Tree diagrams and the binomial distribu2on

Tree diagrams and the binomial distribu2on

Outline for today

Be#erknowaplayer:HonusWagnerQues5onsaboutworksheet6?Reviewofconceptsinprobability:addi5veandmul5plica5verulesTreediagramsandtheanalysisofStrat-o-ma5cThebinomialdistribu5on

Announcement: class final projects

FinalprojectproposaldueonWednesdayShouldfocusonresearchques5on

•  Doesn’thavetobeaboutbaseball,butneedtofinddatasetthatyoucanusetoanswertheques5on

1-2paragraphproposalisduetheWednesdayMarch29thProjectspresenta5onareonMay3rd–goodtostartworkingonthemsoon!

Be>er know a player: Honus Wagner

Probability

Probabilityisawayofmeasuringtheuncertaintyoftheoutcomeofanevent

Defini5ons:Samplespace

•  Allpossibleoutcomes

AnEvent•  Subsetofthesamplespace

Probabilitykeyproper5es:•  0≤Pr(X)≤1•  ΣPr(X=x)=1

Probability rules - Addi2ve rule

IftherearetwoeventsA,andB,thentheprobabilityofAorBhappeningis:

Pr(AorB)=Pr(A)+Pr(B)–Pr(A,B)

EventsarecalledmutuallyexclusiveifeventsAandBcannotbothoccur-i.e.,Pr(A,B)=0

Q:WhatwouldmutuallyexclusiveeventslooklikeintheVenndiagram?A:Thecircleswouldnotoverlap

Mul2plica2ve Rule

Pr(A,B)=Pr(A|B)×Pr(B)

ProbabilityofBhappening×ProbabilityofAhappeninggivenBhappened

Twoeventsareindependentif:Pr(A,B)=Pr(A)xPr(B)i.e.,iftheoccurrenceofBdoesnoteffecttheprobabilityofAhappening

Big League Baseball

Foranyonepitch(notassumingthattheplayisinplay),whatistheprobabilityofgedngahit?

•  Probabilityofasingle: 1/3·7/36+•  Probabilityofadouble: 1/3·1/36+•  Probabilityofatriple: 1/3·1/36+•  Probabilityofahomerun: 1/3·1/36=10/108

1 2 3 4 5 61 Single Out Out Out Out Error2 Out Double Single Out Single Out3 Out Single Triple Out Out Out4 Out Out Out Out Out Out5 Out Single Out Out Out Single6 Error Out Out Out Single Homerun

2ndDie

1st D

ie

Strat-o-ma2c

Muchmorecomplexboardgames•  TakesintoaccountHi#ersandPitchers•  Advancedversionaccountsforaddi5onalfactors(e.g.,ballparksetc.)

Eachplayerisrepresentedbyacard

1.Awhitesingledieisrolledtodeterminewhethertousehi#erorpitcher’scard:

•  1-3->hi#erscard•  4-6->pitcher’scard

2.Then,twodicearerolledandtheirsumdetermineswhichplayinthecardshouldbeused

3.Forsomeplaysaddi5onallya20sideddieisrolledtodeterminethefinaloutcome

•  andothertables/rulesonenneedtobeconsulted

Strat-o-ma2c rules and tree diagrams

Let’scalculatetheprobabilityofdifferentevents…

Calcula5ngtheprobabilityofgedngapar5cularcolumnpitcherorhi#er’scardispre#ysimple•  Answer?

Calcula5ngthesumofthetwodiceisali#lemoreinvolved…• Whatisthesamplespacehere?

•  i.e.,howmanypossibleoutcomesarethere?•  Canyoucalculatetheprobabilitydistribu5on?

Strat-o-ma2c: analysis

Fillinthetablebelowwiththesumofthetwodiceandthencalculatetheprobabilityofrollinga2toa12

Strat-o-ma2c: analysis

1 2 3 4 5 61234 756

1st D

ie

2ndDie

Fillinthetablebelowwiththesumofthetwodiceandthencalculatetheprobabilityofrollinga2toa12

Strat-o-ma2c: analysis

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

1st D

ie

2ndDie

Strat-o-ma2c: analysis

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

1st D

ie

2ndDie

2 3 4 5 6 7 8 9 10 11 121/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Whatistheprobabilityofrollinga1onthewhitedieandthengedngasumof7onthetworeddice?

•  1/6·6/36=6/216

Whatistheprobabilityofrollinga:•  2onthewhitedie….andthengedng…•  Sumof8onthetworeddice…andthen…•  Anumberfor1-8onthe20sideddie?

