working ST Core Reg - Mathematics Vision ProjectMathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 3. Write several other

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2017 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 9

Probability

SECONDARY

MATH TWO

An Integrated Approach

Standard Teacher Notes

SECONDARY MATH 2 // MODULE 9

PROBABILITY

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 9 - TABLE OF CONTENTS

PROBABILITY

9.1 TB or Not TB – A Develop Understanding Task

Estimating conditional probabilities and interpreting the meaning of a set of data (S.CP.6, S.MD.7+)

READY, SET, GO Homework: Probability 9.1

9.2 Chocolate versus Vanilla – A Solidify Understanding Task

Examining conditional probability using multiple representations (S.CP.6)


9.3 Fried Freddy’s – A Solidify Understanding Task

Using sample to estimate probabilities (S.CP.2, S.CP.6)


9.4 Visualizing with Venn – A Solidify Understanding Task

Creating Venn diagram’s using data while examining the addition rule for probability (S.CP.6, S.CP.7)


9.5 Freddy Revisited – A Solidify Understanding Task

Examining independence of events using two-way tables (S.CP.2, S.CP.3, S.CP.4, S.CP.5)


9.6 Striving for Independence – A Practice Understanding Task

Using data in various representations to determine independence (S.CP.2, S.CP.3, S.CP.4, S.CP.5)


SECONDARY MATH II // MODULE 9

PROBABILITY - 9.1


9.1 TB or Not TB? A Develop Understanding Task

Tuberculosis(TB)canbetestedinavarietyofways,includingaskintest.Ifapersonhas

turberculosisantibodies,thentheyareconsideredtohaveTB.Belowisatreediagramrepresenting

databasedon1,000peoplewhohavebeengivenaskintestforturberculosis.

1. WhatobservationsdoyounoticeaboutTBtestsbasedonthetreediagram?

2. Youmayhavenoticedthat380patientshaveTB,yetnotall380patientswithTBtested

positive.Instatistics,thenotation:“Testednegative|TB”means‘thenumberofpatientswhotestednegative,giventhattheyhaveTB’.DeterminetheprobabilitythatapersonwhohasTBcouldreceiveanegativeresultcomparedtootherswhohaveTB.Whatdoesthismean?

Thisisanexampleofconditionalprobability,whichisthemeasureofanevent,giventhat

anothereventhasoccurred.

CCBYhttps://flic.kr/p/xXebu

Testednegative|notTB558

PatienthasT

B380

Testednegative|TB19

PatientdoesNOThaveTB620

Testedpositi

ve|TB361

Testedpositi

ve|notTB6

2

1


PROBABILITY - 9.1


3. Writeseveralotherprobabilityandconditionalprobabilitystatementsbasedonthetree

diagram.

Partofunderstandingtheworldaroundusisbeingabletoanalyzedataandexplainittoothers.

4. Basedontheprobabilitystatementsfromthetreediagram,whatwouldyousaytoafriendregardingthevalidityoftheirresultsiftheyaretestingforTBusingaskintestandthe

resultcamebackpositive?

5. Inthissituation,explaintheconsequencesoferrors(havingatestwithincorrectresults).

6. Ifahealthtestisnot100%certain,whymightitbebeneficialtohavetheresultsleanmore

towardafalsepositive?

7. Isasamplespaceof200enoughtoindicatewhetherornotthisistrueforanentire

population?

2


PROBABILITY - 9.1


9.1 TB or Not TB? – Teacher Notes A Develop Understanding Task

Purpose:Thepurposeofthistaskisforstudentstoanalyzeandmakesenseofdata.Studentswill

connecttheirpriorunderstandingsoftreediagramsandfrequencytables(fromearliergrades)to

analyzedatafromatreediagramandexplaintheresultstoothers.Thefocusofthistaskisto

highlighttheinformationrevealedasaresultoftheconditionalprobabilitystatements.Questions

suchas‘Howdoesthesubgroupinformationtellusamorecompletestory?’shouldbeaddressedin

thistask.

Notetoteacher:Throughoutthemodule,studentswillbeanalyzingdatabutwillhaveotherareas

offocus(suchasrepresentations,theadditionrule,andnotation).Therefore,itisimportantthat

thefocusofthistaskisforstudentstomakesenseofthecontextgiventherepresentationandto

write/communicategeneralprobabilitystatementsandconditionalprobabilitystatements.

CoreStandardsFocus:

S.CP.6:FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelong

toA,andinterprettheanswerintermsofthemodel.

S.MD.7(+):Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,

medicaltesting,pullingahockeygoalieattheendofagame).

RelatedStandards:S.CP.3,S.CP.4,S.CP.5

TheTeachingCycle:

Launch(WholeClass):

Startthistaskbysharinghowimportantitistoknowhowtoreadandmakesenseofdata.Inthe

healthcarefield,therearelotsoftestsusedtodeterminewhetherornotapatienthasaninfection

oradisease.Itisalsoimportanttonotethatnotalltestsareaccurate100%ofthetime,whichis

whythereissometimesmorethanonetesttodetermineadiagnosis.Readthedirectionsfromthe

taskandhavestudentsmakeobservationsaboutthetreediagram(question1).Ifstudentsarenot


PROBABILITY - 9.1


familiarwiththesymbolusedforconditionalprobabilitythatisgivenonthetreediagram,explain

thistothem(Testedpositive|TBmeans“thenumberofpatientswhotestedpositive,giventhatthey

haveTB”).

Explore(SmallGroup):

Givestudentstimetomakesenseofthediagramindependentlyandthenhavethemworkinpairs

tocreatestatements.Ifagroupseemsstumped,youcanaskquestionslike:

• Whatinformationdoesthisdiagramtellyou?

• IsthisTBtestalways100%accurate?Howdoyouknow?

• Domostpeopletestpositiveornegativebasedontheskintestresults?

• Whatotherinformationcanyoudetermine?

Itisbesttostayawayfromspecificquestionsatthistimeaspartofthistaskisforstudentstomake

senseandinterpretthedatathemselves.Thelowthresholdofthistaskisthatmoststudentsshould

beabletocreatestatementssuchas“36.1%ofthepopulationtestedhaveTBantibodiesandtested

positive”.Asyoumonitor,ifyounoticestudentsarenotwritinganyconditionalstatements,probe

groupsthatarefinishingquicklytoanalyzetheresultsofaspecificbranch.Forexample,‘Whatdo

youknowaboutpatientswhotestpositive,giventhattheyhaveTB?’ortoexplainthedifference,in

context,betweentheseprobabilities:620/1000,62/620and62/1000.Besuretohavestudents

writeouttheirstatementsintheirjournalandnotjustsaythem.Thisassistsstudentsin

communicatingconditionalprobabilities.Iftheyneedprompting,youmaywishtogiveanexample

ortwousingphrasessuchas“outof”or“giventhat”or“If…then”.

Asyouprepareforthewholegroupdiscussion,chooseagrouptochartsomeoftheirstatements

relatedtothetotalnumberofpeoplewhotooktheskintest.Alsoselectstudentstosharetheir

conditionalstatementswhohavearticulatedunderstanding.

