Ch05 - Probability Concepts 3 Subjective Probability •If there is little or no past experience or...
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9/5/17 1 Probability Concepts Chapter 5 Dr. Richard Jerz © 2017 rjerz.com 1 Goals • Probability Concepts • Define probability. • Explain the terms experiment, event, and outcome • Graph event probability: probability distribution • Define the terms conditional probability and joint probability. • Calculate probabilities using the rules of addition and rules of multiplication. • Graph multiple events: tree diagram • Describe the classical, empirical, and subjective approaches to probability. • Calculate the number of arrangements, permutations, and combinations. © 2017 rjerz.com 2 What is probability? • Probability is a measure of the likelihood that an event in the future will happen. It can only assume a value between 0 and 1, or 0 and 100% • A value near zero means the event is not likely to happen. A value near 1 means it is likely. © 2017 rjerz.com 3
Transcript of Ch05 - Probability Concepts 3 Subjective Probability •If there is little or no past experience or...
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ProbabilityConcepts
Chapter5Dr.RichardJerz
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Goals
• ProbabilityConcepts• Defineprobability.• Explainthetermsexperiment,event,andoutcome• Grapheventprobability:probabilitydistribution• Definethetermsconditional probabilityandjointprobability.
• Calculateprobabilitiesusingtherulesofadditionandrulesofmultiplication.
• Graphmultipleevents:treediagram• Describetheclassical,empirical,andsubjectiveapproachestoprobability.
• Calculatethenumberofarrangements,permutations,andcombinations.
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Whatisprobability?
• Probability isameasureofthelikelihoodthataneventinthefuturewillhappen.Itcanonlyassumeavaluebetween0and1,or0and100%
• Avaluenearzeromeanstheeventisnotlikely tohappen.Avaluenear1meansitislikely.
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ProbabilityExamples
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ExamplesofBusinessQuestions
• Ifwereviseourproduct,willcustomerspurchaseit?
• Shouldweguaranteeourcompany’stirestolast40,000miles?
• ShouldIhireanewemployee?• IfIamdealtfivecards,willIgetaroyalflush?• Willthenextdicethrowproducea4?• Willthetossofacoinbeheadsortails?• WillHillaryClintonbethenextpresident?• Canastudentguessona10questionmultiplechoicetestandearnan85%?
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TheThreeWaysofAssessingProbability
• Subjective: A Guess,baseduponintuitionorjudgment
• Classical:Baseduponcountingallpossibilitiesthatareequallylikelywithinapopulation(suchasrollingadice)
• Empirical:Baseduponrelativefrequencyofsampleddata(suchasourM&Mdata)
• Classicalandempiricalare“objective”(i.e.,measuredorcalculated)
• InBusinessStatistics:Objective© 2017 rjerz.com6
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SubjectiveProbability
• Ifthereislittleornopastexperienceorinformation(data)onwhichtobaseaprobability,itmaybearrivedatsubjectively.
• Examplesofsubjectiveprobabilityare:• EstimatingthelikelihoodtheNewEnglandPatriotswillplayintheSuperBowlnextyear.
• Estimatingthelikelihoodthatanewcompetitorwillenterthemarketplace.
• EstimatingthelikelihoodtheU.S.budgetdeficitwillbereducedbyhalfinthenext10years.
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EmpiricalProbability
• Don’thaveallthepopulationdata,wecollectsample data
• Biggersamples arebetter(moreaccurate).“Lawoflargenumbers.”
• Theprobabilityofaneventhappeningisthepercentorfractionofthetimesimilareventshappenedinthepast
• Calculatedbydividingtheoccurrencesoftheeventofinterestbythetotalsamplesize.
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ClassicalProbability
• Wecancountallpossibleoutcomes oftheexperiment
• Wecancountallpossibleoccurrences oftheevent(favorableoutcomes)thatweareinterestedin
• Thecalculatedprobabilityisthedivisionofeventoutcomebyallpossibleoutcomes
• Challenge:canyoucountthese“outcomes?”© 2017 rjerz.com9
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SummaryofTypesofProbability
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HelpfulDefinitions
• Anexperiment istheobservationofsomeactivityortheactoftakingsomemeasurement.
• Anoutcome istheparticularresultofanexperiment.
• Anevent isthecollectionofoneormoreoutcomesofanexperiment.
