade: Event Spade:

22, 3, 3, 4, 4, ..., A, ..., A

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ProbabilitiesProbabilities

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What is Probability?What is Probability?

1.1. Numerical Numerical Measure of Likelihood Measure of Likelihood that Event Will Occurthat Event Will Occur

PP(Event)(Event) PP(A)(A) ProbProb(A)(A)

2.2. Lies Between 0 & Lies Between 0 & 11

3.3. Sum of Events is 1Sum of Events is 1

11

.5 .5

00

CertainCertain

ImpossibleImpossible

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Assigning Event Assigning Event ProbabilitiesProbabilities

1.1. a prioria priori Classical Classical MethodMethod

2.2. Empirical Empirical Classical Method Classical Method

3.3. Subjective Subjective MethodMethod

What’s the What’s the probability?probability?

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a prioria priori Classical Classical MethodMethod

1.1. Prior Knowledge of Prior Knowledge of ProcessProcess

2.2. Before ExperimentBefore Experiment

3.3. PP(Event) = (Event) = XX / / TT XX = No. of Event Outcomes = No. of Event Outcomes TT = Total Outcomes in Sample Space = Total Outcomes in Sample Space Each of T Outcomes Is Equally LikelyEach of T Outcomes Is Equally Likely

PP(Outcome) = 1/(Outcome) = 1/TT

© 1984-1994 T/Maker Co.

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Empirical Classical Empirical Classical MethodMethod

1.1. Actual Data CollectedActual Data Collected

2.2. After Experiment After Experiment

3.3. PP(Event) = (Event) = XX / / TT Repeat Experiment Repeat Experiment

TT Times Times Event Observed Event Observed XX

TimesTimes

4.4. Also Called Relative Also Called Relative Frequency MethodFrequency Method

Of 100 Parts Of 100 Parts Inspected, Only Inspected, Only 2 Defects!2 Defects!

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Subjective MethodSubjective Method

1.1. Individual Individual Knowledge of SituationKnowledge of Situation

2.2. Before ExperimentBefore Experiment

3.3. Unique ProcessUnique Process Not RepeatableNot Repeatable

4.4. Different Different Probabilities from Probabilities from Different PeopleDifferent People

© 1984-1994 T/Maker Co.

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1.1. That a Box of 24 Bolts Will Be Defective?That a Box of 24 Bolts Will Be Defective?

2.2. That a Toss of a Coin Will Be a Tail?That a Toss of a Coin Will Be a Tail?

3.3. That Tom Will Default on His PLUS Loan?That Tom Will Default on His PLUS Loan?

4.4. That a Student Will Earn an A in This Class?That a Student Will Earn an A in This Class?

5.5. That a New Store on Rte. 1 Will Succeed?That a New Store on Rte. 1 Will Succeed?

Which Method Should Be Used to Find the Which Method Should Be Used to Find the Probability ... Probability ...

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Compound Event Compound Event ProbabilityProbability

1.1. Numerical Measure of Likelihood that Numerical Measure of Likelihood that Compound Event Will OccurCompound Event Will Occur

2.2. Can Often Use Contingency TableCan Often Use Contingency Table 2 Variables Only2 Variables Only

3.3. Formula MethodsFormula Methods Additive RuleAdditive Rule Conditional Probability FormulaConditional Probability Formula Multiplicative RuleMultiplicative Rule

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EventEventEventEvent BB11 BB22 TotalTotal

AA11 P(AP(A1 1 BB11)) P(AP(A1 1 BB22)) P(AP(A11))

AA22 P(AP(A2 2 BB11)) P(AP(A2 2 BB22)) P(AP(A22))

TotalTotal P(BP(B11)) P(BP(B22)) 11

Event Probability Event Probability Using Contingency Using Contingency

TableTable

Joint ProbabilityJoint Probability Marginal (Simple) ProbabilityMarginal (Simple) Probability

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AceAce 2/522/52 2/522/52 4/524/52

Non-AceNon-Ace 24/5224/52 24/5224/52 48/5248/52

TotalTotal 26/5226/52 26/5226/52 52/5252/52

Contingency Table Contingency Table ExampleExample


P(Ace)P(Ace)

P(Ace AND Red)P(Ace AND Red)P(Red)P(Red)

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EventEventEventEvent CC DD TotalTotal

AA 44 22 66

BB 11 33 44

TotalTotal 55 55 1010


What’s the Probability?What’s the Probability?

