MATHONTRIAL

MATHONTRIAL

HowNumbersGetUsedandAbusedintheCourtroom

LeilaSchnepsandCoralieColmez

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Wededicatethisbooktoallthosewhohavesufferedfrommiscarriagesofjustice,andtoallvictimsofcrimeswhoseperpetratorswentunpunished,duetothemisuseormisunderstandingofmathematicsin

thelegalprocess.

CONTENTS

Introduction

MATHERRORNUMBER1»MULTIPLYINGNON-INDEPENDENTPROBABILITIES

TheCaseofSallyClark:MotherhoodUnderAttack

MATHERRORNUMBER2»UNJUSTIFIEDESTIMATES

TheCaseofJanetCollins:HairstyleProbability

MATHERRORNUMBER3»TRYINGTOGETSOMETHINGFROMNOTHING

TheCaseofJoeSneed:AbsentfromthePhoneBook

MATHERRORNUMBER4»DOUBLEEXPERIMENT

TheCaseofMeredithKercher:TheTestThatWasn’tDone

MATHERRORNUMBER5»THEBIRTHDAYPROBLEM

TheCaseofDianaSylvester:ColdHitAnalysis

MATHERRORNUMBER6»SIMPSON’SPARADOX

TheBerkeleySexBiasCase:DiscriminationDetection

MATHERRORNUMBER7»THEINCREDIBLECOINCIDENCE

TheCaseofLuciadeBerk:CarerorKiller?

MATHERRORNUMBER8»UNDERESTIMATION

TheCaseofCharlesPonzi:AmericanDream,AmericanScheme

MATHERRORNUMBER9»CHOOSINGAWRONGMODEL

TheCaseofHettyGreen:ABattleofWills

MATHERRORNUMBER10»MATHEMATICALMADNESS

TheDreyfusAffair:SpyorScapegoat?

Conclusion

Sources

CreditsIndex

INTRODUCTION

Everywhere we turn we are assailed by numbers. Advertisements, news, price reductions, medicalinformation, weather forecasts, investment, risk assessment: all this and more is communicated to usthroughprobabilitiesandstatistics.But theproblemis that thesefiguresarenotalwaysusedtoconveyinformation.Asoftenasnot,theyareusedtogiveusspin:toinfluence,frighten,andmisleaduswiththecoolauthorityofnumbersandformulas.

Now,youmightthinkthisatrivialmatter.Youmaybeoneofthosepeoplewhoskippastthenumbersinthearticlesyouread,whopaynoattentionto thedeclarationsofsensational increaseordecreaseinwhatever drama is playing out on the front page,whether it be globalwarming, shark infestations, orilliteracy.Atworst, you think, people aremildlymisinformed.But aswe show inMathonTrial, themisuseofmathematicscanbedeadly.Thesamemathematicaltricksthatmisleadthepublicaboutmarkettrendsandriskandsocialproblemshavesentinnocentpeopletoprison.Beingwrongaboutthepriceofoilisonething;beingdeniedjusticeduetomiscalculationisquiteanother.

Despitetheirubiquity,however,mostofthesefallaciesareeasytospot.Thefactisthatanyonecanmake a decent assessment ofmathematical statements that appear in popular publications, on commonproducts, and in everyday activities from investment toDNAanalysis.Anyone can acquire the simplereflexes to cut through the fog ofmathematical deception.All it takes is a little practice to recognizewhat’s going on. It turns out there isn’tmuch variation in these numerical sleights of hand, but publicignoranceallows them topermeateeveryareaofour lives.Wehavechosen theexamples in thisbookbecausewhileillustratingthepitfallsthateveryoneshouldbeawareof,theyalsoshowthatthemisuseofmathematicsisnotmerelyanacademicissuethatwecaneasilyignore.

Weneedtoknowwhenwearebeingmisled.Weneedtobeabletodistinguishwhetherthenumbersbrandishedinourfacesarelegitimatelyprovidinginformationorbeingmisusedfordangerousends.Weneedtogobeyondtheabstractionoftheoryandseetheplaintruthforourselves.

Mathematics hasmade but few appearances in criminal trials throughout history.When it has beenused,ithasbeenforpurposesofidentification,tocalculatetheprobabilitythatagivenidentificationiscorrect.Thesesamecalculationsoccur ina thousandotherdomainsofpublicandprivate life,andonemightwonderwhywehavechosentofocushereonitsrelativelyrareuseintrials.Webelieveitisworthcollecting and examining these cases for the simple reason that many of the common mathematicalfallaciesthatpervadethepublicsphereareperfectlyrepresentedbythesetrials.Thus,theyserveasidealillustrationsoftheseerrorsandofthedrasticconsequencesthatfaultyreasoninghasonreallives.

Thecaseswepresentinthisbookcoverabroadrangeofmathematicsusedinthecourtroom,fromthesimplest handwriting analysis at the end of the nineteenth century to probabilities used in DNAidentificationtoday.Thesecasesarenotorderedchronologically,butaccordingtothecomplexityofthe

probabilityconceptsinquestion.Wediscusscasesinwhichmathematicswaspresentedattrialtojustifyconviction,andothersinwhichitwasemployedtoconvincethepublicthatconvictionwaserroneous.

In spite of mathematics’ disastrous record of causing judicial error, the main conclusion of ouranalysisisnotthatprobabilityisauselesscoginthejudicialmachine.Rather,wefoundthattheinjusticesperpetrated in the name of probability arise from themisuse ofmathematical principles, not from anyinherent inapplicability of mathematics to justice. We believe that mathematics can be useful infundamentalways, and indeed that the futureof criminal justicewillnecessarily contain anelementofmathematicalanalysis,giventheprevalenceofDNAevidenceintrialstoday.Buttoreachthatgoaltheremust be some certainty thatmathematical errorswill be excluded from trials, and the first step in thisdirectionistoidentifythemostimportanterrorsthathaveactuallyoccurred.

Inthisbookwesharethedramasofpeoplewhosawtheirlivesrippedapartbysimplemathematicalerrors—wrongcalculations,orcalculationsthatwerenotmadeornotunderstood—graveinjusticesthatwere committed or only narrowly avoided.We hope that these incredible true stories will show thatmathematicscanreallybeamatteroflifeanddeath.

MATHERRORNUMBER1»MULTIPLYINGNON-INDEPENDENTPROBABILITIES

MOST PEOPLE KNOW that to measure the probability that several events will occur, the separateprobabilitiesofeacheventshouldbemultipliedtogether.Forinstance,ifyouarepregnantwithasinglechild,thereisa1outof2chanceyouwillgivebirthtoagirl.Thusifyouhavetwochildrenatdifferenttimes,theprobabilityofhaving2girlsis½squared,whichis¼,or1chanceoutof4.

