What are the
chances?
Student Worksheet
789101112
TI-Nspire
Investigation
Student
90min
Introduction
Inprobability,weoftenlookatlikelihoodofeventsthatareinfluencedbychance.Considertheeventof
rollingapairofsix-sideddiceandobservingthesumofthetwonumbersthatturnup.Wecanobserve
whatmighthappenifwerollthediceafixednumberoftimes(theexperimentalprobability),andthenwe
canusemathematicstolookatwhatwemightexpecttohappeninthelongrun(thetheoretical
probability).
Part 1: Simulating chance events
Mostcalculatorsalsohavearandomintegercommand,whichgeneratesrandomintegersbetweentwo
limitingvalues.Forexample,thecommandrandInt(1,20)ontheTI-NspireCASwillgeneratearandom
integerbetween1and20(inclusive).
TodothisontheTI-NspireCAS
•
•
•
•
PressHOME-1tocreateanewdocument,andthen
press1toaddaCalculatorpage.
PressMENU-5toselectProbability,thenpress4
(Random)andthen2(Integer).
Typerandint(1,20)andthenpress·todisplaya
randomintegerbetween1and20.
PressENTERafewtimestogeneratesomeextra
randomnumbers.
Anumberofrandomintegerswillbegenerated.
Tobeginwith,wewillsimulatetheprocessofrollingofasingledie.
Question1.
Whatcommandwouldsimulatethisexperiment?randInt(1,6)
Question2.
Nowtrythecommand60times,andrecordtheresultsinafrequencytable(asshownbelow).
Diescore
1
2
3
4
5
6
Tally
Frequency
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
2
Whatarethechances?–StudentWorksheet
Question3.
Observeyourtableresults.Whichscoreoccurredmostfrequently?
Answerswillvary,butthefrequencyofeachdiescorewillbecloseto10.Giventhatonly60trialsare
done,theremaystillbesignificantvariationinthefrequencies.
Question4.
Compareyourresultswithsomeotherstudentsnearyou.Whatdoyounotice?
Answerswillvary,butthefrequencyofeachdiescorewillbecloseto10.Giventhatonly60trialsare
done,theremaystillbesignificantvariationinthefrequencies.
Tosimulatetherollingoftwodice,andaddingtheresultingnumbers,wecanaddtworandomintegers.
Question5.
Whatcommandwouldsimulatethesumofthetwonumberswhentwodicearerolled?
randInt(1,6)+randInt(1,6)
Question6.
Trythiscommandanumberoftimesonyourcalculator.
a) Whatisthesmallestvalueyoucanobtainasasumofthetworandomintegers?2
b) Whatisthelargestvalueyoucanobtainasasumofthetworandomintegers?12
Question7.
Nowtrythecommand100times,andrecordtheresultsinafrequencytable(asshownbelow).
Diesum
Tally
Frequency
Question8.
Commentonthedistributionofresultsyouobtainedhere.
Thediesumfrequencieswillvary,butthemiddlesums(e.g.6,7and8)shouldoccurmostfrequently.
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
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Whatarethechances?–StudentWorksheet
Part 2: Analysing results graphically
Generatingatableofresultsforanexperimentsuchasthiswouldbetimeconsumingifwewereto
conductalargenumberoftrials,andsowewillusethecalculatortohelpusstore,visualiseandanalyse
theresultsusingstatisticalfeatures.Tobeginwith,notethatthecommandrandInt(1,6,600)→diescore
willgenerate600integersbetween1and6,andstoretheresultsinastatisticalvariablenameddiescore.
Thiscanbeusedtogenerateandstoretheresultsofalargenumberoftrialsforfurtheranalysis.
TodothisontheTI-NspireCAS
•
•
PressMENU-5toselectProbabilityandthenpress4
(Random)andthen2(Integer).
Typethefollowingtocompletethecommand
randInt(1,6,600)→diescore
(NotethatthekeyforstoringtoavariableisCTRL-VAR.)The
resultsofthe600trialsaredisplayedasabracketedlistas
shownright.
Question9.
Howmanytimeswouldyouexpectthenumber1tooccurinthese600trials?Explainyourreasoning.
Wewouldexpectascoreof1tooccurabout100trialsoutof600(inthelongterm).
Itisdifficulttoscrollthroughthislistof600numberstocountthefrequencyofeachnumber.
However,thecalculatorhasacountingfeaturethatcanprovidethisinformation.Forinstance,the
commandcountif(diescore,1)willreportthefrequencyofthenumber1inthe600storedtrials.
[Note:ThecountifcommandcanbefoundbypressingtheCATALOGkey–orjusttypeitdirectly).]
Question10. Usethiscommandtocompleteafrequencytableforthe600trials.
Diescore
1
2
3
4
5
6
Tally
Frequency
Question11. Comparethesefrequencieswithyouranswertoquestion9.Aretheseresultswhatyouexpected?
Explainyouranswer.
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
4
Whatarethechances?–StudentWorksheet
Wecanobtainarelativefrequencyofeachscorebydividingthefrequencybythetotalnumberoftrials.
