Probabilistic Thinking: Presenting Plural Perspectives

746 Pages·2014·13.711 MB·English

by  Egan J. Chernoff, Gale L. Russell (auth.), Egan J. Chernoff, Bharath Sriraman (eds.)

Advances in Mathematics Education
SeriesEditors:
GabrieleKaiser,UniversityofHamburg,Hamburg,Germany
BharathSriraman,TheUniversityofMontana,Missoula,MT,USA
InternationalEditorialBoard:
UbiratanD’Ambrosio(SãoPaulo,Brazil)
JinfaCai(Newark,NJ,USA)
HelenForgasz(Melbourne,Victoria,Australia)
JeremyKilpatrick(Athens,GA,USA)
ChristineKnipping(Bremen,Germany)
OhNamKwon(Seoul,Korea)
Forfurthervolumes:
www.springer.com/series/8392
Egan J. Chernoff (cid:2) Bharath Sriraman
Editors
Probabilistic Thinking
Presenting Plural Perspectives
Editors
EganJ.Chernoff BharathSriraman
CollegeofEducation Dept.ofMathematicalSciences
UniversityofSaskatchewan TheUniversityofMontana
Saskatoon,Saskatchewan,Canada Montana,MT,USA
ISSN1869-4918 ISSN1869-4926(electronic)
AdvancesinMathematicsEducation
ISBN978-94-007-7154-3 ISBN978-94-007-7155-0(eBook)
DOI10.1007/978-94-007-7155-0
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2013956368
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Series Preface
TheseriesAdvancesinMathematicsEducationaimsmainlytoproducemonographs
basedonimportantissuesfromZDM–TheInternationalJournalonMathematics
Educationfromthepast;however,“theseriesisopentoproposalsfromthecommu-
nityonothertopicsofinteresttothefield.”Probabilityhas,morerecently,becomea
mainstreamstrandwithinworldwidemathematicscurriculumand,further,isacon-
tinualburgeoningareaofresearchinmathematicseducation.Giventheformerand
latterpoints,theseventhvolumeintheseriesAdvancesinMathematicsEducation
dealswithprobabilisticthinking.
This seventh volume exemplifies that Advances in Mathematics Education ac-
ceptsproposalsfromthecommunityontopicsofinteresttothefieldwhilepreserv-
ingmanyofthecharacteristicsofZDM(sinceitsinceptionin1969).Asmentioned
in the Series Preface to the first book in the series on Theories of Mathematics
Education,thepublicationofthemedissuesascharacteristicofZDMaimstobring
thestate-of-the-artoncentralsub-domainswithinmathematicseducation.Similarly,
this volume is thematic; in that it is based on four different, yet interrelated “per-
spectives”orcentralsub-domainswithinprobability:MathematicsandPhilosophy,
Psychology,Stochastics,andMathematicsEducation.Further,thisvolumecontin-
ues “the usage of the ancient scholarly Chinese and Indian traditions of commen-
taries” (Kaiser and Sriraman 2010, p. vi) and solicits “commentaries from experts
andnovices(ibid.).”Asisalsothecaseinpastvolumes,“prefacestochapterssetthe
stageforthemotivation,purposeandbackgroundofagiven[perspective]”(ibid.).
Lastly,althoughnotdirectlybasedonpreviouslypublishedthemedissuesofZDM,
thisvolumereliesontheworkofmanyauthorswhowerealsoinvolvedinarecently
themedissueofZDM,“ProbabilityandReasoningaboutDataandRisk,”whichwas
guesteditedbyRolfBiehlerandDavePratt.Ultimately,thisvolumedemonstrates
that the first book in this series Theories of Mathematics Education did provide a
prototypeofthebooksseries,whichcanbeappliedtomonographsbasedonimpor-
tant ZDM issues of the past and proposals from the communityon other topics of
interesttothefield.
InwhathasturnedouttobeanimplicitfeatureoftheAdvancesinMathematics
Educationseries,thescopeofthisvolumeissubstantial:28chapters,from(intotal)
v
vi SeriesPreface
56 authors (including luminaries from the fields of mathematics, psychology and
mathematics education) from across the globe. While research has strayed more
towardstherealmofstatisticalthinkingandreasoning,thisbookdeliberatelytakes
thepathlesschosen,thatis,explorestherootsanddifferentfacetsofprobabilistic
thinkingand,intheforwardlookingspiritoftheseries,fertiledirectionsinwhich
theresearchcanbepushed.ProbabilisticThinkingisbothanthologicalandfuture-
oriented, with the explicit purpose of becoming a reference book for mathematics
education.
