3 - 1 © 1998 Prentice-Hall, Inc. Chapter 3 Probability.

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© 1998 Prentice-Hall, Inc.© 1998 Prentice-Hall, Inc.

Chapter 3 ProbabilityChapter 3 Probability

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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 diagram, Use a contingency table, Venn diagram, or tree to find probabilitiesor tree to find probabilities

4.4. Describe & use probability rulesDescribe & use probability rules

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Thinking ChallengeThinking Challenge

You ‘know’ that the You ‘know’ that the probability of getting a head probability of getting a head on a single toss of a fair coin on a single toss of a fair coin is 0.5 or 50%. What does is 0.5 or 50%. What does this mean?this mean?

Ten students Ten students will each toss will each toss a single coin a single coin 20 times20 times. The . The number of number of headsheads will be will be recorded. Compute the recorded. Compute the probability of a head.probability of a head.

AloneAlone GroupGroup Class Class

© 1984-1994 T/Maker Co.

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Probability GraphProbability Graph

0.0

0.2

0.4

0.6

0.8

1.0

20 40 60 80 100 120 140 160 180 200

0.0

0.2

0.4

0.6

0.8

1.0

20 40 60 80 100 120 140 160 180 200

P(Head)P(Head)

Cumulative coin tossesCumulative coin tosses

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Possible Result*Possible Result*

Cumulative coin tossesCumulative coin tosses

Total headsTotal headsNumber of tossesNumber of tosses

0.000.00

0.250.25

0.500.50

0.750.75

1.001.00

00 5050 150150 200200100100

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Experiments, Experiments, Outcomes, & EventsOutcomes, & Events

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Experiments & Experiments & OutcomesOutcomes

1.1. Random ExperimentRandom Experiment 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

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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

ExperimentExperiment Sample SpaceSample Space

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Toss a coin, note faceToss a coin, note face Head, TailHead, TailExperimentExperiment Sample SpaceSample Space

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Toss a coin, note faceToss a coin, note face Head, TailHead, Tail

Toss 2 coins, note facesToss 2 coins, note faces


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Toss 2 coins, note facesToss 2 coins, note faces HH, HT, TH, TTHH, HT, TH, TT


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Select 1 card, note kind Select 1 card, note kind


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Select 1 card, note kind Select 1 card, note kind 22, 2, 2, ..., A, ..., A (52) (52)


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Select 1 card, note colorSelect 1 card, note color


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Select 1 card, note colorSelect 1 card, note color Red, Black Red, Black


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Play a football gamePlay a football game


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Play a football gamePlay a football game Win, Lose, TieWin, Lose, Tie


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Inspect a part, note qualityInspect a part, note quality


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Inspect a part, note qualityInspect a part, note quality Defective, OKDefective, OK


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Observe genderObserve gender


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Observe genderObserve gender Male, FemaleMale, Female


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Outcome PropertiesOutcome Properties

1.1. Mutually exclusiveMutually exclusive 2 outcomes can not occur 2 outcomes can not occur

at the same timeat the same time BothBoth male & female in male & female in

same personsame person

2.2. Collectively Collectively exhaustiveexhaustive

1 outcome in sample 1 outcome in sample space must occurspace must occur Male or femaleMale or female

Experiment: Observe Experiment: Observe gendergender

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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

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 HH, HT, TH, TTHH, HT, TH, TT



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1 head & 1 tail1 head & 1 tail



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1 head & 1 tail1 head & 1 tail HT, THHT, TH



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Heads on 1st coinHeads on 1st coin



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Heads on 1st coinHeads on 1st coin HH, HTHH, HT



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At least 1 headAt least 1 head



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At least 1 headAt least 1 head HH, HT, THHH, HT, TH



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Heads on bothHeads on both



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Heads on bothHeads on both HHHH



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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

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 Mutually exclusive eventevent

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 & & are mutually exclusive are 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, 4, 4, ,

..., 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 measure Numerical measure of likelihood that event will of likelihood that event will occuroccur

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

2.2. Lies between 0 & 1Lies between 0 & 1

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. Requires prior knowledge Requires prior knowledge of processof process

2.2. Can assign before Can assign before

experimentexperiment

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 Each of TT outcomes is equally likely 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. Assign after Assign after experiment experiment

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

TT Times Times Event observed Event observed XX times times

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. Requires individual Requires individual knowledge of knowledge of situationsituation

2.2. Done before Done before experimentexperiment

3.3. Unique processUnique process Not repeatableNot repeatable

4.4. Different probabilities Different probabilities from different peoplefrom different 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 Chris will default on his PLUS loan?That Chris 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 computer store will succeed?That a new computer store will succeed?

Which method should be used to find the Which method should be used to find the probability ... probability ...

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1.1. That a box of 24 bolts will be defective? That a box of 24 bolts will be defective? (E)(E)

2.2. That a toss of a coin will be a tail?That a toss of a coin will be a tail? (C)(C)

3.3. That Chris will default on his PLUS loan?That Chris will default on his PLUS loan? (S)(S)

4.4. That a student will earn an A in this class?That a student will earn an A in this class? (S)(S)

5.5. That a new computer store will succeed?That a new computer store will succeed? (S)(S)

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

Not-AceNot-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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Conditional Conditional Probability Using Probability Using


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SS

BlackBlack

AceAce



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SS

BlackBlack

AceAce


Venn DiagramVenn DiagramBlack ‘happens’: Black ‘happens’: Eliminates all Eliminates all other outcomesother outcomes

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SS

BlackBlack

AceAce


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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S

Black

Ace




(S)(S)BlackBlack

Black ‘happens’: Eliminates all other outcomes

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(S)(S)BlackBlack

New sample spaceNew sample space

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(S)(S)BlackBlack

New sample spaceNew sample space

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

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AceAce 22 22 44




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 green & 6 red. Don’t replace.14 green & 6 red. Don’t replace.

Dependent!Dependent!GG

RR

GGRR

GG

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

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

P(G) = 14/20P(G) = 14/20

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

P(G|G) = 13/19P(G|G) = 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)C) 55

1010

11

55

11

1010

P(B P(B D) D) == P(B)P(B) P(D|P(D|B)B) 44

1010

33

44

33

1010

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

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Random SamplingRandom Sampling

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Simple Simple Random SampleRandom Sample

1.1. Each population Each population element has an element has an equal equal chancechance of being selectedof being selected

2.2. Selecting 1 element Selecting 1 element does not affect selecting does not affect selecting othersothers

3.3. ComputerComputer © 1984-1994 T/Maker Co.

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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

5.5. Defined random sampleDefined random sample

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This Class...This Class...

1.1. What was the most important thing you What was the most important thing you learned in class today?learned in class today?

2.2. What do you still have questions about?What do you still have questions about?

3.3. How can today’s class be improved?How can today’s class be improved?

Please take a moment to answer the following questions in writing:

End of Chapter

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