Question12. Usethismethodtocalculatetherelativefrequencyofobtainingadiescoreof1.Usetheformulathat
followstoassist.
Relativefrequencyofscoring1=
Frequencyofscoring1
Numberoftrials
Theexperimentalfrequencyofscoringa1willbecloseto,butnotnecessarilyexactly100,sothe
relativefrequencywillbecloseto1/6or0.1666...
Question13. Copyandcompletethisfrequencytableforyourpreviousresults.
Diescore
1
2
3
4
5
6
Tally
Frequency
Relative
frequency
(fraction)
Relative
frequency
(decimal
approx.)
Question14. Whatwouldyouexpecttherelativefrequencyofadiescoreof1togetclosetoifyouconductedthis
trialaverylargenumberoftimes?Expressyouranswerasafraction.
Therelativefrequencywillbecloseto1/6or0.1666...
Theresultsoftheexperimentcanbeviewedgraphically.Todothis,wewillcreateahistogramofthe
results.
OntheTI-NpireCAS
•
•
•
PressMENU-5toaddaDataandStatisticspage.
Clickatthebottomofthescreentoaddthevariable
diescoretotheplottingscreen.(Thiscreatesadotplot
oftheresultsofthe600trials.)
PressMENU-1toselectPlotTypeandselect3toselect
Histogram.
Theresultsofthe600trialsaredisplayedasahistogram.
[Notethatthecolumnsarefornumbersintherange1–<2,2–<3,
andsoon,ratherthancentredonthenumbers1to6.
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
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Whatarethechances?–StudentWorksheet
Question15. Howwouldyouexpectthishistogramtochangeinappearanceifyouweredisplayingtheresultsofa
muchlargernumberoftrials?
Itwouldbeexpectedthatthehistogramcolumnsbecomecloserinheight;thatis,theirfrequencies
willconvergetosamenumberasthenumberoftrialsincrease.Thisisbecauseeachscoreonthedie
isequallylikely.
Nowwewillusethismethodtoanalysethesumoftwodice.
Question16. WhatcommandusingtherandIntfunctionwillsimulatetheresultsof1000rollsoftwodice,and
storetheresultsinastatisticalvariablenameddiesum?
randint(1,6,1000)+randint(1,6,1000)→diesum
PressCTRL-LEFTARROWtore-displaytheCalculatorpage,andenterthiscommand.
Question17. Usethecountifcommandtocalculatethefrequencyofasumof2inthe1000storedtrials.
Answerswillvary,buttheexpectednumberwouldbe1/36x1000,orabout28.
Question18. Copythistableandusethiscommandtocompletethetableforthe1000trials.
Diesum
2
3
4
5
6
7
8
9
10
11
12
Tally
Frequency
Relative
frequency
(fraction)
Relative
frequency
(decimal
approx.)
Question19. Aretheserelativefrequencieswhatyouexpected?Explainyouranswer.
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
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Whatarethechances?–StudentWorksheet
Question20. UseyourCAStodisplayahistogramoftheseresults.Drawasketchofthishistogramandcommenton
itsshape.Clickoneachcolumntoverifythefrequencieslistedinthetableabove.
Samplescreenshownbelow
Part 3: Long-term relative frequency
Toworkoutwhatweexpecttohappeninthetwo-dicesumexperimentinthelongrun,wewilllookatall
thepossibleoutcomeswhentwodicearerolled.
Question21. Copyandcompletethefollowingtabletodisplayallthepossibleoutcomeswhenrollingtwodice.
Die2score
Die1score
1
2
3
4
5
6
1
1,1
2,1
3,1
4,1
5,1
6,1
2
1,2
2,2
3,2
4,2
5,2
6,2
3
1,3
2,3
3,3
4,3
5,3
6,3
4
1,4
2,4
3,4
4,4
5,4
6,4
5
1,5
2,5
3,5
4,5
5,5
6,5
6
1,6
2,6
3,6
4,6
5,6
6,6
Question22. Forthisexperiment,howmanypossibleoutcomesarethereintotal?36
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TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan
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Whatarethechances?–StudentWorksheet
Theprobabilityofaparticularsumcanbecalculatedasfollows.
Probabilityofasumof3 = Numberofoutcomeswithasumof3
Totalpossibleoutcomes
2
=
36
≈ 0.056
Question23. Usetheabovemethodtocalculatetheprobabilityofobtainingasumof2,3,4...12,andthen
completethetablebelow.
Sum
2
3
4
5
6
7
8
9
10
11
12
No.ofways
toobtain
thissum
1
2
3
4
5
6
5
4
3
2
1
Probability
(fraction)
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Probability
(decimal)
0.028 0.056 0.083 0.111 0.139 0.167 0.139 0.111 0.083 0.056 0.028
Question24. Comparethesetheoreticalprobabilitieswiththerelativefrequenciesobtainedinyouranswerto
question17and18,andcomment.
©
TexasInstruments2016.Youmaycopy,communicateandmodifythismaterialfornon-commercialeducationalpurposes
providedallacknowledgementsassociatedwiththismaterialaremaintained.
Author:D.Tynan