Math310 Lecture Outline.docx

Math310 Lecture Outline.docx

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Math310LectureOutline

Math310LectureOutline.

ChapterTwo.TheBasicsofProbability.

Vocabulary（Section2.1）.Inyesterday’slab,westudiedasituationthatcouldbemodeledby11blackcardsand12redcardsthatareshuffledandthen12cardsaredealttoGroupA.Describeeachofthefollowingforthissituation.

Experiment.

SampleSpace.

Events.

SimpleEvents.

CompoundEvents.

MoreVocabulary.（HereAandBdenoteevents）.

Union（AÈB）

Intersection（AÇB）

Complement（A'orAC）

MutuallyExclusiveEvents

DefinitionofProbability（Section2.2）

GivenanexperimentwithoutcomespaceS,aprobabilityisafunctionPwhichassignsarealnumberP（A）toeacheventAS,subjecttothefollowingthreerules:

1.

2.

3.

Usingprobabilitiestomodel.

Probabilityrepresents

EmpiricalProbability

Theprobabilitymodel.

InterpretationsofProbability.

ObjectiveProbability

SubjectiveProbability

Note.Themathematicaldevelopmentislimitedtoobjectiveprobabilityandcautionshouldbetakenwhenapplyingresultstosubjectiveprobabilities.

Otherpropertiesoftheprobabilityfunction.AnyfunctionPthatsatisfiestheabovethreepropertiesalsosatisfiessomeotherusefulproperties.

Property1.Allprobabilitiesarebetween0and1.

Property2.Thecomplementrule.

Property3.Theadditionrule.

Example1.Apersonisapplyingfortwojobs.Theyfeelthattheprobabilitytheywillgetanofferforthefirstjobis0.6,theprobabilitytheywillgetanofferforthesecondjobis0.8,andtheprobabilitytheywillgetbothoffersis0.5.

a）Isthissubjectiveorobjective?

b）Calculatetheprobabilitytheywillgetatleastonejoboffer.

c）Calculatetheprobabilitytheywillgetnojoboffers.

Example2.Acertainsystemcanhavethreedifferenttypesofdefects.LetAidenotetheeventthatthesystemhasdefecti.SupposethatP（A1）=0.12,P（A2）=0.07,P（A3）=0.05,P（A1ÇA2）=0.06,P（A1ÇA3）=0.03,P（A2ÇA3）=0.02,andP（A1ÇA2ÇA3）=0.01.

a）Isthissubjectiveorobjective?

b）ExplainwhattheeventsA1ÇA2ÇA3andA1ÈA2ÈA3represent.

c）DrawaVenndiagramrepresentingtheprobabilities.

d）Whatistheprobabilitythatasystemdoesnothaveadefect?

e）Whatistheprobabilitythatasystemhasatleastonedefect?

f）Whatistheprobabilitythatasystemhasatleasttwodefects?

g）Whatistheprobabilitythatasystemhasallthreedefects?

CountingandProbability（Section2.3）.SupposeasamplespaceScontainsfinitelymanyelements,andassumethateachoftheseoutcomesisequallylikely.Inthiscase,givenanyeventAS,wecancalculatetheprobabilitythatAoccursinthisway:

MultiplicationPrincipleandTreeDiagrams.

Example3

a）Howmany1toppingpizzascanbemadeifyouhave3choicesforsize,2choicesforcrusttype,and12pizzatoppingstochoosefrom?

b）Howmanypizzascanbemadeifyoucanincludeanynumberoftoppings?

Permutations.

Example4.Ifthereare12playersonalittleleaguebaseballteam,howmanywayscanthe9fieldpositionsbeassigned?

Combinations.

Example5.Acommitteeof10wouldliketocreateasubcommitteeof3.Howmanydifferentwayscanthisbedone?

Hyper-geometricProbabilities.

Setting

Logic

Formula

Example6.Fifteentelephoneshavejustbeenreceivedatanauthorizedservicecenter（5cellular,5cordless,and5corded）.Supposethatthephonesareservicedinrandomorder.

a）Whatistheprobabilitythatallofthecellularphonesareamongthefirst6tobeserviced?

b）Whatistheprobabilitythatafterservicing6phones,phonesofonlytwoofthethreetypesremaintobeserviced?

c）Whatistheprobabilitythatafterservicing10phones,phonesofonlytwoofthethreetypesremaintobeserviced?

d）Whatistheprobabilitythattwophonesofeachtypeareamongthefirst6tobeserviced?

Example7（Section2.4）.Ashopkeeperphonestotellyouthatshehastwonewbabybeagles.Youaskherifshehasafemalepuppy,andsherepliesthatshedoes.However,onthewaytomeether,youchangeyourmindanddecidethatyouwouldratherhaveamalepuppy.Whatistheprobabilitythatshehasamalepuppy?

a）Guesstheanswer.

b）Doasimulationusingpennies.Recordtheempiricalprobabilityfromthesimulationusingresultsofwholeclass.

c）Giveoriginalsamplespaceandprobabilityassignmentandreducedsamplespaceandprobabilityassignment.Usethistogivetheexactprobability.

d）Notation.

ConditionalProbability.

IntuitionviaVennDiagrams.

FormalDefinition.

Example8.Supposeyourolltwofairsix-sideddice,oneofwhichisredandoneofwhichisblue.Yourecordthenumbersshowingoneachdie.LetA=“thebluediecomesup3”,B=“atleastoneofthedicecomesup3”,andC=“thesumofthediceis7”.Calculatethefollowing:

a）P（A|B）

b）P（B|A）

c）P（A|C）

d）P（C|A）

e）P（B|C）

f）P（C|B）.

