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ProbabilityUsingTwo-wayFrequencyTables

Atwo-wayfrequencytableisawayofrepresentingdatathatfitsintomultiplecategories.Anyprobabilityproblemthathasmorethanoneoverlappingcategorycanbere-writtenasatwo-wayfrequencytable.

Example:“Ifthereare8juniorbaseballplayers,4juniorsoccerplayers,7seniorbaseballplayers,5seniortrack&fieldathletes,and6seniorsoccerplayers…”

Tosetupmytable,Istartwithmytwocategories(whicharecalled“variables”):theirgradelevelandtheirsport,(includingarowandacolumntowriteinthetotals).Then,IfillintheinformationthatIknow.

Grade Grade

Sport

Junior Senior TOTAL

Sport

Junior Senior TOTAL

Baseball Baseball JUNIORBASEBALL

SENIORBASEBALL

Soccer → Soccer JUNIORSOCCER

SENIORSOCCER

Track&Field Track&

FieldJUNIOR

TRACK&FIELDSENIOR

TRACK&FIELD

TOTAL TOTAL

Grade

Sport

Junior Senior TOTAL

Baseball 8 7 15Baseball

Soccer 4 6 10Soccer

Track&Field 0 5 5

Track&Field

TOTAL 12Junior

18Senior

30TOTAL

So,accordingtothetable,theprobabilityofrandomlyselectingajuniorbaseballplayerwouldbe:

𝑃 𝑗𝑢𝑛𝑖𝑜𝑟 𝑏𝑎𝑠𝑒𝑏𝑎𝑙𝑙 𝑝𝑙𝑎𝑦𝑒𝑟 =8 𝐽𝑟 𝐵𝐵30 𝑇𝑜𝑡𝑎𝑙

=415

IfIwantedtheprobabilitythathewasajuniorORabaseballplayer,Iwouldusecountupthosecategories. Grade Acceptableoutcomes:

8Jr.baseball,7Sr.baseball4Jr.Soccer

&0Jr.Track&Field=19Total

𝑃 𝑗𝑢𝑛𝑖𝑜𝑟 𝑂𝑅 𝑏𝑎𝑠𝑒𝑏𝑎𝑙𝑙 𝑝𝑙𝑎𝑦𝑒𝑟

=1930

Sport

Junior Senior TOTAL

Baseball 8 7 15Baseball

Soccer 4 6 10Soccer

Track&Field 0 5 5

Track&Field

TOTAL 12Junior

18Senior

30TOTAL

Icouldalsohavedone15 𝑏𝑎𝑠𝑒𝑏𝑎𝑙𝑙 + 12 𝑗𝑢𝑛𝑖𝑜𝑟 − 8 𝑏𝑜𝑡ℎ = 27 − 8 = 19

Icouldalsofindtheprobabilityofrandomlyselectingabaseballplayergiventhatheisajunior(“giventhat”meanshehastobeajunior).Forthisprobability,Iwouldignorealloptionsthatarenotjuniors:

𝑃 𝐵𝑎𝑠𝑒𝑏𝑎𝑙𝑙 𝐽𝑢𝑛𝑖𝑜𝑟) =8 𝐵𝐵12 𝐽𝑟

=23

ProbabilityUsingTwo-wayFrequencyTables 2

EXAMPLE Color

Clothing

Item

Black Blue TOTAL

Shirts 12 8 20Jackets 3 2 5Pants 9 6 15TOTAL 24 16 40

a.Namethetwovariablesdisplayedinthetable.

Color&Clothingb.Ifanitemisselectedatrandom,whatistheprobabilitythatitisajacket?

𝑃 𝑏𝑙𝑢𝑒 𝑗𝑎𝑐𝑘𝑒𝑡 =𝑡𝑜𝑡𝑎𝑙 𝑗𝑎𝑐𝑘𝑒𝑡𝑠𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑡𝑜𝑡𝑎𝑙

=540

=18

c.Whatistheprobabilitythatarandomlyselecteditemisblackorashirt?

Theitemsthatmeettherequirementsare:12blackshirts,3blackjackets,9blackpants,

and8blueshirts= 12 + 3 + 9 + 8 = 32

𝑃 𝑏𝑙𝑎𝑐𝑘 𝑜𝑟 𝑠ℎ𝑖𝑟𝑡 =3240

=45

d.Whatistheprobabilitythatarandomlyselecteditemisapairofpantsgiventhattheitemisblue?

𝑃 𝑝𝑎𝑛𝑡𝑠|𝑏𝑙𝑢𝑒 =𝑝𝑎𝑛𝑡𝑠 𝑖𝑛 𝑏𝑙𝑢𝑒 𝑐𝑎𝑡𝑒𝑔𝑜𝑟𝑦

𝑏𝑙𝑢𝑒 𝑡𝑜𝑡𝑎𝑙=

616

=38

EXAMPLE Grade

ClubParticipation 9th 10th 11th 12th TOTAL

Isinaclub 105 125 147 101 478

Isnotinaclub 78 92 75 122 367

TOTAL 183 217 222 223 845

a.Namethetwovariablesdisplayedinthetable.Grade&ClubParticipation

b.Ifastudentisselectedatrandom,whatistheprobabilitythatheorshewillbeinaclub?