Discuss(WholeClass):

Forthewholegroupdiscussion,thegoalisforstudentstobeabletoanswer“Howaccurateisthe

tuberculosis(TB)test?”usingthedataprovidedtowriteprobabilitystatements.Thereareseveral

waystodothis,withthefollowingasonepossibility:


PROBABILITY - 9.1


First,havegroupwhochartedstatementsrelatedtothetotalshareandexplaintheirstatements.

Highlightsofthisconversationincluderesultsofcertainaspectsofthesample(percentwhohave

antibodies,percentwhotestpositive,etc.)andthatthetestisaccurate‘most’ofthetime.Whilethis

isgoodinformation,takingadeeperlookcanrevealmoreinformation.Next,sequencethestudents

youselectedearliertosharetheirconditionalstatementsandaskthemwhattheirstatement

reveals.Besureeachstatementisaccurate(notjustcomputation,butalsohowitis

written/spoken),andmaketheappropriateadjustmentsforthosewhoneedit.Duringthispartof

thediscussionisalsoanappropriatetimetointroduceprobabilitynotationasstudentshavelikely

notseenthisinthepast.

Discussionitemsthatyoumaywishtomakesurecomeout:

1. Meaningoferror(falsepositive/falsenegative)andwhythisparticulardataleansmore

towardafalsepositive.

2. Thelawoflargenumbers.Thesamplesizeofthisdatais1,000.Whatdoesthismeanforan

individualwhohasaskintestforTB?

3. Howdoesexploringconditionalprobabilitiesallowforinvestigationoftheaccuracyof

medicaltests?

AlignedReady,Set,Go:ProbabilityRSG9.1


PROBABILITY – 9.1

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

9.1

Needhelp?Visitwww.rsgsupport.org

READY

Topic:VennDiagrams,howtocreateandread.

ForeachVennDiagramprovidedanswerthequestions.

1.Howmanystudentsweresurveyed?

2.Whatwerethestudentsasked?

3.Howmanystudentsareinboth

choirandband?

4.Howmanystudentsarenotineither

choirorband?

5.Whatistheprobabilitythata

randomlyselectedstudentwouldbein

band?

ThisVennDiagramrepresentsenrollmentinsomeof

theelectivecourses.

6.Whatdoesthe95inthecentertellyou?

7.Whatdoesthe145tellyou?

8.Howmanytotalstudentsarerepresentedinthe

diagram?

9.Whichelectiveclasshastheleastnumberof

studentsenrolled?

READY, SET, GO! Name PeriodDate

3


PROBABILITY – 9.1




9.1


SET Topic:Interpretingatreediagramtodetermineprobability

Giventhetreediagrambelowanswerthequestionsanddeterminetheprobabilities.Thediagramrepresentsthenumberofplateappearancesduringthefirstmonthofaminorleaguebaseballseason.

10. Howmanytimesdidabattercometotheplateduringthistimeperiod?

11. Basedonthisdata,ifyouarealeft-handedbatterwhatistheprobabilitythatyouwillfacearight-handedpitcher?

12. Basedonthisdata,ifyouarearight-handedbatterwhatistheprobabilitythatyouwillfacealeft-handedpitcher?

13. Whatistheprobabilitythataleft-handedpitcherwillbethrowingforanygivenplateappearance?

14. Whatistheprobabilitythataleft-handedbatterwouldbeattheplateforanygivenplateappearance?

Whatobservationsdoyoumakeaboutthedata?Isthereanyamountthatseemstobeoverly

abundant?Whatmightaccountforthis?

GO Topic:BasicProbability

Findtheprobabilityofachievingsuccesswitheachoftheeventsbelow.

15. Rollinganevennumberonstandardsix-sideddie.

16. Drawingablackcardfromastandarddeckofcards.

17. FlippingacoinandgettingHeadsthreetimesinarow.

18. Rollingadieandgettingafour.

19. Drawinganacefromadeckofcards.

20. Rollingadietwiceinarowandgettingtwothrees.

21. Fromabagcontaining3blue,2red,and5whitemarbles.Pullingoutaredmarble.

4


PROBABILITY- 9.2


9.2 Chocolate versus Vanilla A Solidify Understanding Task

Danielleloveschocolateicecreammuchmorethanvanillaandwas

explainingtoherbestfriendRaquelthatsodoesmostoftheworld.Raquel

disagreedandthoughtvanillaismuchbetter.Tosettletheargument,they

createdasurveyaskingpeopletochoosetheirfavoriteicecreamflavor

betweenchocolateandvanilla.Aftercompletingthesurvey,thefollowing

resultscameback:

• Therewere8,756femalesand6,010maleswhoresponded.

• Outofallthemales,59.7%chosevanillaoverchocolate.

• 4,732femaleschosechocolate.

1. Uponfirstobservations,whichflavordoyouthink“won”?_____________________.Writea

sentencedescribingwhatyouseeat‘firstglance’thatmakesyouthinkthis.

2. Raquelstartedtoorganizethedatainthefollowingtwo-waytable.Seeifyoucanhelp

completethis(usingcountsandnotpercentages):

3. OrganizethesamedatainaVenndiagramandatreediagram.

4. Usingyourorganizeddatarepresentations,writeprobabilitiesthathelpsupportyourclaim

regardingthepreferredflavoroficecream.Foreachprobability,writeacomplete

statementaswellasthecorrespondingprobabilitynotation.

Chocolate Vanilla Total

Female 8,756

Male 6,010

Total

CCBYhttps://flic.kr/p/dAmJrc

5


PROBABILITY- 9.2


5. Lookingoverthethreerepresentations(treediagram,two-waytable,andVenndiagram),

whatprobabilitiesseemtobeeasiertoseeineach?Whatprobabilitiesarehiddenorhard

tosee?

Highlighted(easiertosee) HiddenTreediagram

Treediagram

Two-waytable

Two-waytable

Venndiagram

Venndiagram

6. Gettingbacktoicecream.Doyouthinkthisisenoughinformationtoproclaimthe

statementthatoneicecreamisfavoredoveranother?Explain.

6


PROBABILITY- 9.2


9.2 Chocolate versus Vanilla – Teacher Notes

A Solidify Understanding Task

Purpose:Thepurposeofthistaskisforstudentstointerpretinformationprovidedthatallows

themtomakesenseofandorganizedatainatreediagram,atwo-waytable,andaVenndiagram.

Studentswillsolidifytheirunderstandingofconditionalprobabilitybywritingstatements

supportedbydatacollectedtojustifytheflavoroficecreampreferredbymost.Inthistask,

studentswill:

● Organizedataintoatreediagram,two-waytable,andaVenndiagram

● CalculateprobabilitiesandconditionalprobabilitiesofAgivenBasthefractionofB’s

outcomesthatalsobelongtoA,andinterprettheanswerintermsofthemodel

● Highlightthedifferentrepresentationsandbecomemorefamiliarwithwhateach

representationhighlightsandconceals.