We do this type of calculation all the time, almost without thinking. But there’s a caveat: thismultiplicationiscorrectonlyiftheeventsyou’recomparingaretotallyindependentfromeachother,likehavingseparatepregnancies.Iftheyarenotindependent,thesituationchanges.Suppose,forexample,thatyouhappentoknowfromanultrasoundthatyouarepregnantwithidenticaltwins.Nowthebirthofyourtwochildrendoesnotconstitutetwoindependentevents,andofcourseitwouldbewrongtosaythattheprobabilityofyourhavingtwogirlsis¼;itisinfact½,becausethetwobabiessharethesamegenes,sotheywillnecessarilybeofthesamesex;thustheycanonlybeeithertwogirlsortwoboys.

If you multiply the probabilities of events that are not independent of each other, you will get asignificantlysmallerprobabilitythanisaccurate.Butit’seasytofallintothetrapofassumingthatasetofseparateeventsoccurredorwilloccurindependentlyofoneanother.Someeventsmayseemindependentbuthaveasingleunderlyingcause.Forexample,acardplayermaygoonawinningstreakthatdefiesallodds—butthereasoncouldbethathe’scheating.

It’sriskytoassumethateventsareindependentwhenallthedataisnotin.Yetithasbeendone,evenbyhighlyrespectedpeople,incourtsoflaw.Andsometimesithasresultedindisaster.

TheCaseofSallyClark:MotherhoodUnderAttack

SteveandSallyClarkwerealovingcoupleofbright,ambitiousyounglawyers.BothworkeddemandingjobsinLondon,buteventuallytheyboughtthemselvesalittlehousecalledHopeCottage,wellawayfromthebustleof thecity,anddecidedtoraiseafamily.OnSeptember22,1996,Sallygavebirth toason,Christopher.Shedecidedtostopworkingforafewmonthsandstayhomewithherchild.

From the beginning the baby appeared fragile and delicate, with the face of an angel. He wasextremelyquiet,sleptagreatdeal,andalmostnevercried.InearlyDecemberhedevelopedwhatseemedtobesnifflesandabadcold,but thedoctor toldSallynot toworry.EverythingseemednormalenoughuntilDecember13,when shewentdown to thekitchen for tenminutes toprepareherself adrink, andreturned to thebedroomtofind thebabygray-faced inhisbasket.Shecalledforanambulanceand thebabywasrushedtothehospital,butsadlyhecouldnotbesaved.Anautopsyindicatedthathehadbeensufferingfromaninfectionofthelungs.

After Christopher’s death Sally returned to work, but although she functioned adequately, sheunderwentaperiodofgrieving,depression,anddespair,occasionallydrinkingheavily.Anewpregnancyhelped her snap out of it, and she underwent therapy to bolster a complete renunciation of alcohol.HealthybabyHarrywasbornonNovember29,1997.

LikeallyoungersiblingsofbabieswhohavediedinEngland,thissecondbabywascloselymonitoredundertheprogramknownasCareofNextInfants(CONI).SteveandSallyweretaughtthebasicgesturesofresuscitation,andHarrywasgivenanapneaalarmtowearpermanently,whichwassupposedtostartringing if he stopped breathing.As amatter of fact, the alarmwent off quite frequently, but the healthvisitors and nurses who stopped by the house both regularly and for random checkups found nothingwrong with the baby, so everyone assumed that the apnea alarm was malfunctioning. Little Harryappearedstrongandhearty,wasnoisyandactive,cried loudly,anddemanded frequent feedings.Sallydevotedherselftohimandkeptacloseeyeonhishealth,fillingoutthemanychartsrequiredbytheCONIprogramandkeepinghimwellawayfromanychanceofinfectionbycontactwithothersickpeople.AsStevewas in a castwith a tornAchilles tendon, theClarks hired daily help during the firstweeks ofHarry’s life togiveSallyahandwithall thehousework.OnJanuary26,1998,Sally tookHarry to thecommunityhealthcenterforhisstandardvaccinations.

Afterthevaccinationshewasmuchquieterthanusual,appearinglethargicandpaleasSallywheeledhimhome.FivehourslaterStevewastryingtoamusethebabyandplaywithhim,butHarrydidn’tseeminterested,soSteveputhimdowninhisbouncychairandwenttothekitchen.NotfiveminuteslaterheheardSallycallinghimdesperately.LittleHarryhadgonelimpandwhite;hisheadwasfallingforward.Steverushedbacktothebedroom,laidthebabyonthefloor,andtriedtoresuscitatehim,firstgentlyandthenwithincreasingstrength,whileSallycalledforhelp.Theambulancearrivedandrushedthefamilytothehospital.Butforthesecondtime,doctorswereunabletosavethelifeofthecouple’sbaby.

This time the autopsy gave surprising, and seemingly contradictory, evidence. The pathologist, Dr.Williams,claimedthathecouldseeretinalhemorrhageinHarry’seyes,afrequentsignofsmothering,andcouldfeelabrokenrib,thoughwhetherrecentoroldhecouldnotsay;itdidnotshowuponX-ray.Harryalsohadlargeamountsofbacteriainhisnose,throat,lungs,andstomach,butnonoticewastakenofthis.Thepathologistbelievedtherewassufficientevidenceofabusetowarrantacompleteinvestigation.

SteveandSallyClarkwerearrestedforthemurderoftheirtwochildren.Afterintensiveinterrogation,during which they answered all questions freely and openly and did not ask for a lawyer, they werereleasedonbailwhiletheinvestigationproceeded.

SteveandSallyClark

Theyreturnedhome,minustheirpassportsandwiththeobligationofregisteringregularlyatthepolicestation,andtriedtopickuptheremainsoftheirshatteredlives.Buttotheirhorror,astheinvestigationcontinuedand theywere repeatedlycalled in forquestioning, they realized that theirdesperateneed tounderstandthemedicalcausesforthedeathoftheirsonswasgraduallybeingovershadowedbythenew,urgentnecessityofdefendingthemselvesfromtheaccusationsofseverechildabusebeingleveledagainstthembythepolice.Theyrealizedthattheyhadnoadequatedefenseagainstsuchanaccusation—thereisnoactualproofthatadeadbabyhasnotbeensmothered!Theycouldhardlybelievethattheinvestigationwould result in a trial, yet on the advice of their friends they eventually went to a criminal lawyer.SolicitorMikeMackeyagreedtotaketheircaseandhelpthem,comewhatmay.

Two important events followed: a third littleboy,bornayear to thedayafterHarry’sbirth, andaformalchargeofdoublemurderagainstSally.

Steve,exoneratedfromanywrongdoingandnotcharged,washelplesstopreventthedestructionofhisfamilyandthepersecutionofhiswife.Thenewbabywasplacedinfostercare,andadatewassetforSally’strialforthemurderofhersons.

