VOL.81,NO.2,APRIL2008QuasigeometricDistributionsandExtraInningBaseball - PDF document

VOL.81,NO.2,APRIL2008QuasigeometricDistributionsandExtraInningBaseballGamesDARRENGLASSGettysburgCollege200N.WashingtonSt.Gettysburg,PA17325PHILIPJ.LOWRYCityUniversityofNewYorkNewYork,NY10016 VOL.81,NO.2,APRIL2008 .Thus,theentiredistributionisdeterminedbythevalueofItisastraightforwardcomputationtoseethatthemeanofthisdistributionis thevarianceis Quasigeometricdistributions.Inthispaper,wewishtodiscussavariationofge-ometricdistributionswhichcanreasonablybereferredtoasquasigeometricdistribu-tions,astheybehaveverysimilarlytogeometricdistributions.Thesedistributionsaredenedsothattheyaregeometricotherthanatastartingpoint.Inparticular,wewanttheretobeacommonratiobetweenforall1butnot(neces-sarily)thesameratiobetween.Tobeexplicit,wemakethefollowingEFINITION2.3.Aquasigeometricdistributionisadistributionsothatforallforsome01.Wecalldepreciationconstantassociatedtothedistribution.Justasgeometricdistributionsarecompletelydeterminedbythevalueof,aquasi-geometricdistributionisentirelydeterminedbythevaluesof(whichwewilloftendenoteby).Inparticular,acomputationanalogoustotheoneaboveshowsthatfor.Giventhis,itispossibletocomputethemeanandvarianceofthedistributionasfollows: 1Šd,2=n=0n2f(n)Šµ2=n=1n2(1ŠaŠd)dnŠ1Šµ2=(1ŠaŠd)n=1n2dnŠ1Šµ2 MATHEMATICSMAGAZINEnine-inninggame.Inordertodoso,werstmaketheassumptionthatallinningsareindependentofoneanother.Whilethisassumptionisalmostcertaintobeoverlystrong—teamsarelikelytofacesimilarpitchers,weather,andparkeffectsinconsec-utiveinnings—itgreatlysimpliestheproblem.Furthermore,wewillseethatitleadstomathematicalresultsthatmatchwithactualgamedata.Wedenotetheprobabilitythatateamscoresrunsinnineinningsas,andnotethatwherethesumrangesoverall9-tuplesofnonnegativeintegers,...,sumtoistheprobabilitythattheteamscoresrunsininningIfateamscoresrunsinagame,thenweknowthattheteammustscoreinbetweenoneandmindifferentinnings.Breakingupbythesecases,wecancomputewheretheinteriorsumisoverallordered-tuplesofpositiveintegerssummingtoIfwenowinvokeourassumptionthattheprobabilityofscoringagivennumberofrunsinaninningisquasigeometric(andindependentoftheinning),andthereforethatforall1,wecancalculatethatInthisformula,representsthenumberofinningsinwhichtheteamscores,representstheprobabilitythatateamgoesscorelessinagiveninning,andthedepreciationconstant,whichweareassumingisequalto0436forallteams.Onewaytoviewthetermisthatitcountsthenumberofwaystodividerunsamonginnings.Itwillbemoreusefultoustotranslatethisresultintermsofthestrengthofagiventeam.Todothis,wenotethatEquationshowedthattomodelateamthatscoresanaverageofrunsperinningweshouldchoose.Doingso,wecompute:whereagainisthedepreciationconstant0436andrepresentstheaveragenumberofrunsperinningthatateamscores.TABLE4computesforateamthatscoresthehistoricalaverageof0487runsperinningandcomparesthesevalueswiththeempiricaldistributionofrunspergamescoredbyNationalLeagueteamsbetween1969and2002.Oneseesthatthisquasigeometricmodelappearstogiveagoodapproximationofreality,andthereforewemightwanttoseehowthistypeofmodelcanbeusedtoanswermanydifferenttypesofquestions.Inthefollowingsection,wewilllookatthequestionofhowoftenweshouldexpectgamestolast20inningsormore,butbeforemovingontothat,wethinkitwouldbeinterestingtonotethatonecouldusethismodeltocomputetheoddsthatateamofagivenstrengthwouldbeatanotherteamofagivenstrength.Inparticular,wenotethatthe2003AtlantaBravesscoredanaverageof0618runsperinning,whereasthe2003NewYorkMetsscoredanaverageof0runsperinning.Whilethisisclearlyalopsidedmatchup,oneofthebeautifulthings