● Continuetobecomemorefamiliarwithprobabilitynotation.

● Makedecisionsaboutmeaningofdata.

CoreStandardsFocus:

S.CP.4:Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesare

associatedwitheachobjectbeingclassified.Usethetwo-waytableasasamplespacetodecideif

eventsareindependentandtoapproximateconditionalprobabilities.



RelatedStandards:S.CP.3,S.CP.5,S.MD.6

TheTeachingCycle

Launch(WholeClass):

Beginbyreadingthecontextoftheproblemandthedataprovidedbythesurvey.Havestudents

individuallymakeadecisiontoquestion1.Thisistheir‘estimation’usingtheirfirstimpression

reasoning.Givestudentswaittimetowritetheirfirstimpression.Tolaunchtherestofthetask,

explaintostudentsthattheywillbeorganizingthedataintothethreerepresentations(tree


PROBABILITY- 9.2


diagram,Venndiagram,andtwowaytable).Theywillusethisinformationtofurthermakeclaims

regardingthe‘favoriteicecreamflavor’.

Explore(SmallGroup):Asyoumonitor,listenforstudentstomakesenseofthedataprovidedsothattheycancompletethe

treediagram.Allowfortimesothatstudentscandeterminewhatinformationisstillneededand

howtogoaboutfindingthis.Ifafteracoupleofminutesyounoticeagroupthatseemstobestuck

astohowtobegin,askprobingquestionssuchas“Whatpartofthetreediagramneedstostillbe

completed?”followedby“Howcouldyoufindthesolutiontothatpartofthediagram(suchasthe

numberoffemaleswhoprefervanillaicecream)or(thenumberofmaleswhoprefervanillaice

cream)?”Alsolookforthecommonerroragroupcouldmakebycalculating59.7%ofthetotalto

determinethenumberofmaleswholikeVanillaicecream.Toclarify,havethegroupexplainthe

diagramandtheircalculation.Likely,theywillnoticetheirmistakeandrecalculatetofind59.7%of

themalesinsteadofthetotal.Ifnot,askthemtoexplainthemeaningofthebulletpointinthegiven

datathatreads“Outofallthemales,59.7%chosevanillaoverchocolate.”Asstudentsmovefrom

completingonerepresentationtotheothers(tree,Venn,table),listenforexplanationstheyhavefor

determiningvaluesineachrepresentationandanyconnectionstheymakebetween

representations(asyoumonitor,selectstudentstosharebasedontheseexplanations).Belowis

thecompletedtwowaytabletoassistinmakingsurestudentcalculationsareaccurateaswellasa

coupleofprobabilitystatementsstudentsmaychoosetouse.

Forthosewhochosevanilla:

P(vanilla)=7614/14766

P(vanilla|male)=3590/6010

BeginthewholegroupdiscussionaftermoststudentshavecompletedpartI.


Sequencestudentstosharetheirstrategiesfororganizingdata.Includeconnectionstheynotice

betweenrepresentations.Usemisconceptionsstudentsmayhavehadduringsmallgroupsto


Female 4,732 4,024 8,756

Male 2,420 3,590 6,010

Total 7,152 7,614 14,766


PROBABILITY- 9.2


highlighthowtodeterminevalues.Foreachsituation,encouragestudentstouseappropriate

academiclanguage.

ExplorePartII(SmallGroups):

Havestudentscompletequestions5and6.Studentsshouldbeabletoarticulatehoweach

representationorganizesthedatatohighlightcertaininformation.Thegoalofthisportionofthe

taskisforstudentstobecomemorecomfortablewitheachrepresentationsotheycanchoose

whichrepresentationtouseinthefuturewhenorganizingdata.Ifstudentsseemstuck,askthemto

writeaprobabilitythatis‘easier’toseeinthetreediagramthanintheothertworepresentations.

Likewise,writeaprobabilitythatis‘easier’toseeinthetwowaytableorthatis‘easier’toseeinthe

VennDiagram.

DiscussPartII(WholeClass):

Completethechartasawholegroupafterstudentshavehadtimetothinkaboutthisthemselves.

Toconclude,answerquestionsixbydiscussingthelawoflargenumbersandrandomness.



PROBABILITY – 9.2




9.2


READY

Topic:AnalyzingdatagiveninaVennDiagram.

UsetheVennDiagramsbelowtoanswerthefollowingquestions.(Hint:youmayusethesamedataprovidedinthetwo-waytablefromquestion3onthenextpagetohelpmakesenseoftheVennDiagram)ThefollowingVennDiagramrepresentstherelationshipbetweenfavoritesport(SoccerorBaseball)andgender(FemaleorMale).

1.Howmanypeoplesaidsocceristheirfavoritesport?

2.Howmanyfemalesareinthedata?

3.Howmanymaleschosebaseball?

4.Whatistheprobabilitythatapersonwouldsaysocceristheir

favoritesport?P(soccer)=

5.Whatistheprobabilitythatafemalewouldsaysocceristheirfavoritesport?(“Outofallfemales,

____%saysocceristheirfavoritesport”)P(soccer|female)=

ThefollowingVennDiagramrepresentstherelationshipbetweenfavoritesubject(MathorScience)andgradelevel(NinthorTenth).Usingthisdata,answerthefollowingquestions.

6.Howmanypeoplesaidmathistheirfavoritesubject?

7.Howmanytenthgradersareinthedata?

8.Howmanyninthgraderschosescience?

9.Whatistheprobabilitythatapersonwouldsayscienceistheir

favoritesubject?P(s)=

10.Whatistheprobabilitythatatenthgraderwouldsayscienceistheirfavoritesubject?(“Ifyouarea

tenthgrader,thentheprobabilityofsciencebeingyourfavoritesubjectis_____%”)P(science|tenth)=


30

25

7


PROBABILITY – 9.2




9.2


SET Topic:Writingconditionalstatementsfromtwo-waytables

11.Completethetableandwritethreeconditionalstatements.

Soccer Baseball Total

Male 30

Female 50 76

Total 85

12.Completethetableaboutpreferredgenreofreadingandwritethreeconditionalstatements.

Fiction

Non-

Fiction

Total

Male 10

Female 50 60

Total 85

13.CompletethetableaboutfavoritecolorofM&M’sandwritethreeconditionalstatements.

Blue Green Red Other Total

Male 15 20 15 60

Female 30 20 10

Total 45 130

14.Usetheinformationprovidedtomakeatreediagram,atwo-waytableandaVennDiagram.

• Datawascollectedatthemovietheaterlastfall.Notaboutmoviesbutclothes.

• 6,525peoplewereobserved.

• 3,123hadonshortsandtheresthadonpants

• 45%ofthosewearingshortsweredenim.

• Ofthosewearingpants88%weredenim.

8


PROBABILITY – 9.2




9.2


GO Topic:BasicProbability

Findthedesiredvalues.

15.Whatishalfofone-third? 16.Whatisone-thirdoftwo-fifths?

