3 - 3 - 1 1 © 2000 Prentice-Hall, © 2000 Prentice-Hall, Inc. Inc. Statistics for Statistics for Business and Economics Business and Economics Probability Probability Chapter 3 Chapter 3
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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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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
Experiment: Toss 2 Coins. Note Faces.Experiment: Toss 2 Coins. Note Faces.
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
Experiment: Toss 2 Coins. Note Faces.Experiment: Toss 2 Coins. Note Faces.
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
Experiment: Toss 2 Coins. Note Faces.Experiment: Toss 2 Coins. Note Faces.
TT
HH
TT
HH
TT
HHHH
HTHT
THTH
TTTT
HH
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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
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
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
Sample Sample Space: Space:
22RR, , 22RR, ,
22BB, ..., , ..., AABB
Event Ace: Event Ace:
AARR, , AARR, , AABB, , AABB
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
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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 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
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
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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
Sample Sample Space: Space:
22RR, , 22RR, ,
22BB, ..., , ..., AABB
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
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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
Sample Sample Space: Space:
22, 2, 2, ,
22, ..., A, ..., A
Event Spade: Event Spade:
22, 3, 3, 4, 4, ..., A, ..., A
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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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Thinking ChallengeThinking Challenge
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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ColorColorTypeType RedRed BlackBlack TotalTotal
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
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
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
Thinking ChallengeThinking Challenge
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
EventEventEventEvent CC DD TotalTotal
AA 44 22 66
BB 11 33 44
TotalTotal 55 55 1010
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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
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
ColorColorTypeType RedRed BlackBlack TotalTotal
AceAce 22 22 44
Non-AceNon-Ace 2424 2424 4848
TotalTotal 2626 2626 5252
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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Thinking ChallengeThinking Challenge
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) =
EventEventEventEvent CC DD TotalTotal
AA 44 22 66
BB 11 33 44
TotalTotal 55 55 1010
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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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ColorColorTypeType RedRed BlackBlack TotalTotal
AceAce 22 22 44
Non-AceNon-Ace 2424 2424 4848
TotalTotal 2626 2626 5252
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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Thinking ChallengeThinking Challenge
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?
EventEventEventEvent CC DD TotalTotal
AA 44 22 66
BB 11 33 44
TotalTotal 55 55 1010
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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
Experiment: Draw 1 Card. Note Kind, Color Experiment: Draw 1 Card. Note Kind, Color & Suit.& Suit.
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
ColorColorTypeType RedRed BlackBlack TotalTotal
AceAce 22 22 44
Non-AceNon-Ace 2424 2424 4848
TotalTotal 2626 2626 5252
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© 2000 Prentice-Hall, Inc.© 2000 Prentice-Hall, Inc.
Thinking ChallengeThinking Challenge
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) =
EventEventEventEvent CC DD TotalTotal
AA 44 22 66
BB 11 33 44
TotalTotal 55 55 1010
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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