statistics, data analysis, and probability activity set 2

Statistics, Data Analysis, and Probability

Activity Set 2

Trainer Guide

StatiSticS, Data analySiS, anD probability—activity Set 2 Int_DAP_02_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

Flip It

In this activity, participants explore probability concepts by using coins.

materials

• Transparency/Page:TreeDiagram• Transparency/Page:FundamentalPrincipleofCounting• Transparency/Page:ModelEPDLE• Transparency/Page:MoreChoices• Transparency/Page:Counting• Transparency/Page:NumberCubeTreeDiagram• Transparency/Page:FlipItOutcomes• Transparency/Page:FlipItOutcomes—Pennies• Transparency/Page:ProbabilityDefinitions• Transparency/Page:FlipItData• Transparency/Page:FlipItDataSummary• 1 nickel and 1 penny per pair of participants

Vocabulary

• theoreticalprobability• experimentalprobability• statistics• samplespace• outcome

time:40minutes

IntroDuCe

•Suggesttoparticipantsthatinlifewefrequentlymakepredictionsbasedoninformationthatwehaveaboutthecircumstancesthatsurroundevents.

StatiSticS, Data analySiS, anD probability—activity Set 2 Int_DAP_02_TG

Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development 1

statIstICs, Data analysIs, anD ProbabIlItyaCtIvIty set #2

NGSSS 4.A.4.2

•Askparticipantstoconsiderthepossibleoutcomesthattheymightexpectiftheyweretoflip2coins, 1 penny and 1 nickel.

•Askthemhowmanyoutcomestherewouldbe.

•DisplayTransparency:TreeDiagram.

•Tracethepathsacrossthetreetoeachpossibleoutcomeforflipping1pennyandthen1nickel.

•Explainthatatreediagramcanbeusedtoshowpossibleoutcomesforeventsthatincludemorechoicesthan2.Onesimplyaddsadditionalbranches to the tree.

•Pointoutthatthelistofelementsattherightshowsthetotalnumberofpossibleoutcomes—inthis case, 4.

•DisplayTransparency:FundamentalPrincipleofCounting.

•Pointouttoparticipantsthatiftherearemoutcomesforthefirsteventand,foreachofthoseoutcomes,there are noutcomesforthesecondevent,thenthereare m•noutcomespossibleinall.

•Pointoutthatthetreediagramshowsthatthereare 2possibleoutcomesforthefirstflip(thepenny):head or tail.

•Explainthatforeachsuccessiveevent(theflipofthenickel)thereare2additionalpossibleoutcomes:heador tail.

•Pointoutthatthissuggeststhattherewillbem•noutcomesor2•2outcomes,4possibleoutcomes in all.

•DisplayTransparency:ModelEPDLE and have participantstakeouttheirmatchingpages.




Coin 1Possibilities

Flip ItOutcomes

penny:head

penny:tail

penny: headnickel: head

penny: tailnickel: head

penny: headnickel: tail

penny: tailnickel: tail

Coin 1 + Coin 2Possibilities

tree Diagram

McGraw-Hill Professional Development StatiSticS, Data analySiS, anD probability/21

The rightmost line of the tree shows the total number ofpossible outcomes: 4.

Transparency: Tree Diagram

When there are m outcomes for the first event and,for each of those outcomes, there are n outcomes forthe second event, then there are m • n outcomespossible in all.

m events • n events = total possible outcomes

Consider the coin toss.

2 possible outcomes on the first flip (m)times

2 possible outcomes on the second flip (n)

m • n outcomes 4 possible outcomes in all

Fundamental Principle of counting


Transparency: Fundamental Principle of Counting

StatiSticS, Data analySiS, anD probability—activity Set 2 Trans_K6_SP_02Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Exterior Interior Sound System

Model EPD LE

Granite Gray

Gracious Gray

radio radio, cd player radio, cd player, extra speakers radio, cd player, extra speakers, Dolby®

Elegant Ivory

radio radio, cd player radio, cd player, extra speakers radio, cd player, extra speakers, Dolby

Silky Sand


Forest Green

Gracious Gray


Elegant Ivory


Silky Sand


Burnished Bronze

Gracious Gray


Elegant Ivory


Silky Sand


Model EPD LE

Transparency: Model EPD LE

•Explaintoparticipantsthatthetreediagramrepresents the various selection options available to acustomerwhoisinterestedinpurchasingaspecificmodelofnewcar.

•Pointouthowtoworkdownthetreetoselectanautomobilebymakingonechoicefromeachoptiondisplayed.

•Askparticipantshowmanypossiblecombinationsofexterior/interior/soundsystemsfortheautomobilesaredisplayedbythetree.(36)

•Explainthatthenumbercanbefoundbycountingfromthetreediagram.

•DisplayTransparency:MoreChoices.