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Experiments,EventsandOutcomes
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M&MExperiment
• Experiment:pickanM&Mfromabag
• Outcomes:red,blue,brown,green,orange,oryellow
• Events:• aredM&M• aredororangeM&M• aprimarycolorM&M
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M&MExperiment:EmpiricalorClassical?
• Itdepends…
• Onebagastheentirepopulation– classical• Treatingthedataasasample- empirical
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ChartTypesforM&M’s“ProbabilityDistribution”
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FrequencyDiagram16%
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PieChart
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RelativeFrequencyDiagram
ProbabilityDistribution
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UsingaProbabilityDistribution
• Experiment:PickanM&Mfromthebag
• Outcomes:Blue,brown,green,orange,yellow,red
• Probabilityofevent:• “Red”=26%• “Redororange”=42%• “Primarycolor”=63%• “Redandorange”=0%
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16% 16%
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UsingaProbabilityDistribution
• Experiment:YellowM&Minabag
• Outcomes:0,1,2…,7,8yellowM&Msinbag
• Probabilityofevents:• Exactly2=16.7%• Lessthan4=22.3%• Greaterthan4=44.5%• Between2and5=33.3%
• Anevennumber=66.7%
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BagCo
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#inBag
#ofBagswithYellowM&M's
5.6%16.7%
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33.3%22.2%
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%ofBagswithYellowM&M's
DiceTossProbabilityDistribution
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AnEvenNumber
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0%10%20%30%40%50%60%70%80%90%100%
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TossanEvenNumber
ANumberGreaterthan4
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0%10%20%30%40%50%60%70%80%90%100%
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ANumberGreaterthan4
ANumberLessthan3
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0%10%20%30%40%50%60%70%80%90%100%
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ANumberLessthan3
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TheComplement Rule
• Thecomplementrule isusedtodeterminetheprobabilityofanevent“notoccurring”bysubtractingtheprobabilityoftheeventoccurringfrom1.
P(A)+P(~A)=1orP(A)=1- P(~A).
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DiagramingtheExperiment
• Multipleformsofquestionscanbeasked:
“not red”,“not redororange”,“not primarycolor”
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16% 16%
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RelativeFrequencyDiagram
Howdowecalculatetheprobabilityofanevent?
• Whatisthe“experiment”(natureofproblem)?
• Whatareallpossibleoutcomes(totalcount)?• Whichoutcomesarecontainedinthe“event”ofinterest?
• Mutuallyexclusive:Arewe“doublecounting”anything?
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ComputingEventProbabilitiesRuleofAddition
RulesofAddition• SpecialRuleofAddition- Iftwo
eventsAandBaremutuallyexclusive,theprobabilityofoneORtheotherevent’soccurringequalsthesumoftheirprobabilities.
P(AorB)=P(A)+P(B)
• TheGeneralRuleofAddition- IfAandBaretwoeventsthatarenotmutuallyexclusive,thenP(AorB)isgivenbythefollowingformula:
P(AorB)=P(A)+P(B)- P(AandB)
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JointProbabilityVennDiagram
• JOINTPROBABILITY:Aprobabilitythatmeasuresthelikelihoodtwoormoreeventswillhappenconcurrently.P(AandB)
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AdditionRule- Example
• Whatistheprobabilitythatacardchosenatrandomfromastandarddeckofcardswillbeeitherakingor aheart?
• P(AorB)=P(A)+P(B)- P(AandB)• =4/52+13/52- 1/52• =16/52,or.3077
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AdditionRule- Example
• Doesjointprobabilityexist?
“red”,“redororange”,“redand orange”
“redor primarycolor”
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16% 16%
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RelativeFrequencyDiagram
ProbabilityofMultipleEvents
• ThespecialruleofmultiplicationrequiresthattwoeventsAandBareindependent.
• Thisruleiswritten:• P(AandB)=P(A)P(B)
• ThegeneralruleofmultiplicationisusedwhentheprobabilityofeventBisinfluenced,orconditioned,bytheoccurrenceofeventA.
• Thisruleiswritten:• P(AandB)=P(A)P(B|A)
(theconditionalprobabilityofB)
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Example:GeneralMultiplicationRule
• Agolferhas12golfshirtsinhiscloset.Suppose9oftheseshirtsarewhiteandtheothersblue.Hegetsdressedinthedark,sohejustgrabsashirtandputsiton.Heplaysgolftwodaysinarowanddoesnotdolaundry.