17.Whatisone-fourthoffour-sevenths? 18.Whatpercentis!!?

19.Whatis35%of50? 20.Seventyis60%ofwhatnumber?

21.Write!!"asapercent. 22.Write

!!asapercent.

23.Whatis52%of1,200? 24.Whatpercentis32of160?

25.Sixtyiswhatpercentof250? 26.Whatpercentof350is50?

9


PROBABILITY- 9.3


9.3 Fried Freddy’s A Solidify Understanding Task

Daniellewassurprisedbytheresultsofthesurveytodetermine

the‘favoriteicecream’betweenchocolateandvanilla(Seetask

9.2Chocolatevs.Vanilla).Thereason,sheexplains,isthatshehadaskedseveralofherfriends

andtheresultswereasfollows:

1. Inthissituation,chocolateismostpreferred.Howwouldyouexplaintoherthatthisdata

maybeless‘valid’comparedtothedatafromtheprevioussurvey?

Usingasufficientlylargenumberoftrialshelpsusestimatetheprobabilityofaneventhappening.If

thesampleislargeenough,wecansaythatwehaveanestimatedprobabilityoutcomeforthe

probabilityofaneventhappening.Ifthesampleisnotrandomlyselected(onlyaskingyourfriends)

ornotlargeenough(collectingfourdatapointsisnotenoughinformationtoestimatelongrun

probabilities),thenoneshouldnotestimatelargescaleprobabilities.Sometimes,oursample

increasesinsizeovertime.Belowisanexampleofdatathatiscollectedovertime,sotheestimated

probabilityoutcomebecomesmorepreciseasthesampleincreasesovertime.

Freddylovesfriedfood.Hispassionfortheperfectfriedfoodrecipesledtohimopeningthe

restaurant,“FriedFreddies.”Histwomaindishesarefocusedaroundfishorchicken.Knowinghe

alsohadtoopenuphismenutopeoplewhoprefertohavetheirfoodgrilledinsteadoffried,he

createdthefollowingmenuboard:


Female 23 10 33

Male 6 8 14

Total 29 18 47

CCBYhttps://flic.kr/p/9a7kMg

10


PROBABILITY- 9.3


Afterbeingopenforsixmonths,Freddyrealizedhewashavingmorefoodwastethanheshould

becausehewasnotpredictinghowmuchofeachheshouldprepareinadvance.Hisbusinessfriend,

Tyrell,saidhecouldhelp.

2. WhatinformationdoyouthinkTyrellwouldneed?

Luckily,FreddyusesacomputertotakeorderseachdaysoTyrellhadlotsofdatatopullfrom.AfterdeterminingtheaveragenumberofcustomersFreddyserveseachday,TyrellcreatedthefollowingVenndiagramtoshowFreddythefoodpreferenceofhiscustomers:

Tomakesenseofthediagram,Freddycomputedthefollowingprobabilitystatements:

3. Whatistheprobabilitythatarandomlyselectedcustomerwouldorderfish?

P(fish)=

Shadethepartofthediagramthatmodelsthissolution.

30%15%20%

Fried Fish

Choose dish: Chicken or Fish

Choose cooking preference: Grilled or Fried

35%

11


PROBABILITY- 9.3


4. Whatistheprobabilitythatarandomlyselectedcustomerwouldorderfried

fish?

P(fried∩fish)=P(friedandfish)=


5. Whatistheprobabilitythatapersonprefersfriedchicken?

P(fried∩chicken)=P(friedandchicken)=


6. Whatistheestimatedprobabilitythatarandomlyselectedcustomerwould

wanttheirfishgrilled?

P(grilledandfish)=P(____________________)=


7. IfFreddyserves100mealsatlunchonaparticularday,howmanyordersoffishshouldhe

preparewithhisfamousfriedrecipe?

8. Whatistheprobabilitythatarandomlyselectedpersonwouldchoosefishor

fried?

P(fried∪fish)=P(friedorfish)=Shadethepartofthediagramthatmodelsthissolution.

9. WhatistheprobabilitythatarandomlyselectedpersonwouldNOTchoose

fishorfried?


12


PROBABILITY- 9.3


9.3 Fried Freddy’s – Teacher Notes A Solidify Understanding Task

Purpose:Apurposeofthistaskisforstudentstogainastrongerunderstandingthelawoflarge

numbersandhowthishelpstoestimateprobableoutcomes.Anotherpurposeisforstudentsto

solidifytheirunderstandingaroundthefollowingideas:

● Whetherornotthereisenoughdatatoestimateoutcomes.

● Distinguishbetweenageneralprobability,aconditionalprobability,andtheadditionrule.

● UseaVenndiagramtoanalyzedataandtowritevariousprobabilitystatements(unions,

intersections,complements).

● ApplytheAdditionRuleandinterprettheanswerintermsofthemodel.

● Useestimatedoutcomestomakerecommendationsanddecisions.

CoreStandardsFocus:

S.CP.1:Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(or

categories)oftheoutcomes,orasunions,intersections,orcomplementsofotherevents(“or”,

“and”,“not”).



eventsareindependentandtoapproximateconditionalprobabilities.



S.CP.7:ApplytheAdditionRuleandinterprettheanswerintermsofthemodel.

RelatedStandards:S.CP.3,S.CP.5

TheTeachingCycle

Launch(WholeClass):

Havestudentsanswerthefirstquestionaboutthevalidityofdataindividually,thendiscussasa

grouphowimportantsampleiswhenestimatingoutcomes.


PROBABILITY- 9.3


ReadthecontextofFriedFreddy,thenhavestudentsanswerthequestionaboutwhatinformation

Tyrellwouldneedtopulltogetherdata.Dependingonyourclass,youmayalsowishtohave

studentslabeleachsectionoftheVenndiagramandmakesureeveryoneunderstandsthispriorto

havingstudentsanswerthequestionsrelatedtothediagram.


Asyoumonitor,listenforstudentstomakesenseoftheprobabilitystatementstheyareanswering

andlookforthesolutionareatobeshadedontheirmodels.Helpstudentswhoarestrugglingby

suggestingtheycreateadifferentrepresentation(suchasatwo-waytable-seebelow).

Sincethepurposeofthistaskisforstudentstorecognizeanduseprobabilitystatementsto

interpretdata,plantohavethewholegroupdiscussionfocusonquestionsthatrelatetostandards

S.CP.1andS.CP.7.


ChoosestudentstoanswerquestionsrelatingtothedatafromtheVenndiagram.Spendmosttime

onquestionsrelatingtounions,intersections,complements,andtheAdditionRule.Besuretouse

thisvocabularyandhavestudent’srecordunfamiliarvocabularyintheirjournal.Themosttime

maybespentonquestionssixthroughnine.Attheendofthelesson,reviewprobabilitynotation

anddiscussthesimilaritiesanddifferencesbetweentherulesofprobability(unions,intersections,

complements,AdditionRule,conditionalprobability)andothervocabulary(mutuallyexclusive,

joint,disjoint).


