3.13 probability workbook 2018 - mathematics with ms walker · 2018-02-04 · theoretical...
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3.13Probability
Name:
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3.13Probability
Credits:4
Assessment:External
FormativeAssessmentdate:
Thisisanexternalexamination;youwillcompleteaformativeassessmentattheendofthisunit,amockexaminationinTerm3andtheactualpaperinTerm4NZQAexaminations.
Achievement AchievementwithMerit AchievementwithExcellence
• Applyprobabilityconceptsinsolvingproblems
• Applyprobabilityconceptsusingrelationalthinking,insolvingproblems.
• Applyprobabilityconcepts,usingextendedabstractthinking,insolvingproblems.
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Introduction
Thisstandardwillrequireyoutoapplyingprobabilityconceptsinsolvingproblems.Thiswillinvolvetheuseof:
•trueprobabilityversusmodelestimatesversusexperimentalestimates•randomness•independence•mutuallyexclusiveevents•conditionalprobabilities•probabilitydistributiontablesandgraphs•twowaytables•probabilitytrees•Venndiagrams
WhatisProbability?
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Thelanguageofprobability
Whatdothesetermsmean?
P(x>3)
P(x<4)
P(x=6)
P(x³2)
P(x£1)
P(3<x<7)
P(1£x£5)
Doesitmatterwhichwaythearrowsgointhelasttwo?
Page6-8
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ProbabilityRevision
Probabilitiescanbeexpressedasaprobability,fractionorapercentage.
Weusuallyuseprobabilitiesbecausetheyareeasytocompare.
____________
Combiningprobabilities
Thisisveryimportantandfrequentlycomesupinexams.
and____x___
or____+____
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=
=________
=________
pg11-12
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MultiplicationPrinciple
1.Withdifferentitems:egJeremiahhas6shirts,2tiesand4pairsoftrousersInhowmanywayscanhebedressed?
2.Whereitemsaresimilar(andallitemsareavailableforpositions)butnorepeatsareallowed:
eg10peoplearerunningarace-inhowmanywayscanthefirst3placesbefilled.
3.Whereitemsaresimilar,butrepeatsareallowed.egnumberofNewZealandlicenseplates
FactorialsHowmanydifferentwayscan4peoplesitonabench?
Howmanydifferentwayscan12peoplesitonabench?
Challenge:Howmanydifferentwayscan4peoplesitaroundacirculartable?
Pg15-16
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Methodsofdisplay
Howdowefindthisvalue?
Thetableneedstoaddto__________
What’sthedifferencebetweenaprobabilitytableandfrequencytable?________________________________________pg18-20
Howdowefindthisnumber?
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ProbabilityNotation
P(A)
P(A’)
P(AÇB)
P(AÇB’)
P(A∪B)
P(A’∪B)
P(A∪B)’orP(A’∩B’)
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2x2VennDiagrams
P(A)
P(A’)
P(A∩B)
P(A∪B)
P(A∪B)’orP(A’∩B’)
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Onyourformulasheet
Questionsfor2x2VennorTwowaycontingencytables
Thedatamaybegiveninfractions,decimals,probabilitiesor
frequencies(ienumbersof).
A A’
B
B’
Checkthatalltheprobabilitiesaddto_____________
FindP(A∪B)=
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A A’
B
B’
pg24-27
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3x3VennDiagrams
pg30
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3x3Frequencytable
pg31-35
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Probabilities from Tables of Counts
pg37-39
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TreeDiagrams
Thesecanbeusedasanalternativetovenndiagramsandtables,theyareparticularlyusefulwhenthereisasequenceofevents.
1. Firstidentifytheeventsareandwhichordertheygoin
vaccinatedandflu
Whichislikelytohappenfirst?vaccinated
2. Addtheeventsandtheprobabilitiestothetreediagrambelow
Nowanswerthequestions
pg42-48
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ComplementaryEvents(AandA’)If A is an event, then A’ (not A) is the complementary event. If A doesn’thappenthenA’doeshappen.
P(A)+P(A’)=1
Example:Pickingaredcardandpickingablackcardfromapack(nojokers)
P(R)+P(B)=1
InGeneral:
P(A)+P(A')=P(AÈA')=1
P(AÇA')=0
Thinkofsomeexamplesofacomplementaryevents:
• _____________________________________________________
• _____________________________________________________
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Minustheintersection,howeverits0
MutuallyExclusiveEvents
Mutuallyexclusiveeventscannotbothoccur.InaVennDiagramthecirclesdon’toverlap.
P(A∪B)=P(A)+P(B)
Example:Heartsandblackcardsaremutuallyexclusive.
Findtheprobabilityofpickingaheartorablackcard.