Sally’s trial took place at Chester Crown Court, before a judge and jury. She was defended bybrilliantlawyerswhoputtheirfingersquarelyoneachandeveryoneofthecontradictionsinthemassiveandcomplexmedicaltestimony,obligingthemedicalexpertstocontradicteachother,andpointingoutaseries of errors of interpretation inHarry’s autopsy.Most of the prosecution’s expertswere forced toadmit that thedeaths of thebabies couldnot bedefinitively attributed to shaking, smothering, or otherabuse,andSally’sbehaviorasamotherwasvouchedforbymanywitnesses,suchasthenannywhohadhelpedherwithHarry,andthehealthcareprofessionalswhohadkepthimunderregularobservationfortheCONIprogram.Listeningtothem,Sallyfeltcertainthatherinnocencecouldonlybeobvioustothejury.Itwasthiscertainty,thisfaithinthejusticesystem,thatgaveherthestrengthtositthroughgruelinghoursofdescriptionoftheautopsiesofhersons,inwhicheverysignofpossibleviolencewasdiscussedin gruesome but unavoidable detail. During those hours of testimony from the prosecution’s medical

experts,Sallywasforcedtolistentothehatefulpicturethatwasbeingpaintedforthejuryofthepersonshewassupposedtobe—obsessivelytidy,professionallyambitious,acontrolfreak,unfittobeamother—andtheactionsshewasaccusedofhavingcommitted.Notonlyshe,butalsothespectatorsatthetrialwere horrified by a system that imposes such torment on parentswho have lost their children.Was itreally necessary for Steve Clark, gagging on the witness stand, to be shown photographs of his deadbabies’medicallydismemberedlittlebodies?

EverythingthemedicalexpertsweresayingseemedwrongtoSally—drastically,obviously,cruelly,horribly,andoffensivelywrong.UntiltherenownedpediatricianSirRoyMeadowtookthestand.

Charming and avuncular, Meadow appeared filled with sympathy for the plight of the accused,pronouncingwords of condemnationwith a seeming reluctance thatmade his allegations all themoreeffective.Heexudedcompetence,experience,ability,andkindness.Listeningtohistestimony,Sallywasstruckdumb.“IfIdidn’tknowIwas innocent,”she latersaid,“listeningtohimIwouldhavebelievedmyselfguilty.”

Anduponthewitnessstand,MeadowspokethewordsthatswungthebalanceofjusticeirrevocablyagainstSally.

Dr.RoyMeadow,pediatrician

INORDERtounderstandwhatRoyMeadowwasdoingwhenhetoldthejudgeandjuryhisownopinionsaboutSallyClarkandthedeathofherbabies,andwhyhisstatementscarriedsuchweight,itisimportanttoknowwhohewas,wherehewascomingfrom,andwherehissphereofcompetencylay.Aspecialistinchildabuse,hehadstudiedunderlegendarychildpsychoanalystAnnaFreud,andwasgreatlyinfluencedbyherteaching.“Achildneedsmothering—notamother,”heusedtoquoteherassaying,thoughitisnotabsolutely certain that Freud ever really pronounced this sentence; perhaps those words weremerelyMeadow’sowninterpretationofwhatshetaught.Inanycase,theyseemtohavelefttheirmark.

WhileRoyMeadowbeganasapediatrician,workingfirstasageneralpractitioner,andlateratGuy’sHospital, theHospital forSickChildren inLondon,and theRoyalAlexandraHospital inBrighton,hismaininterestchangedtochildabuseashiscareeradvanced,andhefocusedhisattentiononthedetection,analysis,andproofofthemisdeedsandcrueltiesofmothers.Itwasin1977,whileworkingasasenior

lecturer and consultant pediatrician at Leeds University, that Meadow came up with the idea thateventually led him to fame. This was the discovery—or the invention—of a new malady, which hebaptizedMunchausenSyndromebyProxy.

MunchausenSyndromeisthenamegivenbyDr.RichardAsherin1951toapsychologicalconditionbywhichapersonwhoisactuallyinperfectlygoodhealthclaimstosufferfromallkindsofsymptomsofillnessthatheimaginestobetrulypresent,orsometimesevenpurposelybringsonthroughactsofself-injury.Thenameisareferencetotheyarnstoldbytheeighteenth-centuryGermansoldierandnoblemanBaronMunchausen,whoastoundedhis listenersuponhis return from thewarsbydescribing flightsoncannonballs,tripstothemoon,andimpressivefeatsofmarksmanshipsuchasshootingfiftybraceofduckswithasinglebullet.ThereisactuallynotalotofresemblancebetweenMunchausen’stalesandthosetoldbysufferersfromMunchausenSyndrome—except,perhaps,fortheirtallness.

Psychological analysis has determined thatMunchausen Syndrome arises from an intense need forsympathy,care,andattention fromacompetentandprotective figure,a role that is ideallyplayedbyadoctor. Exactly the kind ofmedical test or procedure thatmost peoplewould prefer to avoid—bloodtests, biopsies, colonoscopies—is reassuring and consoling to sufferers ofMunchausenSyndrome, andtheytendtoseeksuchproceduresrepeatedlyandunnecessarily.

WhatRoyMeadownoticedinhisseminal1977paperwasthatsomepeopledisplayavariantformofMunchausenSyndrome,seekingconstantmedicalattentionnot for themselves,but foranotherperson,a“proxy.” These people constantly go to doctors and describe symptoms in their proxy that are eithernonexistentorartificiallyinduced.Obviouslytheproxymustbesomeoneunabletoexplainthetruestateofaffairs;forthisreason,proxiestendtobehelplessinvalidsorchildren.

Thiswas themental condition thatMeadow dubbedMunchausen Syndrome by Proxy (MSbP).Hepublishedhis discovery in themedical journalTheLancet, and the title of the article reveals that hisinterestinthesyndromeitselfwasinspiredbyaprofoundconcernwiththefrighteningrealitiesofchildabuse.Thearticle,“MunchausenSyndromebyProxy:thehinterlandofchildabuse,”describedtwocasesinwhichhehadtakenaparticularinterest.Inoneofthem,amotherpersistentlyalteredhersix-year-olddaughter’s perfectly healthy urine samples, leading doctors to perform an endless stream of invasivemedical examinations on the child and subject her to long-term medicinal treatments ranging fromantibioticstochemotherapy.Thedeceptionstoppedonlywhenthedaughterwasadmittedtoahospitalandkept for two or three days in the absence of hermother,who previously had rarely left her side.Thechild’shealthysamplesduringtheperiodofhermother’sabsence,andherinstantrelapsethemomenthermother returned, finally led caregivers to the truth. In the second case, a seemingly devoted motherbroughthertoddlertothehospitalatleastonceamonthwithattacksofillnessthatwerediagnosedassaltpoisoning.Whenthechildwaskeptinthehospital,hebecamehealthy,andwhenhismothervisited,herelapsed.Thehospitalcontactedsocialservicestoorganizesurveillanceandplacementforthechild,butbefore the discussions could lead to a concrete result, the little boywas brought inwith an attack soseverethathedied.