Fish Chicken total

Fried 15% 20% 35%

Grilled 30% 35% 65%

Total 45% 55% 100%


PROBABILITY – 9.3




9.3


READY

Topic:IndependentandDependentEventsInsomeofthesituationsdescribedbelowthefirsteventeffectsthesubsequentevent(dependentevents).Inotherseachoftheeventsiscompletelyindependentoftheothers(independentevents).Determinewhichsituationsaredependentandwhichareindependent.1. Acoinisflippedtwice.Thefirsteventisthefirstflipandthesecondeventisthenextflip.

2. Abagofmarblescontains3bluemarbles,6redmarblesand2yellowmarbles.Twoofthemarblesaredrawnoutofthebag.Thefirsteventisthefirstmarbletakenoutthesecondeventisthesecondmarbletakenout.

3. Anattempttofindtheprobabilityoftherebeingaright-handedoraleft-handedbatterattheplateinabaseballgame.Thefirsteventisthe1stbattertocometotheplate.Thesecondeventisthesecondplayertocomeuptotheplate.

4. Astandarddieisrolledtwice.Thefirsteventisthefirstrollandthesecondeventisthesecondroll.

5. Twocardsaredrawnfromastandarddeckofcards.Thefirsteventisthefirstcardthatisdrawnthesecondeventisthesecondcardthatisdrawn.

SET Topic:AdditionRule,InterpretingaVennDiagram6.SallywasassignedtocreateaVenndiagramtorepresent!(! or !).Sallyfirstwrites!(! or !) = !(!) + !(!) − !(! and !),whatdoesthismean?Explaineachpart.

7.Sallythencreatesthefollowingdiagram.

Sally’sVenndiagramisincorrect.Why?


13


PROBABILITY – 9.3




9.3


TheVenndiagramtotherightshowsthedatacollectedatasandwichshopforthelastsixmonthswithrespecttothetypeofbreadpeopleordered(sourdoughorwheat)andwhetherornottheygotcheeseontheirsandwich.Usethisdatatocreateatwo-wayfrequencytableandanswerthequestions.

8.Two-wayfrequencytable

9. Whatistheprobabilitythatarandomlyselectedcustomerwouldordersourdoughbread?

P(sourdoughbread)=10. Whatistheprobabilitythatarandomlyselectedcustomerwouldordersourdoughbreadwithout

cheese?P(sourdough∩nocheese)=P(sourdoughandnocheese)=

11. Whatistheprobabilitythatapersonpreferswheatbreadwithoutcheese?P(wheat∩nocheese)=P(wheatandnocheese)=

12. Whatistheestimatedprobabilitythatarandomlyselectedcustomerwouldwanttheirsandwichwithcheese?P(sourdoughcheeseandwheatcheese)=P(____________________)=

13. Iftheyserve100sandwichesatlunchonaparticularday,howmanyorderswithsourdoughshouldbepreparedwithoutcheese?

14. Whatistheprobabilitythatarandomlyselectedpersonwouldchoosesourdoughorwithoutcheese?P(sourdough∪nocheese)=P(sourdoughornocheese)=

15. WhatistheprobabilitythatarandomlyselectedpersonwouldNOTchoosesourdourghornocheese?

20%

14


PROBABILITY – 9.3




9.3


GO Topic:EquivalentRatiosandProportionsUsethegivenratiotosetupaproportionandfindthedesiredvalue.

16. If3outof5studentseatschoollunchthenhowmanystudentswouldbeexpectedtoeatschool

lunchataschoolwith750students?

17. Inawelldevelopedandcarriedoutsurveyitwasfoundthat4outof10studentshaveapairofsunglasses.Howmanystudentswouldyouexpecttohaveapairofsunglassesoutofagroupof45students?

18. Datacollectedatalocalmallindictedthat7outof20menobservedwerewearingahat.Howmanywouldyouexpecttohavebeenwearinghatsif7500menweretobeatthemallonasimilarday?

15


PROBABILITY-9.1


Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

9.4 Visualizing with Venn

A Solidify Understanding Task OneoftheattributesofVenndiagram’sisthatitcanbeeasytoseetherelationshipswithinthedata.Inthistask,wewillcreatemultipleVenndiagramsusingdataanddeterminetheeventsthatcreatediagramstoeitherhaveanintersectionorforthemtobemutuallyexclusive.1. ThefollowingdatarepresentsthenumberofmenandwomenpassengersaboardtheTitanicand

whetherornottheysurvived.Fillintheblanksforthistable:

Survived Didnotsurvive Total

Men 659 805

Women 296

Total 442 765 1207

2. Usingthedataabove,createaVenndiagramforeachofthefollowing:a. MenvsWomen b. WomenvsSurvivedc. Youchoosetheconditions

3. CreatetwoprobabilitystatementsusingeachofyourVenndiagramsfromquestion2.

4. CreateandlabelthreedifferentVenndiagramsusingthefollowingdata.Createatleastonethatismutuallyexclusiveandatleastonethathasanintersection.Samplesize:100

P(girl)= !"!""P(girlorart)=( !"!"" + !"!"") −

!"!""

P(art)= !"!""P(notart)=P(boy)=

5. DescribetheconditionsthatcreatemutuallyexclusiveVenndiagramsandthosethatcreateintersections.

6. WhatconjecturecanyoumakeregardingthebestwaytocreateaVenndiagramfromdatatohighlightprobabilities?

CC

BY

htt

ps://

flic.

kr/p

/9a7

kMg

16


PROBABILITY-9.1



9.4 Visualizing with Venn- Teacher Notes A Solidify Understanding Task

Purpose:ThepurposeofthistaskistohavestudentscreateandanalyzeattributesofVenndiagrams,

thenmakesenseofthedata.Studentswilladdacademicvocabulary(mutuallyexclusive,joint,disjoint)

andwilldistinguishbetweenconditionalprobabilityandusingtheadditionrule.Attheendofthetask,

studentswillbeableto:

● CreateVennDiagramsthathighlightspecificdata.

● Understandmutuallyexclusive,joint(intersection),andtheAdditionRuleusingVenn

diagrams.

● DistinguishbetweenconditionalprobabilityandtheAdditionRule.

● Getoutvocabularysuchasjoint,disjoint,mutuallyexclusive,AdditionRule,andconditional

probability.

● AnalyzingdatafromatwowaytableandfromprobabilitynotationtocreatevariousVenn

Diagrams

CoreStandardsFocus:S.CP.6:FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongto

A,andinterprettheanswerintermsofthemodel.

S.CP.7:ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerinterms

ofthemodel.

RelatedStandards:S.CP.3,S.CP.4,S.CP.5

TheTeachingCycle:

Launch(WholeClass):

HavestudentssharewhattheyknowaboutVenndiagramstoaccessbackgroundknowledge,thenhave

thembeginworkingonthetaskinsmallgroups.