• Whatisthelikelihoodbothshirtsselectedarewhite?
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GeneralMultiplicationRule-Example
• TheeventthatthefirstshirtselectediswhiteisW1andthesecondshirtisW2
• TheprobabilityisP(W1)=9/12• P(W2)isP(W2|W1)=8/11• Todeterminetheprobabilityof2whiteshirtsbeingselectedweuseformula:P(AandB)=P(A)P(B|A)
• P(W1andW2)=P(W1)P(W2|W1)=(9/12)(8/11)=0.55
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IndependentEventsConditionalProbability
• Questions,withorwithoutreplacement
red,thenanotherred
red,thenanorange
threeredsonthreetries
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16% 16%
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RelativeFrequencyDiagram
TreeDiagramsforIndependentEvents
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CalculatingProbabilities:WhatistheQuestion?
• Aking• Aheart• Akingoraheart• Akingandaheart• Aheart,ifthekingisnotreplaced• Aking,thenaheart(withoutreplacement)• Aking,thenaheart(withreplacement)• Many“not”questions,i.e.,notaking
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ClassicalProbability
• Consideranexperimentofrollingasix-sideddie.Whatistheprobabilityoftheevent “anevennumber ofspotsappearfaceup”?
• Thepossibleoutcomes are:
• Therearethree“favorable”outcomes(atwo,afour,andasix)inthecollectionofsixequallylikelypossibleoutcomes.
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“Randomness”StillExists!• Supposewetossafaircoin.Theresultofeachtossiseitheraheadoratail.Ifwetossthecoinagreatnumberoftimes,theprobabilityoftheoutcomeofheadswillapproach.5.Thefollowingtablereportstheresultsofanexperimentofflippingafaircoin1,10,50,100,500,1,000and10,000timesandthencomputingtherelativefrequencyofheads
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CountingallOutcomes
• Forclassicalprobability,weneedtocountthetotalnumberofoutcomes.
• Waysofcounting• Arrangements• Permutations• Combinations
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ArrangementExample
• Anautomobiledealerwantstoadvertisethatfor$29,999youcanbuyaconvertible,atwo-doorsedan,orafour-doormodelwithyourchoiceofeitherwirewheelcoversorsolidwheelcovers.Howmanydifferentarrangementsofmodelsandwheelcoverscanthedealeroffer?
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Arrangements:MultiplicationFormula
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AnotherExample:Dell
• Websiteforlaptops• HowmanydifferentmodelsdoesDellhaveforitsInspironlaptop?
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AnotherExample:RollTwoDice
• Whatarealltheoutcomes?• Whatistheprobabilityofrollinga5?A7?
• Solution:6possibilitiesforthefirstdie,and6forthesecond,therefore6x6or36.
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Results
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PaintingaRoom
• Ourroomhasfourwalls• Wehave4differentcolorsofpaint
• Howmanywayscanwepainttheroom?
• Howwouldthisdifferifwehad6colorsofpaint?
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Factorialofn
• Denotedasn!• n!=n(n-1)(n-2)…1
• Thus4!=4*3*2*1=24
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Counting- Permutation
Apermutation isthenumberofwaystochooser objectsfromagroupn possibleobjectswheretheorderofarrangementsisimportant(abcandcbaaredifferent).
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CountingArrangementsforSubgroupsw/oReplacement
• Howmanywayscanwepick4colorsfromatotalof6colors?
• Permutations(withorder)• Combinations(withoutorder)
• Note:#ofpermutations>=#combinations
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Counting- Combination
• Acombination isthenumberofwaystochooserobjectsfromagroupofnobjectswithoutregardtoorder(abcandcbaarenotdifferent).
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Example:Combination
Thereare12playersontheCarolinaForestHighSchoolbasketballteam.CoachThompsonmustpickfiveplayersamongthetwelveontheteamtocomprisethestartinglineup.Howmanydifferentgroupsarepossible?
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792)!512(!5
!12512 =
-=C
Example:Permutation
Supposethatinadditiontoselectingthegroup,hemustalsorankeachoftheplayersinthatstartinglineupaccordingtotheirability.
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040,95)!512(!12
512 =-
=P
UsingExceltoCount
• CombinationsandPermutations
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