Ifthehospitalworkerstookaslongastheydidtodetectwhatwasgoingon,RoyMeadowexplained,it was partly because bothmothers seemed to be agreeable, intelligent women and loving and tenderparents(albeitwithahistoryofhystericalbehavior,hadanyoneonlythoughttocheck).Noonesuspectedthem, because noone had the habit of suspectingmothers.RoyMeadow stressed the fact thatmothersmustbesuspected.“Wemayteach,andIbelieveshouldteach,thatmothersarealwaysright,”hewrote,“butatthesametimewemustrecognizethatwhenmothersarewrongtheycanbeterriblywrong.”

For tenor twelveyears afterMeadowwrotehis article,MunchausenSyndromebyProxy receivedlittle or no attention, either in the profession or from the public.And then suddenly, hewas given theopportunity to put his theory into practice, because of a grisly and terrible murder case that finally

allowedthewholeideatospringforthandcapturetheattentionofacountry.

INFEBRUARY1991,ayoungnursecalledBeverleyAllittwasengagedinChildren’sWard4oftheseverelyunderstaffed Grantham and Kesteven Hospital in Lincolnshire. Although she appeared kind andcompetent,sheinexplicablywentonakillingspreethat,inthespaceofbarelytwomonths,tookthelivesoffourtinychildrenandseverelyinjuredonemore.

Recalling what it was like to work alongside Beverley Allitt for those two months, nurse andcoworkerMaryReetexpressedan intuitive feel forAllitt’spsychologicalmotivation.“Partof thekickshewould’vehadwasthatwhenthosebabieswerebroughtbacktolife,shewasthere,andshewasthesavior,” she laterwrote. Itwasn’t thebabies’deaths thatAllittwanted: itwasattention.And thiswasexactlyhowRoyMeadowpresenteditwhenhetestifiedasamedicalexpertathertrial.HeshowedhowshedisplayedallthesymptomsofbothMunchausenSyndromeandMSbP,andexplainedthatBeverley’scoldnessinthefaceofthedeathofhervictimswastypical;peoplewithMSbParenotabletograsptheharmtheyare inflicting; theyareclosedoff to it.RoyMeadowstated thathedidnotbelieveBeverleyAllittcouldbecured.Shewouldalwaysbeadangertoothers.Allittwasconvictedandgiventhirteenlifesentences.

RoyMeadow’sdiagnosismadealotofsense.Ontopofthat,thevisibilityofthecase,theterribleandshockingnatureofthecrimes,andhisroleasexpertwitnessatthetrialconferreduponhimnotjustfame,butagreatdealofinfluenceandpoweraswell.

From the moment Beverley Allitt was condemned to life in prison, Roy Meadow’s theory ofMunchausen Syndrome by Proxy took a tremendous leap into both the public and the medicalconsciousness. It is perhaps not fully realized how rapidly the notion took hold and the number ofdiagnosedcasesgrew.MSbPbecameabywordinsocialwork,whereitwascitedasareasontointerferein the lives of innumerable families. Thousands of childrenwere removed from their parents, and thepractice soon spread fromBritain to theUnitedStates, then toAustralia,NewZealand,Germany, andCanada,andasfarasNigeriaandIndia,whereitwasverypopular.

In the name of the new diagnosis, horrific mistakes were made. One example was the “legalkidnapping”of littlePhilipP. in the autumnof1996.Philipwasa child less thanoneyearoldwhosemother,JulieP.,hadmadecountlesstripstohospitalsinthestateofTennessee,whereshelived,seekingtreatment for her baby’s severe birth defects and chronic gastrointestinal troubles. Seeing the child’slengthymedicalhistory,doctorsconcludedthatJuliemustsufferfromMSbP,andshortlyafterthebaby’sarrivalatthehospitaltheycontactedtheDepartmentofChildren’sServicestohavecustodyofthebabyremovedfromhisparentsandgiventothehospital.Thefamilywaskeptatadistanceandthenurseswereeven instructed togive themnomedicaldetailsover thephone.Unfortunately,as ithappened, thebabywas reallyvery ill; separation from themotherhadnocurativeeffects, andexactlyonemonth laterhedied,aloneandfarfromhisparents.ItwasoneofmanycasesinwhichthenotionofMSbPwasabused.

ALLITTWASconvictedin1993,andby1996somedoctorswerealreadybeginningtosoundawarningbell,even as allegations ofMSbP continued to increase at a terrifyingpace.Dr.C. J.Morleypublished anarticle titled “Practical concerns about the diagnosis ofMunchausen syndromebyproxy,” inwhich hewarnedthatafterthecondemnationofBeverleyAllitt,thediagnosishadbecome“chargedwithemotion”and that“thosewhoareaccusedare tarnishedwithher reputation.” In thearticle,hediscussed theso-calledsymptomsofMSbPonebyone,showing thateachof themmightarise foraperfectly legitimatereason.HeevenwarnedagainstadiagnosisofMSbP in the instancewhereachildseparated fromhismother is cured of his illness, as there can be any number of reasons for such an event, including thenaturalrecoveryfrommanyinfantileillnessesthattendtooccuraroundtheageofoneyear.

Intheir1995article“IsMunchausensyndromebyproxyreallyasyndrome?”G.FisherandI.Mitchellalso analyzed the weaknesses of the diagnosis, ending by suggesting it be dropped altogether: “It isrecommended that pediatricians abandon making diagnoses of Munchausen Syndrome by Proxy andinsteaddiagnosethespecificfabricatedorinducedmedicalillness(es)orcondition(s)theyencounter.”

Butsuchcallsforrestraintwerenotheeded.AllegationsofMSbPcontinuedapace,andinfactanewaspect appeared and soon came to occupy a central position in allegations of child abuse: the role ofMunchauseninunexplaineddeathsofbabies,oftenreferredtoascotdeath,cribdeath,orSIDS—SuddenInfantDeathSyndrome.

BABIESHAVEalwaysbeenfragilebeings.Thedeathrateofchildrenundertheageofoneyearoldinthenineteenthcenturywasstartlinglyhigh—asmanyasonehundredperthousandbabiesintheuppersocialclassesandthreehundredperthousandamongthepoor.Evenintheearlytwentiethcenturythenumbersremained significant. Only after the Second World War did doctors and hospitals begin to maketremendousstridesininfantcare,andratesbegantodecrease.

Yeteventoday,asmallnumberofseeminglyhealthyveryyoungbabiesdiesuddenlyfromunexplainedcauses.The phenomenon of crib deathwas not treated as amedical phenomenon in and of itself until1963, when a first conference on the subject was organized in Seattle,Washington. The official term“Sudden InfantDeathSyndrome” (SIDS)was adopted at a second conference in 1969.Obviously, thequestion ofwhat proportion of SIDSdeathsmight be attributed to child abuse or outrightmurderwasraisedatbothconferences,buttherewassimplynoinformationavailabletodrawanyrealconclusionsastotheanswers.AsuggestionthatSIDSwasconnectedtoapnea(babiesstoppingbreathingfornoreason)led to the adoption of huge numbers of apnea alarms being installed. These detectors are attached bysmallsensorstothebaby’sbodyandaresettogooffloudlyifnobreathingmotionisdetected.Intheend,however,theirwidespreaduseservedmainlytoshowthatapneaisnottheonlyormajorcauseofSIDS;therewere toomanycaseswhere thealarmneverwentoff.Andstudies showed that, likeother socialphenomena,SIDSiscorrelatedwithsuchfactorsasfamilybackground,poverty,mentalillness,smoking,ordruguse.

