working st core reg - mathematics vision project · 9.4 visualizing with venn – a solidify...

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

working st core reg - mathematics vision project · 9.4 visualizing with venn – a solidify...

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