•Explaintoparticipantsthatthisissimplyonebranch of the existing tree that has been expanded to include tire choices.

•Pointouttoparticipantshowdifficultitwouldbetoadd this extension to each existing branch of the treediagram.

•PointoutthattheFundamentalPrincipleofCountingwouldallowthemtofindmoreeasilythetotalpossibleoutcomesregardlessofthenumberof options offered.

•Theycan,usingtheFundamentalPrincipleofCounting,countthenumberofchoicesforeachoptionandmultiplythesetogether.

•DisplayTransparency:Counting.

•Writeanequationintheblanksonthetransparencyshowingexteriorchoices(3)timesinteriorchoices(3)timessoundsystemchoices(4)equals36.

•Askhowmanytotalconfigurationswouldbeavailable if 4 tire choices were offered on the car. (33 4 4=144)




StatiSticS, Data analySiS, anD probability—activity Set 2 trans_K6_SP_02Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development


Model EPD LE

Granite Gray

Gracious Gray

maxi radial all weather all terrain




Tires

radio

radio, CD player

radio, CD player, extra speakers

radio, CD player, extra speakers, Dolby®

More Choices

Transparency: More Choices

StatiSticS, Data analySiS, anD probability—activity Set 2 trans_K6_SP_02Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

� � � =

Exterior Interior Sound System Tires

radio

radio, CD player

radio, CD player, extra speakers

radio, CD player, extra speakers, Dolby®

maxi

all weather

radial

all terrain

Model EPD LE

Forest Green

Granite Gray

Burnished Bronze

Gracious Gray

Elegant Ivory

Silky Sand

Counting

Transparency: Counting

•DisplayTransparency:NumberCubeTreeDiagram.

•Pointouttoparticipantshowcomplexthediagrambecomes.

•Askparticipantshowmanyoutcomesthereare.SuggestthattheyusetheFundamentalPrincipleofCountingtodeterminetheanswer.(36=6outcomesfornumbercube 1 6outcomesfornumbercube2.)

•AskparticipantstosuggestsomeothersituationsinwhichtheuseoftheFundamentalPrincipleofCountingwouldbehelpful.Somepossibleanswersinclude counting configurations of clothing in a child’s closet(shirts slacks jackets...)anddetermininghowmanyuniquemealscanbeorderedfromamenu.

•DisplayTransparency:FlipItOutcomes.

•Haveparticipantstakeouttheirmatchingpages.

•Suggesttoparticipantsthatthisisanotherwaytodeterminetheoutcomesofflippingtwocoins.

•Tellparticipantsthattheshadedboxesofthegridshows the samplespace,orthepossibleoutcomes.

•Explaintoparticipantsthattheprobabilityofaneventis the chance of that event occurring. Forexample,inthegrid,whatistheprobabilityofpenny:head/nickel:headoccurring?(1chanceout of4possibleoutcomes)

•Pointouthowprobabilitiesarewritten.Theyarewrittenasfractions:Thenumeratorrepresentsthenumberofwaysaspecificevent can occur, and the denominatorrepresentsthenumberofallpossible outcomes.

•Explainthatthissameinformationalsocanbewrittenas1:4,wherethenumber1representsthenumberofways a specific event can occur and the 4 represents thenumberofallpossibleoutcomes.

•Write 14 inthefirstratiocolumn.




1

12

456

3

2

12

456

3

3

12

456

3

4

12

456

3

5

12

456

3

6

12

456

3

Cube 1 Cube 2 Total

Number Cube Tree Diagram


Transparency: Number Cube Tree Diagram

Flip it outcomes


Penny Flip

Nic

kel

Fli

p

Probabilities

Outcome Ratio Decimal Percent

penny: head/nickel: head

penny: head/nickel: tail

penny: tail/nickel: head

penny: tail/nickel: tail

Total

Transparency: Flip It Outcomes

•Write1:4inthesecondratiocolumn(abovetheblackbox).

•Askparticipantsiftheyhaveseenprobabilitieswritteninotherformsaswell.(percentsanddecimals)

•Explainthattheprobability 14 also can be written as 0.25and25%.

•Writethesenumbersinthecorrectcolumns.

•Tellparticipantstodeterminetheprobabilityof:

◆ penny:head/nickel:tail( 14 ,1:4,0.25,25%)

◆ penny:tail/nickel:head( 14 ,1:4,0.25,25%)

◆ penny:tail/nickel:tail( 14 ,1:4,0.25,25%)

•DisplayTransparency:FlipItOutcomes—Pennies.

•Gothroughthesamestepsaswiththeprevious transparency.

•Pointoutthatbecauseorderdoesnotmatterinthisexample(wecannotdistinguishbetweenthepennies),head/tailandtail/headmustbeconsideredtogether.Therefore,weaddtheprobabilitiesforthe 2occurrences.Theprobabilityof1headand1tailbecomes1

4 + 14 , or 1

2 .