Improvements in baby care methods led to a significant drop in SIDS throughout the 1990s,particularly in families considered “low risk,” meaning stable families with good incomes and goodphysical and mental health. The improvement spurred further study, and during the early 1990s thereoccurredakindofmedicalSIDSfrenzy,withdoctorspublishingresearchbasedonasfewastwocases,rising to stellar heights in their careers, and, evenworse, encouraging parentswhomight be prone toMunchausen-type behavior to give way to it completely by calling for them to bring their babiesrepeatedlytothehospitalforexaminationandcare,exactlythekindoftreatmentthatMSbPpatientsthriveon.

Thenthebubbleburst.Ateamofsuspiciouslawenforcementofficialsdemandedandobtainedtheexhumationandautopsyof

threesiblingsfromNewYorkStatewhohadallsupposedlydiedofSIDS.Adetailedmedicalandlegalinvestigationeventuallyproved that in fact theyhadallbeen suffocatedby their father.The same teamthen investigateda family inwhich two infantswhohaddiedofSIDShadbeen thesubjectofahighlyrespectedmedicalpublicationonSIDSandapnea.Theydiscoveredthatthreeoldersiblingsofthesetwobabieshadalsodied.Theirmothereventuallyconfessedtoallthemurders.Thiseventandothersimilaronesweretheflashpointsthatdrewtogethertwostrandsthathadbeenunconnecteduntilthen:SIDS(cribdeath)andMunchausenSyndromebyProxy.

Untilthemid-1990s,MunchausenSyndromebyProxyhadbeenstudiedinparentsorcaregiverswhoharmed children in order to gain attention and care. The children sometimes died, but this had not

appeared as the goal of the caregivers’ actions. But then came the first diagnosis of MunchausenSyndromebyProxyinacaseofrepeatedSIDS.

ROYMEADOW’Sintention,atfirst,wastojointheswellingranksofdoctorsconcernedwithfindingcausesandpreventiveremediesforSIDS,andtofindfeaturesthatcoulddistinguishbetweennaturalSIDSandthedeathofachildcausedbysuffocationorothermistreatment.Since it isnearly impossible todetectsignsofsuffocation—theparents’desperatevigorousattemptsatresuscitationmaycausethesameslightbruisesorcrackedribsasintentionalabuse—thepossibilitiesfordetectingthedifferenceseemedslim.Yet theywereofparamount importance,bothfor thosebabieswhosuffered“near-miss”cribdeathandfor the siblings of those whose abuse had gone undetected. Like many other doctors involved in themovement,Dr.Meadowwantedtofindsomesignsthatcouldhelphimtellonefromtheother.Hedevotedhimself to this subject in the 1997 book he edited, The ABC of Child Abuse, and in a study titled“UnnaturalSuddenInfantDeath,”inwhichhesurveyedeighty-onecasesofsuddeninfantdeath,collectedoveraneighteen-yearperiod, in familieswhere theparentshadactuallybeenconvictedofmurder.Heattemptedtooutlinesomegeneral typesofscenario todistinguishSIDSfrommurder.Unfortunately, theproblemproveddifficult;inhalfofthecases,autopsiesshowedphysicalsignsofsuffocation,butintheotherhalftherewerenone.

ItisafactthatsomeproportionofSIDSisunquestionablycausedbyparentalabuse,butnooneknowswhatthatproportionis,anditispracticallyimpossibletodistinguishthosecaseswithcertainty.Untilthe1990s the attitudeofpediatricians toward thisproblemwas to “let ill alone lestworsebefall,” in thewordsofDr.JohnDavies,whoworriedthatinnocentparentswouldbeaccused,familiesbrokenfornoreason,andsiblingssentintofostercare.

ButRoyMeadowwasconvincedthattherewerefarmoreparentsmurderingtheirbabiesthananyonehadeverrealized—oratleastadmitted.HecametobelievethatMunchausenSyndromebyProxy,hithertoregardedasaphenomenon leadingmothers toharmtheirchildren,wasactuallyclaiming the livesofamuchlargernumberoftinyvictimsthananybodyrealized.Fromthatpointon,hisbook,hisstudy,andallthe rest of hiswork became focused on developing a newly hawkish “interventionist strategy,”whichmeantmakingsurethatmotherswhosechildrendiedwouldbeaccusedofhavingkilledthemifnoothermedical cause could be found. And his reputation as a specialist of the MSbP phenomenon lenttremendousweighttohiswords.

Partly because of that reputation and partly because of his vocal attitude toward child abuse, RoyMeadow’s career skyrocketed.Hebecamepresidentof theBritishPaediatricAssociation in1994andpresidentoftheRoyalCollegeofPaediatricsandChildHealthin1996.IntheNewYearhonorslistof1997,Dr.Meadowwasknightedfor“servicestopaediatricsandtotheRoyalCollegeofPaediatricsandChildHealth.”Hisstarwasontherise,andhebecameoneofthemostcalled-uponmedicalexpertsintrialsofmothersinallofGreatBritain.

Atthesetrials,hewouldusethewitnessstandasapodiumtopromotehisviews,displayingadistincttalentforcatchyphrasesthatthepresslovedtoquote,suchas“thereisnoevidencethatcotdeathsruninfamilies, but there is plentyof evidence that child abusedoes,” or “one cot death is a tragedy, two issuspicious, three is murder.” His views gained incredible notoriety, and on the strength of his highlyrespectedwordsome250mothersweresenttoprison.

UPONthewitnessstandatSallyClark’strial,Meadowwaseagertosharehisknowledgeandexperience,and the conclusions he drew from them, with the judge and jury. Statistical studies showed that “thechanceofacotdeathinafamilyofthesocialstatusoftheClarkfamilyisabout1in8,543,”heexplainedinhiswarmvoice.“Thatmeansthatthechanceoftwosuchdeathsoccurringinthesamefamilyisequalto

thesquareofthatnumber:onechanceinabout73million.”Sally’scounselbeggedtodiffer.RecordsfromtheCONIprogram,whichfollowedbabiesbornaftera

SIDSdeathinafamily,showedthatoffivethousandbabiesmonitored,eighthaddied.Surelythatprovedthattheprobabilityofsuchaneventwasmuchhigherthan1in73million,sincethelatterfigurepredictedthatadoublecribdeathwouldoccurinEnglandaboutonceinacentury.TheCONIstatisticsshowedthatin reality this sadevent actuallyoccurs inEnglandevery coupleofyears. Indeed, theClarks receivedmanylettersofsupportfromfamilieswhohadlosttwo,sometimeseventhreechildren,toSIDS.