P(A) =P(A) =

P(D) =P(D) =

P(C P(C B) = B) =

P(A P(A D) = D) =

P(B P(B D) = D) =

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Solution*Solution*

The Probabilities Are:The Probabilities Are:

P(A) = 6/10P(A) = 6/10

P(D) = 5/10P(D) = 5/10

P(C P(C B) = 1/10 B) = 1/10

P(A P(A D) = 9/10 D) = 9/10

P(B P(B D) = 3/10 D) = 3/10


AA 44 22 66

BB 11 33 44


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Additive RuleAdditive Rule

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Additive RuleAdditive Rule

1.1. Used to Get Compound Probabilities for Used to Get Compound Probabilities for UnionUnion of Events of Events

2.2. P(A P(A OROR B) B) = P(A = P(A B) B) = P(A) + P(B) - P(A = P(A) + P(B) - P(A B) B)

3. 3. For Mutually Exclusive Events:For Mutually Exclusive Events:P(A P(A OROR B) B) = P(A = P(A B) = P(A) + P(B) B) = P(A) + P(B)

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Additive Rule Additive Rule ExampleExample



AceAce 22 22 44

Non-AceNon-Ace 2424 2424 4848


P(Ace OR BP(Ace OR Black) lack) == P(Ace)P(Ace) ++ P(Black)P(Black) -- P(Ace P(Ace Black)Black)

44

5252

2626

5252

22

5252

2828

5252

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Using the Additive Rule, What’s the Using the Additive Rule, What’s the Probability?Probability?

P(A P(A D) = D) =

P(B P(B C) = C) =


AA 44 22 66

BB 11 33 44


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Solution*Solution*

Using the Additive Rule, the Probabilities Using the Additive Rule, the Probabilities Are:Are:

P(A P(A D) D) == P(A)P(A) ++ P(D)P(D) -- P(A P(A D)D)

66

1010

55

1010

22

1010

99

1010

P(B P(B C) C) == P(B)P(B) ++ P(C)P(C) -- P(B P(B C)C)

44

1010

55

1010

11

1010

88

1010

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Conditional ProbabilityConditional Probability

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Conditional Conditional ProbabilityProbability

1.1. Event Probability Event Probability GivenGiven that Another that Another Event OccurredEvent Occurred

2.2. Revise Original Sample Space to Revise Original Sample Space to Account for Account for NewNew Information Information Eliminates Certain OutcomesEliminates Certain Outcomes

3.3. P(A P(A || BB) = ) = P(A and B)P(A and B) P( P(BB))

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SS

BlackBlack

AceAce

Conditional Conditional Probability Using Probability Using

Venn DiagramVenn DiagramBlack ‘Happens’: Black ‘Happens’: Eliminates All Eliminates All Other OutcomesOther Outcomes

Event (Ace AND Black)Event (Ace AND Black)

(S)(S)BlackBlack

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AceAce 22 22 44



Conditional Conditional Probability Using Probability Using

Contingency TableContingency TableExperiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.