Hamburg,Germany GabrieleKaiser
Missoula,USA BharathSriraman
To
Kristenand Scoutfortheirunconditional
support
&
Sarah, Jacob and Miriam—for letting me
imagineprobabilitiesagainst allodds
The Most Common Misconception
About Probability?
DISCLAIMER:Ihavenotcarriedoutasystematicstudytocatalogmisconceptions
aboutprobability,soIcan’tclaimtoknowifthereisasinglemostcommonone,and
ifsowhatitis.ButIcanofferalikelycandidate,andithassignificantimplications
for the way we teach probability. I would put a high probability on being able to
deletethequestionmarkfrommytitleandproduceanaccuratedescription.
Misunderstandingsaboutprobabilityarelegion.Itis,afterall,afiendishlycom-
plex and elusive notion to wrap your mind around. Yet as the only reliable means
we have to predict—and plan for—the future, it plays a huge role in our lives, so
wecannotignoreit,andwemustteachittoallfuturecitizens.Thepurposeofthis
articleistohighlightoneimportantgoalofthatteaching.
Why only one? Because it addresses a widespread confusion at a very funda-
mentallevel,anditis,Ithink,aconfusionwecanavoidifwearecarefulhowwe
introducethenotionofprobability.
Sincetheearly1980s,Ihavewrittenregularlyonmathematicaltopicsforvarious
newspapers, magazines, and in more recent years various online publications and
blogs,andintheprocesshavereceivedagreatdealofcorrespondence.WheneverI
writeaboutprobability,asmallfloodofcorrespondencegenerallyensues.
SomeofthefeedbackIgetispredictable,bothinamountandthenatureofthe
comments. Famous probability puzzles such as the “What is the probability that
my other child is a girl?” question and the Monty Hall teaser always generate a
lot of controversy, as a new group of innocents encounter them for the first time.
But puzzles such as those are designed to generate such a response, and depend
on a carefully contrived set of background assumptions (only some of which are
usuallyarticulated)inordertoguaranteethecounter-intuitive“correct”answerthe
questionerhasinmind.Givingthe“wrong”answertosuchapuzzledoesnotneces-
sarilymeantherespondentdoesnotunderstandprobabilitytheory.Infact,aperson
whodoeshavesuchunderstandingcaninvariablyprovideasoundrationaleinsup-
port of their answer by articulating a background condition the questioner did not
explicitlyexclude.
For example, in the child gender problem, I say that “I have two children and
one of them is a girl, what is the probability that my other child is a girl?” The
ix
x TheMostCommonMisconceptionAboutProbability?
question is phrased so that the listener will likely answer, incorrectly, “One half.”
The “correct” answer is supposed to be “One third,” arrived at as follows. Since
childrenarebornserially,therearefourpossibilities,orderedbybirth,BB,BG,GB,
GG. (Many people make a type-token error at this point and say I should list BB
and GG twice each, but that is a separate issue.) My statement eliminates the BB
possibility,leavingBG,GB,GG.Ofthesethree,onlyGGresultsinmyotherchild
beingagirl,givingtheprobability1/3.
That,atleast,ishowthisexampleisusuallypresentedandsolved.Butitispos-
sibletoargueforotheranswers.Forexample,whatifIamacompulsivelylogical
personwhoalwaysprovidesinformationaboutmylifeinchronologicalorder,and
mylistenerknowsthataboutme?Thenmystatementimpliesthatmyfirstchildisa
girl,sothepossibilitiesareGBandGG,andthentheansweris1/2.
Alternatively,supposemylistenerandIcomefromastrictlypatriarchalsociety
where parents always give priority to information about sons. Then my statement
impliesthatIhavenoboys,andtheansweris1.
True, these two contextual circumstances are contrived, but then so is the orig-
inalpuzzle.Inreallife,theacceptednormsandmaximsofcommunicationgener-
ally prohibit making a statement of the form “At least one of my children is...”
Sinceprobabilitytheoryisaneminentlypracticalsubject,intentionallyusedwidely
inreal-world situations,confusionovercontrivedpuzzlesdoes notmeansomeone
cannotmakereliablepracticaluseofprobability.
Leavingasidesuchmind-benders,however,overtheyears,exchangeswithread-
ershaveshownmethatthereisacommonmisconceptionabouttheverynatureof
probability. It is this. Many people believe that events—that is, things that happen
intheworld—haveauniqueprobability.
Forexample,inexchangesaboutthechild-genderpuzzle,whereIhavediscussed
theeffectsofdifferentbackgroundassumptions,readersoftenwouldsay,“Yes,Isee
that,butwhatistheactualprobability”?Towhich,ofcourse,Isaythattheactual
probability is 1. I actually do have two children, both girls. Their gender is not a
probabilistic matter. They were born long ago, and their genders factually estab-
lishedatbirth,bothinrealityandinlaw.