Yes,buttheCONIprogramdata,explainedMeadow,hadnotbeencollectedwiththekindofscientificprecisionandstandardsofaproperlyconductedstudy.Thefiguresthathewasusing,bycontrast,camefrom the CESDI report, which was more comprehensive than the CONI information, calculating thenumberofcribdeathsinvarioussectorsofthepopulation.

TheCESDIreport,whosefulltitlewas“ConfidentialEnquiryintoStillbirthsandDeathsinInfancy,”wasacontrolledstudycommissionedbytheBritishDepartmentofHealth.Inittheauthor,PeterFleming,aprofessoratBristolUniversity,identifiedthreemajorriskfactorsforcribdeath:asmokerinthefamily,anunemployedparent,andamotherundertheageoftwenty-six.ThestudyprovidedprobabilitiesfortheoccurrenceofSIDS in thepresenceofoneormoreof these factors and in the absenceof all three. Infamilieswhereallthreeriskfactorswerepresent,theprobabilityroseto1in214;infamilieswhereallthree factors were absent, it fell to 1 in 8,543, the figure cited byMeadow. The overall figure is aprobabilityof1in1,300.Thus,thefigureof1in73milliononlyconcernedfamiliesoftheClarks’habitsandincome;intheoverallpopulationMeadowwouldhaveexpectedthechanceofadoublecribdeathtobe1in1,300squared,orabout1 in1.5million,fifty timesmorethantheprobabilityhewasusingforSally.Inotherwords,Meadowagreedthattherecouldbealegitimatedoublecribdeathaboutonceeverycoupleofyears inEngland,correspondingtoobservedfact.But thesedeathswouldtypicallynotoccuramongpeopleliketheClarks;suchafamily,accordingtohisreasoning,wouldsufferacribdeathonlyaboutonceinacentury.Itwasjusttoounlikelyanevent:theClarksdidnothaveanyofthethreemainriskfactorsforSIDS,sowhyshouldbothoftheirbabiesdiebypurechance?

Asamatteroffact,theCESDIstudymakesitveryclearthatotherfamilyfactorsexistthatmayaffectthe probability of SIDS, but are not yet understood. Meadow’s essential error was to ignore thisobservation and to treat SIDS as a phenomenon that befalls babies as a consequence of completelyrandomchance,likethelottery.

“SowhenHarrywasborn,thechanceofhisbeingacotdeathwasthesameasChristopher’s?Onein8,543—liketossingacoin?It’sthesameoddseachtime?Headsortails?”questionedSally’slawyer.

“It’s the chance of backing the long outsider at the Grand National,” replied Meadow, calmlydisplayingahorrificlackoftaste.“Let’ssayit’san80to1chanceyoubackthewinnerlastyearandnextyearthere’sanotherhorseat80to1andyoubackitanditwins.Togetoddsof73millionto1youhavetobackthat80to1chance,fouryearsrunning.It’sthesamewiththesedeaths.”

ThisresponseclearlyindicatesthatMeadowviewedcribdeathsasarandomoccurrence—inthefaceof theveryCESDIstudyhewasquoting,whichwarnsthat theremaybeunknownfactors,evengeneticones,thatincreasetheriskincertainfamilies.Thechoiceofthefigureof1in8,543fortheprobabilityofacribdeathoccurringonceinafamilylikeSally’siscorrect,sincethenumberisobtainedbyobservationofmillionsoffamilies.ButMeadow’scalculationoftheprobabilityoftwocribdeathsbysquaringthatnumber relieson the totallyunjustifiedassumption thatcribdeath is apurely randomevent. If, in fact,thereisagenetictraitthatcancausecribdeath,thentwocribdeathsinafamilymaybothbetraceabletothistraitandthusnotbeindependentatall.Meadow’scalculationisanexampleofMathErrorNumber1:multiplyingnon-independentprobabilities.

SowhydidRoyMeadowtreatcribdeathasarandomoccurrence?Whenyouthinkaboutit,itreallymakesnosense tonote that therearefactors that increase theriskofcribdeathwhileat thesametimeconsideringeachoccurrenceof it asbeingabsolutely random. Ithas tobeoneor theother; it can’tbeboth.Ifitisrandom,itwillstrikeindependentlyofanyriskfactors.ButtheCESDIstudyclearlyshowsthatthatisnotthecase.Ifthereareknownriskfactors,thentherecanbeunknownonesaswell;infactthere almost certainly are some above and beyond the three identified in the CESDI report, probablyseveral.

Moreover,cribdeathisnotasinglephenomenon,butanumbrellatermusedtodescribeinfantdeathsthatarenotmedicallyunderstood.Thesedeaths,infact,dohavecauses—it’sjustthatdoctorshavebeenunabletoascertainthem.Itsometimeshappensthatexplanationsariselateron,becauseofgeneticfeaturesthatcontinuetoarisewithinthefamilyoruponamoreseriousexaminationoftheautopsyrecords.Oncethe cause of death is known and identified, the baby has no longer died of SIDS, and the statisticsconcerningSIDSaremodifiedby the removalof the case from thedatabases.SIDS isnot anabsoluteeventthateitherhasorhasnotoccurred,norisitapurelyrandomevent,somultipleoccurrencescannotbe assumed to be independent events. Unfortunately for the Clarks, however, Meadow’s figure wasacceptedwithoutquestionbyjudgeandjury.

Andnotonlywasitaccepted,butalsoitwasmisconstrued.Thesecondproblemwithafiguresuchas1in73millionisthatevenifitwerecorrect,ittendsnottobeunderstoodcorrectly.Thepublic,andnodoubtmanymembersof the jury, took it to be theprobability thatSallyClarkmight be innocent—thattherewas,infact,achanceofjust1in73millionthatshemightbeinnocent.Inotherwords,thereasoningwasasfollows:“SuchaneventhappenedtoSallyClark,there’sachanceofjust1in73millionforthatevent tohappennaturally; therefore it ispracticallycertain that itdidnothappennaturally;SallyClarkmusthavemadeithappen.”

This logic,which is almost irresistible, is alsowrong (another example appears in “the incrediblecoincidence”discussionofMathErrorNumber7inthisbook).Thefallacybecomesimmediatelyobviousin the analogous statement: “Onemillion lottery tickets were sold andMr. X won; there was only 1chancein1millionforthattohappennaturally,sotheprobabilityistoolowtobelievethatithappenednaturally; thereforeMr.Xmusthavecheated.”Ofcourse, in lotterysituationsweknowthis isnot true;someonealwayswinsthelottery,andnoonesuspectstheluckypersonofhavingcheated.