PROBABILITY-9.1




Asyoumonitor,checkforstudentunderstandingofhowtowriteaVenndiagramwhengiventhe

constraints(questionstwoandthree).Thesequestionsscaffoldstudentsbytellingthemwhich

categoriestousewhencreatingthediagram,thenhavingtheminterpretwhatkindofinformationthey

canseefromeach.Theintentionofthesemodelsisforstudentstorecognizethatwhentheyare

choosingcategoriesforaVenndiagram,thatthedataismoremeaningfulwhentheyselectfrom

differentcategorytypes.ListenforstudentstomakethefollowingconjecturesregardingcreatingVenn

diagrams:Iftwoeventsarerelatedandhaveoverlappingdata,theintersectioniseasytoseeand

severalprobabilitystatementscanbemade.Likewise,ifthetwoeventsbeingcompareddonothave

overlap,thenthosetwoeventsaremutuallyexclusive.Alsolistenforstudentstobecomeclearastothe

probabilitiesthatareeasiertoseeinaVenndiagramthaninothermodels.


Vocabularytosolidifyduringdiscussionusingstudentwork:mutuallyexclusive,AdditionRule,

intersections,complements,disjoint,andjoint.

Forthewholegroupdiscussion,havestudentssharewhattheyhavelearnedaboutcreatingVenn

diagrams.Chartthisinformation,andthenpressstudentstoexplainhowtheywouldusedatainthe

futuretocreateaVenndiagramthatwouldproducethedatatheywereseeking.Askstudentstoshare

someprobabilitystatementstheycreatedandhavethemexplainhowtheVenndiagramwashelpfulin

seeingtheprobability.Focusonconditionalprobabilitystatements,theAdditionRule,anddatathat

highlightsintersectionsandcomplements.



PROBABILITY – 9.4




9.4


READY

Topic:Productsofprobabilities,multiplyinganddividingfractionsFindtheproductsorquotientsbelow.

1. 12 ∙23

2. 35 ∙13

3. 710 ∙

25

4. 87 ∙34

5. 1312

6. 25 ÷

23

7. P(A)=!!P(B)=

!!

P(A)∗P(B)=

8. P(A)=!!P(B)=!!

P(A)∗P(B)=

SET Topic:ConnectingrepresentationsofeventsforprobabilityForeachsituation,oneoftherepresentations(two-waytable,Venndiagram,treediagram,contextorprobabilitynotation)isprovided.Usetheprovidedinformationtocompletetheremainingrepresentations.9.AreyouBlue?

Notation 2-wayTableKey:Male=MFemale=FBlue=BNotBlue=NSamplesize=200P(B)=84/200P(M)=64/200P(F|B)=48/84P(B|F)=P(M∩B)=P(M∪B)=

Blue NotBlue

Total

Male

Female

Total

(Continuedonthenextpage)


17


PROBABILITY – 9.4




9.4


(Continued from the last page)

VennDiagram TreeDiagram

Writethreeobservationsyoucanmakeaboutthisdata.

10.Rightandlefthandednessofagroup.Notation 2-wayTable

Key:Male=MFemale=FLefty=LRighty=RSamplesize=100peopleP(L)=P(M)=P(F)=P(L|F)=P(L|M)=

Lefty Righty Total

Male

Female

Total


Writethreeconditionalstatementsregardingthisdata.

18


PROBABILITY – 9.4




9.4


11.Themostimportantmealoftheday.Notation 2-wayTable

Key:Male=MFemale=FEatsBreakfast=EDoesn’tEatBreakfast=DSamplesize=P(E)=P(E|M)=P(E∩M)=P(E|F)=P(E∩F)=

Eats Doesn’t Total

Male

Female

Total 685


Doesthisdatasurpriseyou?Whyorwhynot.

GO Topic:Writingconditionalstatementsfromtwo-waytables12.Completethetableandwritethreeconditionalstatements.

Biking Swimming TotalMale 50 Female 35 76Total 85

13.Completethetableaboutpreferredgenreofreadingandwritethreeconditionalstatements.

IceCream

Cake Total

Male 20 Female 10 60Total 85

14.Completethetableabouteyecolorandwritethreeconditionalstatements. Blue Green Brown Other Total

Male 55 20 15 100Female 20 10 Total 75 230

19


PPOBABILITY- 9.5


9.5 Freddy Revisited A Solidify Understanding Task

Intask9.3FriedFreddy’s,TyrellhelpedFreddyin

determiningtheamountandtypeoffoodFreddyshouldprepareeachdayforhisrestaurant.Asa

result,Freddy’sfoodwastedecreaseddramatically.Astimewentby,Freddynoticedthatanother

factorheneededtoconsiderwasthedayoftheweek.Henoticedthathewasoverpreparingduring

theweekandsometimesunderpreparingontheweekend.TyrellandFreddyworkedtogetherand

startedcollectingdatatofindtheaveragenumberofordershereceivedofchickenandfishona

weekdayandcomparedittotheaveragenumberofordershereceivedofeachontheweekend.

Aftertwomonths,theyhadenoughinformationtocreatethetwowaytablebelow:

Fish Chicken TotalWeekday 65 79 144Weekend 88 107 195Total 153 186 339

1. Whatobservationscanbemadefromthetable(includeprobabilitystatements)?

2. Whatdoyounoticeabouttheprobabilitystatements?

3. Basedonthedata,ifFreddyhadasalespromotionandanticipated500ordersinagivenweek,howmanyofeach(chickenandfish)shouldheorder?

CcbyNicoleAbaldi

http://flic.kr/p/dBMGid

20


PPOBABILITY- 9.5


9.5 Freddy Revisited – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskisforstudentstodetermineiftwoeventsareindependent.In

thistask,studentsareaskedtointerprettheamountoffishandchickenFreddyshouldprepareon

anygivenday.ThegoalisforstudentstorecognizethatFreddysellsmorefoodonaweekendday

thanhedoesonaweekday,however,thepercentageofeachfoodtypestaysthesame.Inother

words,thelikelihoodthatarandomlyselectedcustomerwouldorderchickenisindependentasto

whetherornotitisaweekdayoraweekend.

CoreStandardsFocus:

S.CP.2:UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurring

togetheristheproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyare

independent.

S.CP.3:UnderstandtheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpret

independenceofAandBassayingthattheconditionalprobabilityofAgivenBisthesameasthe

probabilityofA,andtheconditionalprobabilityofBgivenAisthesameasthe

probabilityofB.



eventsareindependentandtoapproximateconditionalprobabilities.Forexample,collect

datafromarandomsampleofstudentsinyourschoolontheirfavoritesubjectamongmath,science,

andEnglish.Estimatetheprobabilitythatarandomlyselectedstudentfromyourschoolwillfavor

sciencegiventhatthestudentisintenthgrade.Dothesameforothersubjectsandcomparethe

results.


PPOBABILITY- 9.5


S.CP.5:Recognizeandexplaintheconceptsofconditionalprobabilityandindependencein

everydaylanguageandeverydaysituations.Forexample,comparethechanceofhavinglungcancer

ifyouareasmokerwiththechanceofbeingasmokerifyouhavelungcancer.