•  1/6·5/36·8/20=.00926

Strat-o-ma2c: analysis

2 3 4 5 6 7 8 9 10 11 121/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Whatistheprobabilityofrollinga1onthewhitedieora6onthewhitedie?•  1/6+1/6=2/6

Whatistheprobabilityofrollinga:•  5onthewhitedie…andthengedng…•  asumof8orasumof10onthetworeddice?•  1/6·(4/36+3/36)=.0324

Strat-o-ma2c: analysis

2 3 4 5 6 7 8 9 10 11 121/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Whatistheprobabilityofrollinga3onthewhitedie•  andthengedngasumof8tworeddice•  orasumof10onthetworeddice

•  andthena1-10onthe20sideddie?

Treediagram!

Strat-o-ma2c: analysis

2 3 4 5 6 7 8 9 10 11 121/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

3

1/6

5/36

3/361-10

1/6·5/36=.0231

10/20

8

10 1/6·3/36·½=.0069

.0231+.0069=.0300

Strat-o-ma2c: Pujols vs. Kershaw

Whatistheprobabilityofahidngadouble?

Strat-o-ma2c: Pujols vs. Kershaw

Whatistheprobabilityofahidngadouble?

Strat-o-ma2c: Pujols vs. Kershaw

2 3 4 5 6 7 8 9 10 11 121/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Tree diagram

1

6

6

1/6

1/6

5/36

5/361-146

14/20

Tree diagram

1

6

6

7

8

1/6

1/6

5/36

6/36

5/36

5/361-146

1-2

18-203/20

2/20

14/20

1/6·(5/36·3/20+6/36+5/36·2/20)+1/6·5/36·14/20=.04977

Strat-o-ma2c: Pujols vs. Kershaw

Whatistheprobabilityofahidngahomerun?

Strat-o-ma2c: Pujols vs. Kershaw

Whatistheprobabilityofahidngahomerun?Treediagram!1/6·[3/36+4/36+5/36·17/20]+1/6·(4/36·2/20)=.0594

Parametric probability models

Wehaveexploredprobabilityusing:•  Probabilityrulestocalculatetheprobabilityofanevent•  Dice/spinners

Weonenusemathema5calformulas,calledprobabilitydistribu5ons,tocalculatetheprobabilityofdifferentevents

Random variates

Wecanthinkofarandomvariateasarandomnumber•  Typicallyrandomvariatesaredenotedwithcapitalle#ers,e.g.,X

•  Xcantakeonvaluesinthesamplespace•  Whichinthiscaseisasetofnumbers

Wecanuseprobabilitydistribu5onstoassesstheprobabilitythatarandomvariateXwillhaveavaluebetweentwoothernumbers•  Nota5on:Pr(a<X<b)

Random variates

Randomvariatescanbeeither:•  Discrete:Xtakesonintegervalues•  Con5nuous:Xtakesonrealvalues

Sameproper5esofprobabilitydistribu5onsapply:•  Pr(a<X<b)≥0•  ΣPr(X=xi)=1

Bernoulli Distribu2on

Modelstheprobabilityoftwooutcomes:•  Success:X=1•  Failure:X=0•  E.g.,gedngahead(X=1)oratail(X=0)forflippingacoin

Modelhasoneparameterπ,whichistheprobabilityofgednga1•  E.g.,theprobabilityofgedngheadonacoinflip•  Pr(X=1)=?Pr(X=0)=?

Bernoulli Distribu2on

ProbabilitymassfuncFonstelltheprobabilityofeachoutcomeinthesamplespaceofadiscretedistribu5on

ForBernoulliDistribu5onswecanwritetheprobabilitymassfunc5onas:

Wecanalsoplottheprobabilitymassfunc5on•  Ifπ=.5whatwoulditlooklike?•  Ifπ=.9whatwoulditlooklike?

Parameterπ

Bernoulli Distribu2on

Canyouthinkofbaseballexample?•  Example:Theprobabilitythataplayergetonbaseforagivenatbat•  IfweweremodelingDavidOr5z,whatwouldagoodes5mateofπbe?•  OBP=.355,soagoodnumbertousewouldbeπ=.355

Binomial distribu2on

Modelstheprobabilityofhavingksuccessesoutofthentrials

•  Probabilityofsuccessoneachtrialisπ

Example:ifaplayercomestobat45mesinagame,whatistheprobabilityofgedngon-basek5mes

•  Whatisthesamplespace?

Assumesthesameprobabilityofgedngon-baseeachplateappearance(π)

•  Aretheassump5onsreasonableforthismodel?•  Nostreakiness

Binomial distribu2on

Probabilityofgedngksuccessesoutofntrialsis:

where

WhatvaluescanXtakeon(i.e.,whatisthesamplespace)?

•  krangesfrom0,1,…,n

Parameters:πandn

n choose k

Nchoosekfunc5ontellsushowmanywaystherearetoorderkitemsoutofntotal

Q:Howmanywaysaretheretochoose3thingsoutofatotalof8?

A:(8·7·6)/(3·2)=56

R:choose(n,k)