Thepoint is thatdoublecribdeaths,whileextremely rare,dohappen, and someunfortunate familysomewherewillfallvictimtoit,justassomewheretherewillbeafortunateMr.Xwhowinsthelottery.Oncetheeventhasalreadyoccurred,youcannotretroactivelycalculatetheprobabilitythatitcouldhavehappenedandthensuspect that the likelihoodis toosmall for it tohavereallyhappened.Whenit’s thelottery,nooneeverhasadoubt.

In addition to the possibility that Sally’s babies died by pure chance and the possibility that shemurdered them, therewas a third possibility, by far themost likely of the three: that they died of anactualmedicalcausethatthedoctorshadbeenunabletodetermine.Butthejurymemberswerenevertold this. Theywere left to choose between “1 chance in 73million that it happened by chance” and“otherwiseshekilledthem.”Howcouldtheyhesitate?

SallyClarkwasconvictedofmurderbya10–2majorityverdictonNovember9,1999,andgiventhemandatorysentenceoflifeimprisonment.Thepresshadafielddayrevilingherasachildmurderer.OnherarrivalatStyalPrisonshecouldheartheotherprisoners,whohadfollowedthenewsontelevision,screaming,“Herecomesthemurderer!”and,“Die,woman,die!”astheystrainedandclamberedtogetabetterlookather.

Thanksto“1in73million,”SallyClarkhadsuddenlybecomethemosthatedcitizenofGreatBritain.Nearly a year later, onOctober 2, 2000, theCourt ofAppeal upheld her conviction. They denied the

influence of the statistic on the jury,writing, “The point on statisticswas ofminimal significance—asideshow—andthereisnopossibilityofthejuryhavingbeenmisled.”

ArequesttotheHouseofLordsforleavetoappealagain—Sally’sfinalchanceforjusticeinGreatBritain—wasrejected.Shefacedlifeinprison,withouteventhehopeofearlyrelease,whichwouldhavebeenpossibleonlyifsheacceptedtoadmitguiltandexpressremorse.ButSallywasinnocent.Notforfreedom—notevenforherlife—wouldshesaythatshehadkilledhersons.

THEONLY ray of hope in themonths that followedwas that the FamilyCourt granted SteveClark fullcustodyoftheirremainingson,makingitpossibleforSallytoseethechildeachweekandevenspendafulldaywithhimonceamonth.StevesoldHopeCottage,movednearwhereSallywasimprisoned,anddevotedhimselftocontinuingthestruggleforherfreedomwhilelearningtobeasinglestay-at-homedad.Steve’sprofessionalandfamilylifehadbeenshattered,andnothingremainedtohimbuthislittleboy.

HeandIbecomeateam—heismylittlemate,andwedevelopastrongbondoflove;Igetclosertohimthanmost dads, butwhy did it have to be like this?Wemanage tomuddle ourway through together.SometimesIsitoutsidehisnurseryallnight,justincaseheneedsme...ThencomesthemorningItakehimforhisfirstday[atnurseryschool].Icannotbearit.Icannothandlethethoughtofleavinghimwithstrangers.Butwewalk there together,heholdingmyhand,sometimesquite tightly.Suddenly,muchtoosoon,wearethere.Idon’twanthimtoseemecrying.ButIcan’thelpit.Ikisshimgoodbyeandhandhimovertoalovelylady,tearscoursingdownmyface.Icryallthewaybacktothehouse;itfeelsstrangelyempty.WhathaveIdone?Isitthere,desolate,terrifiedthatsomethingmayhappentohim.

Fortunately,Steve refused togiveup.With thehelpof the lawyerwhohadworkedonSally’scasefrom thebeginning,andofmanyotherpeoplewhogenerouslygave their time for free,hecontinued tochaseupeveryavenuethatmightleadtoanything—anappealtotheEuropeanCourtofHumanRights,asubmissiontotheCriminalCasesReviewCommission,apublicrelationsspecialisttohelpthetruthfilterout to the public, and as a last resort, further analyses of the medical examinations performed onChristopherandHarry,theresultsofwhichwerebeingkeptunderlockandkeybythehospitalwheretheydied.

ThemedicalresultshadnotinitiallybeenatthetopofSteve’spriorities.Hewasconvincedthattheexpertshadseeneverythingtherewastosee.Strugglingwithbillsandatime-consumingnewjob,Stevehadotherthingsonhismind.However,asthecase’sexposuregrew,peopleappearedoutofnowheretohelpthecause.Oneofthesevolunteerswasalawyerwhowantedtoobtainthemedicalrecordsfromthehospital,convinced thatSteve’s teamneeded them.Amongother things, theywanted toget theoriginalapneaalarmthathadoftenrungwhenHarrywaswearingit.Theythoughtitpossiblethatthealarmwasnotdefectiveatall,andthatinfactthechildhadundergonerepeatedepisodesofabnormalapneathathadnotbeenrecognizedbythehealthprofessionalscheckingonhim.

Instead,aftermonthsof legalefforts,when therecordswerefinally, reluctantlymadeavailable, theteam found something else—something completely different, and shocking; something that had beenoverlooked by every single doctor who had had direct access to the records, meaning every singlemedicalexpertfortheprosecution.Notbythedefense,though.Asamatteroffact,thesedocumentshadneverbeendisclosedtothedefense.

NofewerthaneightdifferentcoloniesofthelethalbacteriumStaphylococcusaureushadbeenfoundinHarry’sbody,someappearingwithpolymorphs,whicharethecellsthatourbodiesdeveloptofightoffdisease.Theyshowedthatthebabyhadbeensufferingfromaseriousbacterialinfectionwhenhedied,onethatevencouldhaveledtomeningitis.Confrontedwiththeserecords,adozennewandindependent

medicalexpertswrotereportsstatingthatHarrymostcertainlycouldhavedied,andveryprobablydiddienaturally,fromaseriousinfection.Hisdeathnevershouldhavebeenconsideredanunexplainedcribdeath.

Ataroundthesametime,Meadow’smathematicalassertionswereputunderscrutinywhenonOctober23, 2001, theRoyal Statistical Society sent a public complaint to the LordChancellor, inwhich theyexposedhiserrorsandharshlyexpressedtheirgravity.

This approach is, in general, statistically invalid. It would only be valid if SIDS cases aroseindependentlywithinfamilies,anassumptionthatwouldneedtobejustifiedempirically.Notonlywasnosuchempiricaljustificationprovidedinthecase,butthereareverystrongapriorireasonsforsupposingthat the assumption will be false. There may well be unknown genetic or environmental factors thatpredisposefamiliestoSIDS,sothatasecondcasewithinthefamilybecomesmuchmorelikely.