RelatedStandards:S.CP.6,S.MD.7+

TheTeachingCycle:

Launch(WholeClass):

BeginwithremindingstudentsaboutthetaskFriedFreddy’sandhowTyrellhelpedFreddy

determinetheamountofchickenandfishheshouldprepareeachday.Launchintothistaskby

sharingthescenario,thenhavestudentsanswerthethreequestions.


Insmallgroups,havestudentsmakeasmanyobservationsaspossibleastotheamountoffood

Freddyshouldprepare.Someobservationsmaybesimple(suchas‘Freddyservesmorechicken

thanfish’),butotherobservationsmayincludethefollowing:

• Additiveobservation:Freddyservesmorechickenthanfish:thereare14moreorders

ofchickenthanfisheachweekday.

• Probabilitystatement(multiplicativeobservation):Giventhatitisaweekday,55%of

Freddy’sbusinessordersarechicken.

• Probabilitystatement(multiplicativeobservation):Giventhatitisaweekend,55%of

Freddy’sbusinessordersarechicken.

• Probabilitystatement(multiplicativeobservation):55%ofFreddy’sbusinessordersare

chicken(regardlessofwhetheritisaweekdayorweekend).

• Additive:Thereare23moreordersoffishontheweekendthanduringtheweek.

• Additive:Thereare28moreordersofchickenontheweekendthanduringtheweek.

• Additive:Thereare51moreordersontheweekendthanduringtheweek.


PPOBABILITY- 9.5


• Multiplicative:Thereisa35.4%increaseinthenumberofordersforfishonthe

weekendthanduringtheweek.

• Multiplicative:Thereisa35.4%increaseinthenumberofordersforchickenonthe


• Multiplicative:Thereisa35.4%increaseinthenumberofordersforfoodonthe


• Probabilitystatement(multiplicativeobservation):45%ofFreddy’sbusinessorders

fish,regardlessofwhetherornotitisaweekdayoraweekend.

Therearemanyobservationsstudentscanmakeatthispoint.Thegoalistodistinguish

observationsthatareprobabilitystatementsversus‘additive’observations.


Beginthediscussionphasebyhavingstudentsshareobservationsingeneral,thenhoneinonthe

probabilitystatements.Selectastudenttosharewhohasnoticedthattheconditionalprobabilityis

thesameasageneralprobabilitystatement.Atthispoint,introducethevocabularyofindependent

eventsandexplainthattwoeventsareconsideredindependentwhentheconditionalprobabilityof

AgivenBisthesameastheprobabilityofA.Next,showtheconditionalprobabilitytestfor

independenceusingnotation(forexample:P(fish|weekday)=f(fish)).Askstudentstoconsiderthe

examplejustsharedandtodetermineiftheexampleshowstwoindependentevents.Havestudents

sharewiththeirpartnerandexplainwhytheeventsareindependentornot.

Haveanotherstudentshareadifferentobservationusingaprobabilitystatement,andthenaskthe

classtofindoutiftheseareindependentevents.Attheendofthistask,allstudentsshouldbeable

torecognizethatfishisindependentfromthedayoftheweekbecausethepercentofpeoplewho

preferfish,giventhatitisaweekday(orweekend)isthesameasthepercentofpeoplewhoprefer

fishingeneral.

Usenotationtoshowthisrelationship.Dothesamewithchickengiventhatitisaweekday.Show

thisdatausingatreediagramandaVenndiagram.Whatdostudentsnoticeregardingindependen

cewhendataispresentedinaparticularrepresentation?Again,usenotationtoshowindependence

andthenconnectthistoeachrepresentationyouhavedrawn.Concludethetaskbywriting


PPOBABILITY- 9.5


independencestatements,drawingarepresentationmodelthatmatchesandconnectingto

standardsS.CP.2,S.CP.3,S.CP.4,andS.CP.5.



PROBABILITY – 9.5




9.5


READY

Topic:QuadraticfunctionsFindthex-intercepts,y-intercept,lineofsymmetryandvertexforthequadraticfunctions.

1.!(!) = !! + 8!– 9

2.! ! = !!– 3! – 5 3.ℎ ! = 2!! + 5! − 3

4.!(!) = !! + 6! – 9 5.!(!) = (! + 5)!– 2 6.!(!) = (! + 7)(! – 5)

SET Topic:IndependenceDeterminingtheindependenceofeventscansometimesbedonebybecomingfamiliarwiththecontextinwhichtheeventsoccurandthenatureoftheevents.Therearealsosomewaysofdeterminingindependenceofeventsbasedonequivalentprobabilities.

• Twoevents,AandB,areindependentifP(AandB)=P(A)∙P(B)• Additionally,twoevents,AandB,areindependentifP(A|B)=!(! !"# !)

!(!) =P(A)

Usethesetwowaysofdeterminingindependenteventstodetermineindependenceintheproblemsbelowandanswerthequestions.

7.P(AandB)=!!

P(A)=!!P(B)= !!"

8.P(A)=!!

P(AandB)=!!

P(B)=!!

9.P(A)=!!

P(AandB)=!!

P(B)=!!

10.P(AandB)=!!

P(A)=!!P(B)=!!


21


PROBABILITY – 9.5




9.5


GO Topic:FindProbabilitiesfromatwo-waytableThefollowingdatarepresentsthenumberofmenandwomenpassengersaboardthetitanicandwhetherornottheysurvived.

Survived Didnotsurvive Total

Men 146 659 805

Women 296 106 402

Total 442 765 1207

11. P(w)=

12. P(s)=

13. P(s|w)=

14. P(wors)=

15. P(worm)=

16. P(ns|w)=

17. P(m∩ns)=

22


PPOBABILITY- 9.6


9.6 Striving for Independence A Practice Understanding Task

Answerthequestionsbelowusingyourknowledgeofconditional

probability(theprobabilityofAgivenBasP(AandB)/P(B))as

wellasthedefinitionofindependence.Twoevents(AandB)aresaidtobeindependentif

! !|! = ! ! !"# ! !|! = !(!).Keeptrackofhowyouaredeterminingindependenceforeachtypeofrepresentation.

1. Outofthe2000studentswhoattendacertainhighschool,1400studentsowncellphones,

1000ownatablet,and800haveboth.CreateaVenndiagrammodelforthissituation.Use

properprobabilitynotationasyouanswerthequestionsbelow.

a) Whatistheprobabilitythatarandomlyselectedstudentownsacellphone?

b) Whatistheprobabilitythatarandomlyselectedstudentsownsbothacellphoneanda

tablet?

c) Ifarandomlyselectedstudentownsacellphone,whatistheprobabilitythatthis

studentalsoownsatablet?

d) Howarequestionsbandcdifferent?

e) Aretheoutcomes,ownsacellphoneandownsatablet,independent?Explain.