•Haveparticipantswritetheratioanddecimalequivalents next to the fractional probabilities on the transparency.

•Pointouttoparticipantsthatineachcolumn,theprobabilitiesofallpossibleoutcomesaddupto1—or100%whenusingpercents.(Addthefractionsor decimalsineachcolumn.)

•Explaintoparticipantsthatbecauseallpossibleoutcomesareequaltothe“whole”samplespace,thefractionalamountsthatexplaineachpossibleoutcomemustalwaysadduptothewhole,or1.





Flip it outcomesPennies

Probabilities


head/head

head/tail

tail/tail

Total

Head Tail

Ta

ilH

ead

Transparency: Flip It Outcomes—Pennies

•Explaintothegroupthattheyhavebeenconsideringtheoreticalprobability, the possibleoutcomesofanexperiment—inthiscase,4.

•DisplayTransparency:ProbabilityDefinitions.

•Pointouttoparticipantsthatthespinnerisdividedinto2halves—thereare2possibleoutcomes—2colorsonwhichthespinnermayland,andeachoutcomehas an equal chance of occurring.

•Readoverthedefinitionoftheoreticalprobability andpointoutthatbecausethereare2possibleoutcomes,thedenominatorofthefractionis2. Thefavorable,ordesiredoutcome,isyellow—1of the2,1

2 .P(yellow)=12 .

•Readoverthedefinitionofexperimentalprobability—thenumberofactualoutcomesdividedbythe numberoftrials.

•Suggesttoparticipantsthatiftheyweretospinthespinner100times,itmightnotlandonyellowexactly50times.Thenumberoftimesitdoesland onyellowdividedbythenumberoftrials(100) wouldbetheexperimentalprobability.Ifthespinnerlandsonyellow47times,theexperimentalprobability would be 47

100.

•Askparticipantswhatthetheoreticalprobabilityoflandingonredis.(P(red)=1

2 —thereisthesameprobabilityofgettingredasthereisofgettingyellow.)

•Remindparticipantsthatinreality,outcomesseldomexactlymatchtheoreticalprobability.

•Tellparticipantsthatnowtheywilltestthe theoretical probabilities that were just defined.





Theoretical Probability—the number of favorableoutcomes divided by the number of possibleoutcomes. P(yellow) = 12. (This may also be written as the ratio 1:2.)

Experimental Probability—the number of actualoutcomes divided by the number of trials.

Probability Definitions

Transparency: Probability Definitions

DIsCuss anD Do

•Askparticipantstoworkinpairsandtotakeouttheir FlipItDatapages.

•DisplayTransparency:FlipItData.

•Haveeachpairflip1pennyandthen1nickel.

•Havethemrecordatallymarkonthedatasheetundertheappropriateoutcome.

•Haveparticipantsrepeatthisprocedure9moretimes.

•Haveparticipantswritethetotalnumberoftalliesineachcolumnaftertheyhavecompleted10flips.

•Call the group together.

•DisplayTransparency:FlipItDataSummary.

•Collectthesummarynumbersfromthefirst5groups.Enterthesenumbersonthesummarytransparency.

•Addthesubtotalsandshowthemasthenumeratorson the transparency.

•Pointoutthatthesenumbersrepresenttheexperimentaloutcomes.

•Askparticipantswhatthetheoreticalprobabilitieswerefortheseoutcomes.

◆ penny:head/nickel:head

◆ penny:head/nickel:tail

◆ penny:tail/nickel:head

◆ penny:tail/nickel:tail

•Ifparticipantsdonotremember,redisplayTransparency:FlipItOutcomes.

•Comparethesefractionswiththetheoreticalprobabilities,emphasizingplaceswheretheydo notmatch.

•Addthedatafromthenext5groupsandrepeattheprevious steps.




Flip it Data


Penny: Head/ Penny: Head/ Penny: Tail/ Penny: Tail/Nickel: Head Nickel: Tail Nickel: Head Nickel: Tail

Total Numberof Tallies

Transparency: Flip It Data

Flip it outcomes


Penny Flip

Nic

kel

Fli

p

Probabilities






Total

Transparency: Flip It Outcomes


Flip it Data summary

Penny: Head/ Penny: Head/ Penny: Tail/ Penny: Tail/

Nickel: Head Nickel: Tail Nickel: Head Nickel: Tail

Group 1

Group 2

Group 3

Group 4

Group 5

subtotal oftallies over

50

Group 6

Group 7

Group 8

Group 9

Group 10

subtotal oftallies

(If all 10 groups,then show

n100 .)

Group 11

Group 12

Group 13

Group 14

Group 15

total Number of tallies

(Multiply thetotal number

of groups by 10 and

make this thedenominator.)