P(H)+P(B)=¼+½=0.75
B B' H 0 .25 .25H' .5 .25 .75 .5 .5 1.0
InGeneral:
P(AÈB)=P(A)+P(B)
and
P(AÇB)=Ø
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Onyourformulasheet
Thisboxwouldbe0formutuallyexclusiveevents
EventswhichareNOTMutuallyExclusive
EventswhichareNOTmutuallyexclusivehave
overlappingsets
A B
P(AÈB)=P(A)+P(B)–P(AÇB)
Youcan’tcounttheintersectiontwice.
Example:Theprobabilityofdrawingaheartora6fromapackofcards.
Hearts 6 of Hearts
Sixes
P(Hand6)=¼x1/13=0.019
6 6’ H 0.019 0.231 0.25H’ 0.058 0.692 0.75 0.077 0.923 1.0
InGeneral:
P(AÈB)=P(A)+P(B)–P(AÇB)
Complementaryeventsaremutuallyexclusive
Howevermutuallyexclusivearenotnecessarilycomplementary.
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QuestionOne
Onaparticularday,P(rain)=0.27,P(heavywind)=0.24,P(neither)=0.64.
Aretheseeventsmutuallyexclusive?
I I' B .15 .09 .24B' .12 .64 .76 .27 .73 1.0
QuestionTwo
Theprobabilityofapersonenjoyinggardeningis0.43andtheprobabilityofapersonenjoyingplayingcomputergamesis0.32.Theprobabilitythatapersonenjoysneithergardeningnorcomputergamesis0.38.Aretheseeventsmutuallyexclusive?
QuestionThree
16peoplestudyFrench,21studySpanishandthereare30altogether.AretheeventsstudyingFrenchandSpanishmutuallyexclusive?
HowmanystudyonlySpanish?
QuestionFour
Inasampleof135shoppers,29boughtapples,34boughtcolaand72boughtneither.Aretheevents‘boughtapples’and‘boughtcola’mutuallyexclusive?
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ConditionalProbabilityP(A/B)
Conditional probabilities are those that are influenced by the occurrence or non-occurrence of previous events.P(A/B)=InwordsthismeanstheprobabilitythatAhasoccurredgiventhatB
hasoccurred
P(B/A)=InwordsthismeanstheprobabilitythatBhasoccurredgiventhatAhasoccurred
Theformulaforcalculatingthiscanbeobtainedfromaprobabilitytree:
P(BÇA)
Onyourformulasheet
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pg55-58
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Ifthisistruethetwoeventsareindependent
IfthisisthecasethenthetwoeventsareNOT
independent
IndependentEvents
Independenteventsarewhentheoccurrenceononehasnoeffectontheother.
P(AÇB)=P(A)xP(B)
P(AÇB)≠P(A)xP(B)
Example:
P(havingabirthdayinDecember) P(becomingaprefect)
Wewouldexpectthesetwoeventstobeindependent.
InGeneral:
KnowingthatBhasoccurredmakesnodifferencetotheprobabilityofAoccurring.
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pg61-64
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Exercises on Probability Tables
1. P(A) = 0.4, P(B) = 0.5 P(A∩B) = 0.1
A A’
B 0.1 0.5
B’
0.4 1
Complete the table and answer the following questions.
(a) What is the probability that A or B occurs?
(b) What is the probability that neither A nor B occurs?
(c) Are A and B independent?
(d) Find P(A|B)
(e) Are A and B mutually exclusive?
2. P(A) is 0.4, P(B) is 0.6, and A and B are independent events.
A A’
B 0.6
B’
0.4 1
Complete the table and answer the following questions.
(a) What is the probability that A or B occurs?
(b) What is P(B|A’)?
(c) Are A and B mutually exclusive?
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3. P(A) is 0.6, P(A∩B) = 0.3, and A and B are independent events.
A A’
B 0.3
B’
0.6 1
Complete the table and answer the following questions.
(a) What is the probability that of A or B occurs?
(b) What is the probability that neither A nor B occurs?
(c) What is P(A|B)?
4. P(A) = 0.45, P(B) = 0.6, P(AÈB) = 0.9
A A’
B
B’
Complete the table and answer the following questions.
(a) Are A and B mutually exclusive?
(b) What is the probability that neither A nor B occurs?
(c) What is P(A|B)?
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5. P(AUB) = 0.7, P(A) = 0.5, P(B) = 0.4
A A’
B
B’
Complete the table and answer the following questions.
(a) What is the probability that both of A and B occur?
(b) What is the probability that neither A nor B occurs?
(c) What is P(A|B)?
(d) Are A and B independent events?
6. P(A) = 0.4, P(B) = 0.5, P(A|B) = 0.5
A A’
B
B’
Complete the table and answer the following questions.
(a) What is the probability that A or B occurs?
(b) What is the probability that neither A nor B occurs?
(c) What is P(B|A)?
(d) Are A and B independent?
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AbsoluteRisk(Risk)Riskorabsoluteriskisthechanceofaeventoccuring.Thisisusuallyexpressedasadecimalorpercentage.
Riskisusuallyassociatedwithsomethingbad(cancer,death,foodpoisoningetc).