2. BelowisapartiallycompletedtreediagramfromthetaskChocolatevsVanilla.

a) Circlethepartsofthediagramthatwouldbeusedtodetermineifchoosingchocolateis

independentofbeingamaleorfemale.

b) Completethediagramsothatchoosingchocolateisindependentofbeingmaleorfemale.

http://www.flickr.com

/photos/ronw

ls/

23


PPOBABILITY- 9.6


3. UsethedatafromtheTitanticbelowtoanswerthefollowingquestions.

Survived Didnotsurvive TotalMen 146 659 805Women 296 106 402Total 442 765 1207

a) Determineifsurvivalisindependentofbeingmaleforthisdata.Explainorshowwhyor

whynot.Ifitisnotindependentdeterminehowmanymenwouldneedtosurvivein

ordertomakeitindependent.

4. Determinewhetherthesecondscenariowouldbedependentorindependentofthefirst

scenario.Explain.

a) Rollingasix-sideddie,thendrawingacardfromadeckof52cards.

b) Drawingacardfromadeckof52cards,thendrawinganothercardfromthesamedeck.

c) Rollingasix-sideddie,thenrollingitagain.

d) Pullingamarbleoutofabag,replacingit,thenpullingamarbleoutofthesamebag.

e) Having20treatsinfivedifferentflavorsforasoccerteam,witheachplayertakinga

treat.

24


PPOBABILITY- 9.6


9.6 Striving for Independence – Teacher Notes A Practice Understanding Task

Purpose:Thepurposeofthistaskisforstudentstopracticedeterminingwhetheroneeventis

independentofanotherevent.Studentswillusedatafromdifferentrepresentations,plusmake

senseofwhetherornotonescenariowouldbeindependentofanother.Intheend,studentswill

explainhowtoquicklydetermineindependencefromaVenndiagram,atreediagram,andatwo-

waytable.

CoreStandardsFocus:

S.CP.2:UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurring

togetheristheproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyare

independent.

S.CP.3UnderstandtheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpret

independenceofAandBassayingthattheconditionalprobabilityofAgivenBisthesameasthe

probabilityofA,andtheconditionalprobabilityofBgivenAisthesameasthe

probabilityofB.

S.CP.4Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesare


eventsareindependentandtoapproximateconditionalprobabilities.Forexample,collect

datafromarandomsampleofstudentsinyourschoolontheirfavoritesubjectamongmath,science,

andEnglish.Estimatetheprobabilitythatarandomlyselectedstudentfromyourschoolwillfavor

sciencegiventhatthestudentisintenthgrade.Dothesameforothersubjectsand

comparetheresults.

S.CP.5Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineveryday

languageandeverydaysituations.Forexample,comparethechanceofhavinglungcancerifyouare

asmokerwiththechanceofbeingasmokerifyouhavelungcancer.


PPOBABILITY- 9.6




RelatedStandards:S.CP.1,S.CP.7

TheTeachingCycle:

Launch(WholeClass):

Startthistaskbyreviewingthedefinitionofindependence,thenhavestudentsworkinpairsto

practiceusingprobabilityanddeterminingiftwoeventsareindependentusingdatafromdifferent

representations.


Asyoumonitor,lookforstudentstousetheirknowledgeofconditionalprobabilityandtheformula

forindependencetodeterminewhethertwoeventsareindependentforeachsituation.Select

studentstosharehowtherepresentationandnotationareconnected,andexplainwhetherthetwo

eventsareindependent.


Thegoalofthewholegroupdiscussionisforstudentstobeabletofluentlywriteconditional

probabilitystatements,tomakesenseofconditionalprobabilityusingthedifferentrepresentations

theyhavebeenusingthroughoutthismodule,andtodetermineiftwoeventsareindependent.

Usingthegroupsyouselectedduringtheexplorephase,sequencetheorderofhowtheysharetheir

problemtoreachthegoalsofthetask.Foreachsituation,connecttherepresentationbeingusedto

theformulaforindependence.



PROBABILITY – 9.6




9.6


READY

Topic:SolvingquadraticsSolveeachofthequadraticsbelowusinganappropriatemethod.

1.m2+15m+56=0 2.5x2–3x+7=0

3.x2−10x+21=0 4.6x2+7x–5=0

SET Topic:RepresentingIndependentEventsinVennDiagramsIneachoftheVennDiagramsthenumberofoutcomesforeacheventaregiven,usetheprovidedinformationtodeterminetheconditionalprobabilitiesorindependence.ThenumbersintheVennDiagramindicatethenumberofoutcomesinthatpartofthesamplespace.5.

a.Howmanytotaloutcomesarepossible?b.P(A)=c.P(B)=d.P(A∩B)=e.P(A|B)=

f.AreeventsAandBindependentevents?Whyorwhynot?


25


PROBABILITY – 9.6




9.6


6.

a.Howmanytotaloutcomesarepossible?b.P(E)=c.P(F)=d.P(E∩F)=e.P(E|F)=

f.AreeventsEandFindependentevents?Whyorwhynot?7.

a.Howmanytotaloutcomesarepossible?b.P(X)=c.P(Y)=d.P(X∩Y)=e.P(X|Y)=

f.AreeventsXandYindependentevents?Whyorwhynot?8.

a.Howmanytotaloutcomesarepossible?b.P(K)=c.P(L)=d.P(K∩L)=e.P(K|L)=

f.AreeventsKandLindependentevents?Whyorwhynot?

26


PROBABILITY – 9.6




9.6


GO Topic:ConditionalProbabilityandIndependenceDatagatheredontheshoppingpatternsduringthemonthsofAprilandMayofhighschoolstudentsfromPeanutVillagerevealedthefollowing.38%ofstudentspurchasedanewpairofshorts(callthiseventH),15%ofstudentspurchasedanewpairofsunglasses(callthiseventG)and6%ofstudentspurchasedbothapairofshortandapairofsunglasses.9.Findtheprobabilitythatastudentpurchasedapairofsunglassesgiventhatyouknowtheypurchasedapairofshorts.P(G|H)=10.Findtheprobabilitythatastudentpurchasedapairofshortsorpurchasedanewpairofsunglasses.P(H∪G)=11.Giventheconditionthatyouknowastudenthaspurchasedatleastoneoftheitems.Whatistheprobabilitythattheypurchasedonlyoneoftheitems?12.ArethetwoeventsHandGindependentofoneanother?WhyorWhynot?Giventhedatacollectedfrom200individualsconcerningwhetherornottoextendthelengthoftheschoolyearinthetablebelowanswerthequestions.

For Against NoOpinion Youth(5to19) 7 35 12 Adults(20to55) 30 27 20 Seniors(55+) 25 16 28

20013.Giventhatconditionthatapersonisanadultwhatistheprobabilitythattheyareinfavorofextendingtheschoolyear?P(For|Adult)=14.GiventheconditionthatapersonisagainstextendingtheschoolyearwhatistheprobabilitytheyareaSenior?P(Senior|Against)=15.Whatistheprobabilitythatapersonhasnoopiniongiventhattheyareayouth?P(noopinion|youth)=

27

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working ST Core Reg - Mathematics Vision ProjectMathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 3. Write several other