MultiplicationRule.

Example9.Supposeourexperimentistodraw2cardsfromastandarddeckwithoutreplacementandrecordthesequenceofcardsselected（anorderedsamplewithoutreplacement）.LetB=“thefirstcardisaSpade”andletA=“thesecondcardisaSpade”

a）FindP（A|B）andP（A|B'）

b）ComputeP（AÇB）andP（AÇB'）.

Example10.Supposeyouknowthatexactly4lightbulbsoutofaboxof10bulbsaredefective.Youplantolocatethedefectivebulbsbydrawingthebulbsoutoneatatimeandtestingthem（withoutreplacement）.Findtheprobabilitythat

a）Youfindthe4thdefectivebulbonorbeforetheseventhtest.

b）Youfindthe4thdefectivebulbontheseventhtest.

WorkingwithConditionals.

▪ForsomeexperimentsyoucancalculateP（AÇB）andP（B）.ThiscanbeusedtofindP（A|B）.Whatexamplesdidwedothison?

▪OthertimesyoucancalculateP（B）andP（A|B）.ThiscanbeusedtofindP（AÇB）.Whatexamplesdidwedothison?

Example11.Supposethatapersonisselectedatrandomfromalargepopulationofwhich1%aredrugusersandthatadrugtestisadministeredthatis98%reliable.Ifapersontestspositivefordrugs,whatistheprobabilitythattheyareadruguser?

a）Whatdoyouthinktheansweris?

b）IntroduceNotation.

c）SimulationviaMaple.

d）Calculation.

IndependentEvents（Section2.5）.EventsAandBaresaidtobeindependentifP（A|B）=P（A）.Inotherwords,“ifweknowthatBoccurs,itdoesnotchangetheprobabilitythatAoccurs.”

MultiplicationRuleforIndependentEvents.

P（A⋂B）=P（A）P（B）.

Notes.

1）IfeventsAandBareindependent,itfollowsthatP（B|A）=P（B）.Thiscanbeshownusingdefinitionofindependence,conditionalprobability,and（general）multiplicationrule.（Thus,therolesofAandBaresymmetricinthedefinitionofindependence.）

2）IfAandBareindependentevents,thenthefollowingpairsofeventsarealsoindependent:

a）AandB'

b）A'andB

c）A'andB'

DeterminingDependenceorIndependence

Example12.•Considerexperiment“Drawonecardfromastandarddeck.”

a）LetA=“thecardisanace”andB=“thecardisaheart”.

AreAandBindependent?

b）LetA=“thecardistheaceofhearts”andB=“thecardisaheart”.AreAandBindependent?

Example13.Considertheexperiment“Rolltwosixsideddie.”

a）LetA=“Rollasumof5”andB=“Thefirstdieisathree”.AreAandBindependent?

b）LetA=“Rollasumof7”andB=“Thefirstdieisathree”.AreAandBindependent?

Example14.

a）Drawtwocardsfromastandarddeckwithoutreplacement,andrecordthesequenceofcardsdrawn.LetA=“thesecondcardisaspade”andB=“thefirstcardisaspade”.AreAandBindependent?

b）Whatissamplingwasdonewithreplacement?

Usingindependencetocalculateprobability.

Example15.•ConsideranelectricalcircuitwithcomponentsA,B,C,andD.Eachoperatesindependentlyandis90%reliable.ComponentsAandBareconnectedinseriesasarecomponentsCandD;thesetwounitsarethenconnectedinparallel.Findtheprobabilitythattheelectricalcircuitworks.

Example16.Theprobabilitythatagraderwillmakeamarkingerroronanyparticularquestionofatruefalseexamis0.1.Thereare10questionsandquestionsaremarkedindependently.

a）Whatistheprobabilitythatnoerrorsaremade?

b）Whatistheprobabilitythatexactlytwoerrorsaremade?

GeneralizationofExample16.•Supposeinsteadthattherearentruefalsequestionsandtheprobabilityofmakingamarkingerrorisp.

a）Whatistheprobabilitythatnoerrorsaremade?

b）Whatistheprobabilitythatexactlytwoerrorsaremade?

c）Whatistheprobabilitythatexactlykerrorsaremade?

BinomialProbabilities.

Setting

Logic（previousexample）

Formula

TheLastExample.A10questiontruefalsetestneedstobere-gradedifanyofthequestionsaregradedincorrectly.（Questionsaregradedindependentlyofeachother,eachwiththesameprobabilityofgradedincorrectly.）

a）If15%ofalltestsneedtobere-graded,whatistheprobabilitythataquestionisgradedincorrectly?

b）Howsmallshouldtheprobabilityofmis-gradingaquestionbetoensurethatonly5%ofexamsneedtobere-graded?

SummaryofChapter2

CountingRules

MultiplicationPrinciple

Permutations

Combinations

ProbabilityRules

Alwaysbetween0and1

RuleforComplements.

AdditionRuleForMutuallyExclusiveEvents.

GeneralAdditionRule

MultiplicationRuleforIndependentEvents

GeneralMultiplicationRule

ConditionalProbabilities

FindConditionalProbabilitiesfromthoseofsingleeventsandintersections.

FindProbabilitiesofintersectionsfromthoseofsingleeventsandconditionals.

IndependentEvents

Usemathematicaldefinitiontocheckifeventsareindependent.

Assumeeventsareindependentandcalculateprobabi