MATHEMATICSMAGAZINEaretheprobabilitiesthatTeamAandTeamBscorerunsinnineinnings,theformulaforwhichwasgivenabove.Wenotethattheformulaabovetellsusthatifweassumebothteamsscorethemajorleagueaverageof0487runsperinning,then103,sothatwewouldexpectjustover10%ofgamestogointoextrainnings.Inreality,9.22%—18,440ofthe199,906majorleaguegamesplayedbetween1871and2005—havegoneintoextrainnings.Thediscrepancybetweenthisnumberandwhatourmodelpredictslikelyarisesfromtwofacts.First,ourmodelassumesthattheteamsarescoringindependentlyofoneanother.Inreality,thisassumptionislikelytobenotquitetrue,asexternalfactors(humidity,altitude,pitching,etc.)maycausegamestobeeitherhighorlowscoring,andtheremaybeapsychologicalfactorthatpromotesteamstoscoremoreiftheotherteamisafewrunsahead,ortostoptryingoncetheyareblowingouttheotherteam.Theotherfactorthatwecanthinkofistrickiertogetahandleon.Theabovecalcu-lationassumesthatbothteamsareaverage,butinmostgamesoneteamwillbebetterthantheother.Foranextremeexample,welookattheALEastin2003,wheretheDetroitTigersscoredanaverageof0405runsperinningandtheBostonRedSoxscoredanaverageof0659runsperinning.Thisisthelargestdiscrepancybetweentwoteamsinthesameleagueinover25years.Inthiscase,theformulapredictsthatonly84%ofgameswillgointoextrainnings.Whilethisspecicexampleisanex-treme,itsuggeststhatwhentwoteamshavedifferingabilitiestoscoreruns,weshouldexpectfewerextrainninggameseveniftheoverallaveragenumberofrunsscoredisheldconstant.ThisexpectationisconrmedbythedatainTABLE5,wheretherowsandcolumnsrepresentthestrengthsofthetwoteamsplaying,andistheprobabilitythattheywillbetiedafternineinnings,accordingtoourmodel.Giventhatalargenumberofgamesareplayedbetweenteamswithwidelydifferingabilitiestoscoreruns,thiswouldsuggestthatourmodelwillpredictalargernumberofextrainninggamesthanactuallyoccur.Aftertheninthinning,thegamewillconcludeattheendoftherstinningafterwhichthescoreisnottied.Therefore,ifweletbetheprobabilitythatthetwoteamsscorethesamenumberofrunsinagiveninning,thentheprobabilitythatagameisstilltiedafterinningsisandfor9theprobabilitythatitendsafterWenotethatwearemakingseveralassumptionshere.First,weareassumingthatthereisnoeffectivedifferencebetweenthetenthinningandanylaterinningasfarasoffensiveproductionisconcerned.Wealsoassumethat,atleastasfarasextrainningsgo,ifistheprobabilitythatthetwoteamsscorethesamenumberofrunsinagiveninningthentheprobabilitythattheyscorethesamenumberofrunsineachofTABLE5:Probabilityofatiegamebetweentwoteamsofvariousstrengths 0.437 0.487 0.537 0.617 0.659 0.405 0.1119 0.1065 0.1006 0.0903 0.0848 0.1097 0.1056 0.1007 0.0918 0.0869 0.1056 0.1033 0.1000 0.0932 0.0892 0.1007 0.1000 0.0982 0.0936 0.0905 0.0918 0.0932 0.0936 0.0921 0.0904 0.0869 0.0892 0.0905 0.0904 0.0896 MATHEMATICSMAGAZINETABLE6:Numberofgamesofagivenlengthpredictedvs.actualnumber #Innings Probingivengame ActualMLB ExpectedMLB 9 7.163E-05 3.982E-05 2.213E-05 1.230E-05 6.839E-06 3.802E-06 2.113E-06 1.174E-06 6.530E-07 3.630E-07 Total 199,906 be0939majorleaguegamesthatwouldhavelasted27ormoreinningsbynow.Infact,wehavenotyethadsuchagamein135yearsofmajorleagueplay.Theseresultsindicatethatthe26-inninggameinBostonisnotanoutlierfromwhatonewouldexpectfromourmodel.Ifweassumethatthescoringpatternsinminorleaguegamesaresimilartothoseinmajorleaguegames(anassumptionforwhichthereissomeevidence),andinpar-ticularthatscoringisquasigeometricwiththesamevaluesof,thenweshouldexpect6.68minorleaguegamestohavegone27ormoreinnings.Infact,wehavehad6suchgames.Ifwelookfurtherweseethatthemodelpredictsthatwewillhavehadonly0.087minorleaguegameswhichlasted33innings.Infact,wehavehadonesuchgame.Furthermore,thereisa99.3%chancewewillhaveaminorleaguemarathonof20ormoreinningsinanygivenseason,a0.13%chancewewillhaveaminorleaguegameof34ormoreinningsinanygivenseason,a1.32%chanceofseeingaminorleaguegameof34inningsormoreinanygivendecade,anda9.4%chanceofseeingaminorleaguegameof34inningsormoreinalifetimeof75years.Ourmodelallowsustoestimatetheprobabilityofgameslastingacertainnumberofinningsorlonger.Thisisanalternativemethod,andperhapsamoreeasilyunderstoodwaytoexpresshowunlikelyaremarathonsofacertainlength.Wewillnowusethisapproachtocomparerelativeprobabilitiesofbreakingthecurrentrecordsformajorleagueandminorleaguegames.Assumingthatmajorleaguebaseballcontinuestohave30teamsplaya162-gameseason,thereisa50%chancewewillseeamajorleaguegamego27inningsormoreinthenext60years.Thereisa95%chancewewillseeamajorleaguegamego27