100 100 100 100

xxx xxx xxx xxx

50 50 50 50

Transparency: Flip It Data Summary

◆ Addalltheresults.

◆ Comparefractionstotheoreticalprobabilities.

•Pointoutthatasthenumberofexperimentalresultsincreases,itismorelikelythattheresultswillbecloser to the theoreticalprobability.

•Repeatwiththeremaininggroups.

ConCluDe

•Askparticipantstodefinethefollowing:

◆ theoreticalprobability—thechanceofaneventoccurringorthenumberoffavorableoutcomesdivided by the numberofpossibleoutcomes

◆ experimentalprobability—thenumberofactual outcomesdividedbythenumberoftrials

◆ outcome—theresultofanexperiment;ageneric element

◆ event—aspecificoutcome

◆ trial—eachrepetitionofanexperiment

•Clarifyanymisconceptions.

•Explaintoparticipantsthatthetermssuccessfuloutcomeandfavorableoutcome refer only to the occurrenceoftheeventthatisbeingconsidered.Forexample,ifyouwantanoutcomeofhead/head,thengettinghead/headwouldbeafavorableoutcome.Neithertermimpliesquality.

end of Flip It




StatiSticS, Data analySiS, anD probability—activity Set 2 Int_DAP_02_PMCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

Coin 1Possibilities

Flip ItOutcomes

penny:head

penny:tail

penny: headnickel: head

penny: tailnickel: head

penny: headnickel: tail

penny: tailnickel: tail

Coin 1 + Coin 2Possibilities

Tree Diagram

The rightmost line of the tree shows the total number of possible outcomes: 4.


When there are m outcomes for the first event and, for each of those outcomes, there are n outcomes for the second event, then there are m • n outcomes possible in all.

m events • n events = total possible outcomes

Consider the coin toss.

2 possible outcomes on the first flip (m)times

2 possible outcomes on the second flip (n)

m • n outcomes 4 possible outcomes in all

Fundamental Principle of Counting



Model EPD LE

Granite Gray

Gracious Gray

radio radio, cd player radio, cd player, extra speakers radio, cd player, extra speakers, Dolby®

Elegant Ivory


Silky Sand


Forest Green

Gracious Gray


Elegant Ivory


Silky Sand


Burnished Bronze

Gracious Gray


Elegant Ivory


Silky Sand


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Flip It Outcomes

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Total

Flip It Outcomes


Flip It OutcomesPennies

Probabilities


head/head

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Total

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Theoretical Probability—the number of favorable outcomes divided by the number of possible outcomes. P(yellow) = 12. (This may also be written as the ratio 1:2.)

Experimental Probability—the number of actual outcomes divided by the number of trials.

Probability Definitions

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100 100 100 100

xxx xxx xxx xxx

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n

Penny: Head/ Penny: Head/ Penny: Tail/ Penny: Tail/

Nickel: Head Nickel: Tail Nickel: Head Nickel: Tail

Group 1

Group 2

Group 3

Group 4

Group 5

Subtotal of Tallies over 50

Group 6

Group 7

Group 8

Group 9

Group 10

Subtotal of Tallies (If all 10 groups, then show

100 .)

Group 11

Group 12

Group 13

Group 14

Group 15

Total Number of Tallies (Multiply the total number of groups by 10 and make this the denominator.)


GlossaryStatistics, Data Analysis, and Probability

bar graph A graph in which information is shown using rectangular bars or objects.

circle graph A graph in which information is shown as parts of a circle.

dependent event An event in which the outcome is affected by the outcome of a previous event.

event A specific outcome of an experiment.

experimental probability In an experiment, the ratio of actual outcomes to the number of possible outcomes.

frequency chart A data chart that shows the number of times that each item or number appears.

independent event An event in which the outcome is not affected by the outcome of a previous event.

mean The sum of a set of numbers divided by the number of numbers in the set; average (e.g., in the set {7, 3, 2, 4}, sum = 16 and mean = 16

4 = 4).

measures of central tendency Mean, median, and mode.


median In an odd set of ordered numbers, the middle number (e.g., {2, 3, 9, 12, 16}, median = 9); in an even set of ordered numbers, the average of the two middle numbers (e.g., {2, 3, 9, 11, 12, 16}, median = 10).

mode The number or element that occurs most frequently in a set of data (e.g., {2, 3, 4, 3, 1, 3, 2}, mode = 3).

outcome The result of an experiment.

probability The chance of an event occurring.

range The difference between the greatest and least values in a data set.

sample A subset of items taken at random from a complete set.

sample space The set of all possible outcomes.

sampling The method or process used to collect a sample.

statistics The use of mathematics to analyze a collection of quantitative data.

theoretical probability The ratio of the number of favorable outcomes to the total number of possible outcomes.

Glossary (continued)

statistics, data analysis, and probability activity set 2

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