𝑃 𝑖𝑛 𝑎 𝑐𝑙𝑢𝑏 =𝑡𝑜𝑡𝑎𝑙 𝑖𝑛 𝑐𝑙𝑢𝑏𝑠𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑡𝑜𝑡𝑎𝑙

=478845

c.Whatistheprobabilitythatarandomlyselectedstudentisinacluborin11thgrade?

Thestudentsthatmeettherequirementsare:1059thinclubs,12510thinclubs,14711thinclubs,

10112thinclubsand7511thnotinclubs= 105 + 125 + 147 + 101 + 75 = 553

𝑃 𝑖𝑛 𝑐𝑙𝑢𝑏𝑠 𝑜𝑟 11𝑡ℎ =553845

d.Whatistheprobabilitythatarandomlyselectedstudentisinaclubgiventhatheorsheisin10th?

𝑃 𝑖𝑛 𝑐𝑙𝑢𝑏|10𝑡ℎ =𝑖𝑛 𝑐𝑙𝑢𝑏 𝑖𝑛 10𝑡ℎ10𝑡ℎ 𝑡𝑜𝑡𝑎𝑙

=125217

1. Employment

Gender

Hasajob Doesnothaveajob TOTAL

Male 27 36 63

Female 28 24 52

TOTAL 55 60 115

a.Namethetwovariablesdisplayedinthetable.b.Ifapersonisselectedatrandom,whatistheprobabilitythatheorshehasajob?c.Whatistheprobabilitythatarandomlyselectedpersonismaleorhasajob?d.Whatistheprobabilitythatarandomlyselectedpersonisfemalegiventhatthepersonhasajob?

2. Color

Item

Yellow Pink Silver TOTAL

Post-it 58 17 0 75

PaperClip 7 25 78 110

TOTAL 65 42 78 185

a.Namethetwovariablesdisplayedinthetable.b.Ifanitemisselectedatrandom,whatistheprobabilitythatitisapaperclip?c.Whatistheprobabilitythatarandomlyselecteditemispinkorapost-it?d.Whatistheprobabilitythatarandomlyselecteditemissilvergiventhatit'sapaperclip?

ProbabilityUsingTwo-wayFrequencyTables 3

3. Grade

Gender

9th 10th 11th 12th TOTAL

Male 204 179 165 202 750

Female 170 246 143 131 690

TOTAL 374 425 308 333 1440

a.Namethetwovariablesdisplayedinthetable.

b.Ifastudentisselectedatrandom,whatistheprobabilitythatheorshewillbein9thgrade?

c.Whatistheprobabilitythatarandomlyselectedstudentisin10thgradeorfemale?

d.Whatistheprobabilitythatarandomlyselectedstudentisin11thgradegiventhatthestudentisfemale?

4. SportsParticipation

ClubParticipation Tennis Soccer Notin

Sports TOTAL

Isinaclub 145 106 138 389

Isnotinaclub 123 164 108 395

TOTAL 268 270 246 784

a.Namethetwovariablesdisplayedinthetable.

b.Ifapersonisselectedatrandom,whatistheprobabilitythatheorsheplaystennis?

c.Whatistheprobabilitythatarandomlyselectedstudentplayssoccerorisnotinaclub?

d.Whatistheprobabilitythatarandomlyselectedstudentisnotinaclubgiventhatheorshedoesnotplaysports?

5. Employment

Age

Hasajob Doesnothaveajob TOTAL

13-15 38 126 164

16-17 109 80 189

18-20 211 48 259

TOTAL 358 254 612

a.Namethetwovariablesdisplayedinthetable.

b.Ifapersonisselectedatrandom,whatistheprobabilitythatheorshehasajob?

c.Whatistheprobabilitythatarandomlyselectedpersonis16-17yearsoldorhasajob?

d.Whatistheprobabilitythatarandomlyselectedpersonhasajobgiventhatheorsheis18-20yearsold.

6. Grade

PhonePreference 9th 10th 11th 12th TOTAL

Android 250 341 266 286 1143

iPhone 294 277 332 276 1179

TOTAL 544 618 598 562 2322

a.Namethetwovariablesdisplayedinthetable.

b.Ifastudentisselectedatrandom,whatistheprobabilitythatheorshepreferstheiPhone?

c.WhatistheprobabilitythatarandomlyselectedstudentprefersAndroidorisin12thgrade?

d.WhatistheprobabilitythatarandomlyselectedstudentprefersAndroidgiventhatheorsheisin12thgrade?

“ProbabilityUsingTwo-wayFrequencyTables“Answers

1𝑎. Employment&Gender

1𝑏. 1123

1𝑐. 91115

1𝑑. 2855

2𝑎. Color&Item

2𝑏. 2237

2𝑐. 2037

2𝑑. 1

3𝑎. Grade&Gender

3𝑏. 187720

3𝑐.8691440

3𝑑.143690

4𝑎. Sports&ClubParticipation

4𝑏.67196

4𝑐.501784

4𝑑.1641

5𝑎. Employment&Age

5𝑏.179306

5𝑐.73102

5𝑑.211259

6𝑎. PhonePreference&Grade

6𝑏.131258

6𝑐.14192322

=1118

6𝑑.143281