Thepointis,Igoontoexplain,probabilitiesapplynottoeventsintherealworld
wearefamiliarwith,buttoourinformationaboutthatworldatanygivenmoment
in time.1 This crucial aspect of the notion of probability is often ignored, if not
obscured, by the way we normally introduce the concept to students, using eas-
ily understood, and readily implemented,experimental procedures such as tossing
coins,rollingdice,orselectingplayingcardsfromadeck.
That pedagogic examples such as these, having well defined, quantifiable
outcomes-spaces, do not provide an adequate basis for understanding subjective
probabilities,wherethereisnowell-definedoutcomes-space,isself-evident.2
1Iwordedthissentencecarefullytosidestepissuesofthequantumnatureofreality.
2Thereisnouniversalagreementamongprobabilityexperts,ofwhichImakenoclaimtoclub
membership, as to what constitutes subjective probability, epistemic probability, etc. Readers
shouldfeelfreetosubstitutetheirownfavoritedescriptor.
TheMostCommonMisconceptionAboutProbability? xi
[ASIDE:Thedistinctionbetweenprobabilitycalculationswherethereisawell-
defined set of outcomes, and the fascinating blend of judgment and mathematics
thatconstitutessubjectiveprobability,seemstobeunknowntomanyjournalistsand
radio producers. I and others with a publicmath identity frequently receivephone
callsfrommediaprofessionalswhowanttoknowtheprobabilityofsomesurprising,
newsworthyoccurrence.“Justwhatistheprobabilitythatamanwouldbiteadog?”
would be the iconic question of this type, though I confess I have never actually
beenaskedthatone.Igenerallyplayalongwiththerequest,byprovidinga“Fermi-
type”answer,butIlongagogaveupanyexpectationthatthepublishedorbroadcast
version of my answer would be accompanied by a statement of the assumptions I
madeinarrivingatthenumberIgave,oreventhatImadeanyassumptionsatall.
Allthejournalistwantsisanumberwithalotofzeros.]
Unfortunately, those quasi-experimental, pedagogic scenarios (tossing coins,
rolling dice, selecting playing cards from a deck) lead to a belief that probabili-
tiesareempiricalpropertiesofthescenario,ratherthanameasureofourknowledge
oftheoutcomes.Ithappensthisway.
We(i.e.,teachersofonekindoranother)typicallyintroducethenotionofproba-
bilitybyfirstgettingstudentstotosscoins,rolldice,orselectcardsfromadeck,or
whatever,andtabulatetheresults.Thenumberstherebyobtainedare,ofcourse,em-
piricallydeterminedfactsaboutthatspecificsequenceofactions.Theyaremeasures
ofthefrequencieswithwhichvariouskindsofoutcomesactuallyoccurred.
We go on to say that, absent the inconceivable event that the world will start
tobehavedifferentlytomorrowfromthewayithasdonethroughouthistory,those
frequencies provide a reliable numerical assessment of what is likely to happen
whenweperformthesameactionsometimeinthefuture.Wesay,forinstance,that
ifwerollanhonestdietenminutesfromnow,theprobabilityitwillland6upis1/6
andtheprobabilitythatitwilllandwithanevennumberuppermostis1/2.
That is certainly true. But what seems to get lost when we make this step is
that the probability refers to our knowledge—to our expectation of what is likely
to happen. There is, after all, no event to quantify until it occurs. There can be no
empiricalfact-of-the-matterabouttheoutcomeofafutureevent.Thefactthatwedo
nothavedefinitiveknowledgeoffutureeventsispreciselywhatmakesprobability
sousefulinplanningforthefuture.Itisnotthefactthatthedierollisinthefuture
thatmakescalculationoftheprobabilityuseful;itisthefactthatwedonothavethe
informationabouttheoutcome.Afutureeventforwhichtheoutcomeisknownin
advancedoesnotrequireaprobabilitycalculation.
Thedistinctionbecomesclearerifweperformthepedagogicexperimentdiffer-
ently. Suppose we split the students into two groups. One group rolls the die and
seestheoutcome,theothergroupdoesnot.I,asinstructor,askthesecondgroupto
tell me how confident they are that the die landed 6-up. If they have been paying
attentiontotheexperimentalworkearlier,theywillanswer“One-sixth.”(Thereare
variouswaysIcouldframethisquestion,suchastheoddstheywouldwantinorder
to place a bet on being right, etc.) But if I ask the first group the same question,
and force them to answer, they will say “Zero” or “One.” They can see how the
dielanded.Forthem,thereisnouncertainty.(Infact,ratherthananswer“Zero”or