Riskcanbearatio
EgTheriskofbeingstruckbylighteningis4.5per1,000,000people
Thereforetheriskis0.0000045or0.00045%
AsitisaprobabilityRiskwillalwaysbebetween0and1
What is the absolute risk of getting heart disease in each group?
Risk of heart disease in an overweight
male
___________________________________
___________________________________
Risk of heart disease in an smoking
male
___________________________________
___________________________________
Risk of heart disease in a non-
overweight male
___________________________________
___________________________________
Risk of heart disease in a non-smoking
male
___________________________________
___________________________________
What conclusion can you come to from this?
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RelativeRiskRelativeRiskisusedtocomparetheriskfortwogroups.Itgivesameasureoftheimpactofthebehaviourortreatmentgroupisexposedto.
Usuallythebaselinegroup(denominator)isthenon-treatmentgroup(thiswillbeinthewordingofthequestion).
RelativeRisk = Riskfortreatmentgroup
Riskfornontreatmentgroup
Relativeriskdoesnothavetobebetween0and1
RRof2.5means=youare2.5timesmorelikelyto...(ora150%increase)
RRof1means=youareequallylikelyto......
RRof0.8means=thereis0.8timesthechanceof....(adecreasedriskegexerciseonthechanceofaheartattack)(Thisis20%LESSriskforthosethatexercise)
or0.8timesaslikely.....
What is the Relative Risk of heart
disease in an overweight male
compared with a non-overweight male.
___________________________________
___________________________________
What can we conclude?
___________________________________
___________________________________
___________________________________
___________________________________
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What is the Relative Risk of heart
disease in an smoking male compared
with a non-smoking male.
___________________________________
___________________________________
What can we conclude?
___________________________________
___________________________________
___________________________________
___________________________________
RelativerisktellsnothingofACTUALrisk
Struckbylightning
NotStruck Total
Hat 3 1000000
NoHat 1 2000000
Whatistherelativeriskofbeingstruckbylightningwhilewearingahatcomparedwithnotwearingahat?
Whatdoyounowconclude?
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pg67-72
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TheoreticalvsExperimentalvsActualProbability
TheoreticalProbability
TheoreticalProbabilityalsoknownasmodelprobabilityiswhenweuseatooltofindoutthetheoreticalprobabilityofaneventoccurring
Example:
Thetheoreticalprobabilityofrollinga3onadiceis56or
0.16666or16%
ExperimentalProbability
Experimentalprobabilityisbasedonthenumberoftimestheeventoccursoutofthetotalnumberoftrials.
Example:
Samrolledadice50times.A3appeared10times.
Thentheexperimentalprobabilityofrollinga3is10outof50or20%.Inprobabilityanexperimentisoneormoretrialsofaprobabilitysituation.
ActualProbability
Alsoknownastrueprobabilityisthe(almostalways)unknownactualprobabilitythatan
eventwilloccurinagivensituation.
Example:
Theactualprobabilityofacoinlandingheadsupisaffectedbythepositionfromwhichitistossed,theasymmetryofthetwofacesofthecoinetc,soisnotexactly0.5,thoughtheprobabilityofafaircoinlandingheadswillbeverycloseto0.5.
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NZQA Sample Paper Question Emma was given a set of six different keys to her new house. One of the keys opened the dead lock on the front door, and a different key opened the door lock on the front door. Emma did not know which keys were the correct keys for each lock, and was able to open both locks after four key attempts. Emma’s friend Sene thought this was a low number of key attempts, and wondered what process Emma used to find the correct keys. Sene designed and carried out a simulation to estimate how many key attempts it would take before both locks were open, using a ‘trial and error’ process, to see if this might have been the process Emma used.
For the ‘trial and error’ process, Sene assumed:
• that a key was selected at random to try to open the dead lock
• once Emma had tried a key for the dead lock, she did not try it again
• once Emma found the correct key for the dead lock, she removed this from the set of keys and tried the same process with the door lock.
The results of Sene’s simulation are shown below:
Numberofkeyattemptsbefore
bothlocksareopen Frequency
2 29
3 70
4 103
5 136
6 164
7 176
8 134
9 74
10 76
11 38
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(a) Calculate the theoretical probability of a person using a 'trial and error’ process taking more than two key attempts before both locks are open. Compare this probability with the results from Sene’s simulation and discuss any differences.
(b) Sene used the central 90% of her simulation results to check if the number of key attempts Emma
took (four) was likely if the ‘trial and error’ process was used, and concluded it was. Discuss if the results of Sene’s simulation support this conclusion.
(c) Calculate the theoretical probability that a person using Emma’s process takes four key attempts before both locks are open.
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Randomness
ActivityOne
Plot3'randompoints'onpaper
Plot5'randompoints'onpaper
Howdotheycompare?
ActivityTwopg74
Ineedtopickwhichoneisthefakeandwhichisreal.