P(Ace | Black) = P(Ace AND Black)

P(Black)

2 52

26 52

2

26

/

/

Revised Revised Sample Sample SpaceSpace

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1.1. Event Occurrence Event Occurrence Does Does NotNot Affect Probability Affect Probability of Another Eventof Another Event

Toss 1 Coin Twice Toss 1 Coin Twice

2.2. Causality Not ImpliedCausality Not Implied

3.3. Tests ForTests For P(P(AA | B) = P( | B) = P(AA)) P(A and B) = P(A)*P(B)P(A and B) = P(A)*P(B)

Statistical Statistical IndependenceIndependence

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Tree DiagramTree Diagram

Experiment: Select 2 Pens from 20 Pens: Experiment: Select 2 Pens from 20 Pens: 14 Blue & 6 Red. Don’t Replace.14 Blue & 6 Red. Don’t Replace.

Dependent!Dependent!BB

RR

BBRR

BB

RRP(R) = 6/20P(R) = 6/20P(R|R) = 5/19P(R|R) = 5/19

P(B|R) = 14/19P(B|R) = 14/19

P(B) = 14/20P(B) = 14/20

P(R|B) = 6/19P(R|B) = 6/19

P(B|B) = 13/19P(B|B) = 13/19

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Using the Table Then the Formula, What’s Using the Table Then the Formula, What’s the Probability?the Probability?

P(A|D) =P(A|D) =

P(C|B) = P(C|B) =

Are C & B Are C & B Independent?Independent?


AA 44 22 66

BB 11 33 44


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Solution*Solution*

Using the Formula, the Probabilities Are:Using the Formula, the Probabilities Are:

DependentDependent

P(A |P(A | D) D) == P(A P(A D)D)

P(D)P(D)

22 101055 1010

2255

////

P(C |P(C | B) B) == P(C P(C B)B)

P(B)P(B)

P(C) P(C) == 55

1010

11 101044 1010

1144

11

44

////

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Multiplicative RuleMultiplicative Rule

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Multiplicative RuleMultiplicative Rule

1.1. Used to Get Compound Probabilities for Used to Get Compound Probabilities for IntersectionIntersection of Events of Events Called Joint EventsCalled Joint Events

2.2. P(A and B) = P(A P(A and B) = P(A B) B)= P(= P(AA)*P(B|)*P(B|AA) ) = P(= P(BB)*P(A|)*P(A|BB))

3. 3. For Independent Events:For Independent Events:P(A and B) = P(A P(A and B) = P(A B) = P(A)*P(B) B) = P(A)*P(B)

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Multiplicative Rule Multiplicative Rule ExampleExample


52

2

4

2

52

4

Ace)|P(BlackP(Ace) = Black) AND P(Ace

52

2

4

2

52

4

Ace)|P(BlackP(Ace) = Black) AND P(Ace


AceAce 22 22 44



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Using the Multiplicative Rule, What’s the Using the Multiplicative Rule, What’s the Probability?Probability?

P(C P(C B) = B) =

P(B P(B D) = D) =

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


AA 44 22 66

BB 11 33 44


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Solution*Solution*

Using the Multiplicative Rule, the Using the Multiplicative Rule, the Probabilities Are:Probabilities Are:

P(C P(C B) B) == P(C)P(C) P(B|P(B|C) = 5/10 * 1/5 = 1/10 C) = 5/10 * 1/5 = 1/10

P(B P(B D) D) == P(B)P(B) P(D|P(D|B) = 4/10 * 3/4 = 3/10 B) = 4/10 * 3/4 = 3/10

P(A P(A B) B) == P(A)P(A) P(B|P(B| A)A) 00

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ConclusionConclusion

1.1. Defined Experiment, Outcome, Event, Defined Experiment, Outcome, Event, Sample Space, & ProbabilitySample Space, & Probability

2.2. Explained How to Assign ProbabilitiesExplained How to Assign Probabilities

3.3. Used a Contingency Table, Venn Used a Contingency Table, Venn Diagram, or Tree to Find ProbabilitiesDiagram, or Tree to Find Probabilities

4.4. Described & Used Probability RulesDescribed & Used Probability Rules

End of Chapter

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3 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Probability Chapter 3.

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Transcript of 3 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Probability Chapter 3.