AllofthesefactswereaddedtoSally’sfilewhenitcameupinfrontoftheCriminalCasesReviewCommission,andherconvictionwasquashedonJanuary29,2003.Sallywasfinallyfree.Butshehadspentmorethanthreeyearsinprison,andinspiteofthejoyofbeingreunitedwithherhusbandandchild,shefounditterriblydifficulttorecoverthehabitsofanormallife.Havingbeenaccusedofmurderingherchildren because she was obsessed with her career, she could not contemplate going back to work.HavingbeentoldthatshekilledHarrybecausehewasmessyanddisruptive,shecringedwhenfriendsadmiredhertidiness.Everythingshehadbeen,everythingshewasproudofinherlife,hadbeenheldupasamodelofhorrortotheentirecountry.Andontopofthis,shehadbeendeprivedoftheabilitytomakeasingledecisionforherselfthroughoutheryearsinprison.

Sallysufferedfromarecognizedpsychologicalphenomenonknownas“enduringpersonalitychangeaftercatastrophicexperience.”Indesperation,shesoughtconsolation inalcohol,asshehadforashortperiodafterChristopher’sdeath.ShediedofacutealcoholintoxicationinherhomeonMarch16,2007,justfouryearsafterherrelease.Shewasforty-twoyearsold.

THETESTIMONYofRoyMeadow,recognizedexpertonMunchausenSyndromebyProxy,childabuse,andtheevilsofmotherhood, sentdozensofmothers toprison.AfterSally’s successful appeal,othercaseswere quickly forwarded on to the Criminal Cases Review Commission, and other mothers wereexculpatedandfreed.OneofthemwasAngelaCannings.

Angelahad lostnot twobut threebabies to inexplicable suddendeath, the third literallydaysafterSally’sfirstconviction.AlthoughtherewasahistoryofinfantdeathinpartofAngela’sfamily,whichmayhave indicated an unknown genetic factor, she was accused of murder after the third occurrence andbroughttotrial.SirRoyMeadowwasthekeyexpertwitnessfortheprosecution.NoonehasdescribedMeadow’sparticularstyleonthestandaspowerfullyasAngeladidinherbookAgainstAllOdds,whichtellsthestoryofhertragedy.Quotingsomeofhistestimony,shewrites:

Irememberoneexchangelateinthedaywhichmademeshudder.Mr.Mansfield[Angela’slawyer]yetagain insisted that looking at the whole picture—me, our family, the lack of injuries on Jason andMatthew,thefeaturesconsistentwithcotdeath—itwasarealpossibilitythatmychildrencouldhavediedfromnatural,butasyetunknown,causes.

“Ithinktheproblemwiththatstatement[isthat]Mr.Mansfieldissayingbecausethefamilyisnormal,childabusedoesn’thappen,”ProfessorMeadowreplied.“Itisabsolutelyrighttosaythatchildabuseandsmotheringaremorecommonincertainfamilies,butnevertheless,mostabuse,mostsmotheringhappens

infamilieswhoonordinarymeetingseemnormalandcaringandthatisso,andmostofthemotherswhosmotherchildren,whenyoumeetthem,arenormal.ThesecondpointistostarttalkingaboutthefeaturesofSIDS.SIDSmeansthatyoudon’tknowwhythebabyhasdied.Itmeansthatanunnaturalcausesuchassmotheringwasn’tfound,andnorwasanaturalcause,sothatinanygroupofSIDSbabiestherearesomewhohavebeensmothered.”

Iwastrapped.IfIappearednormal,Icouldbeachildabuser;ifmybabieswerethoughttohavediedofcotdeath,Icouldhavesmotheredthem.TheremightnotbeanyactualproofagainstmebutProfessorMeadowhadcreatedaworldofsmokeandmirrorsfromwhichIcouldnotescape.

LikeSally,Angelawassentencedtolifeinprison.Sheimmediatelyappealed.Whileshewaswaiting,sheheardaboutthetrialofTruptiPatel,anothermotherwhohadlostthreebabiestoSIDS.Trupti’strialtookplacesixmonthsafterSally’srelease.Meadowtestifiedagainsther, too,and listednofewer thanfour different reasons proving that she must be guilty of murder. Fortunately for Trupti, however, thestinging judicialcriticismofMeadow’serrors thathadbeenpublishedat thequashingofSally’sguiltyverdicthadrippledoutintopublicconsciousnessbythen,andwhenotherpossiblecausesforthedeathofTrupti’sbabieswerediscussed—ageneticdefectinparticular—thejurylistenedcarefullyandacquittedher.Followingthis,theSolicitorGeneralofEnglandandWaleseffectivelybarredSirRoyMeadowfromtestifying for the prosecution in any further trials. Angela Cannings’ conviction was overturned justmonthslater,andfollowingherrelease,casesofimprisonedmotherselsewherewerereviewed.ButaswasthecaseforSally,itwastoolateforAngelatoputthebrokenpiecesofherfamilylifebacktogether.Asshedescribedmovingly inherbook, findingherhusbandsunk indepressionandherone remainingdaughterinastateofpsychologicaldisturbance,Angelastruggledformanymonthsbeforefinallyleavingherhometotrytobeginanewlife.

InJuly2005theBritishGeneralMedicalCouncil(GMC)foundthatSirRoyMeadowhadbeenguiltyofseriousprofessionalmisconductinhismisuseofstatisticsatthetrialofSallyClark,andhisnamewasstruck from the medical register. The GMC’s decision was later overturned and Dr. Meadow wasreinstated,butbythattimehehadretired.

Meadow always denied any wrongdoing, admitting only that what he had done was perhaps“insensitive.”ButtheGMCtermedhisactions“fundamentallyunacceptable.”ThepanelstatedthatwhileSirRoyMeadowwasrecognizedasaneminentpediatrician,“heshouldnothavestrayedintoareasthatwerenotwithinhisremitofexpertise.”Hiscalculationcouldbevalidonlyifitwereknownthattwocribdeaths within a family must necessarily be independent of each other, but in fact there is no knownmedicaljustificationforthatassumptionandmanyreasonstobelieveitfalse.

Meadow stood by his reasoning.But he regretted having used the example of betting at theGrandNationaltoillustratetheprobability.Itwastheonlythingheregretted,apparently.Noneoftheinnocentmotherswhospentyearsinprisonbecauseofhim,noneofthefamilieswhoseliveswereshattered,everheardawordofapology.

MATHERRORNUMBER2»UNJUSTIFIEDESTIMATES

IT IS DIFFICULT TO OVERESTIMATE the extent towhichwe are bombardedwith figures on a daily basis.Intended to informus, toenlightenus, tohelpus, thesefiguresalso, farmorefrequently thanonemightlike,misleadus.Ashockingproportionofthenumericalestimateswereceivearesimplywrong,whetherbyintention,byaccident,orbecauseofignoranceortypographicalerror.Worse,theeffectofthiskindoferrorisfrequentlyminimized,asthoughtheonlyimportant thingishavinganumberatall,onethatcanlendascientificauratowhateverstatementisbeingmade.

ABritish report on the failings of theLabour government fromFebruary2010 cited a figure of 54percentfortheproportionofgirlsinthetenmostdisadvantagedareasofEnglandwhobecamepregnantbefore the age of eighteen.When challenged by an alert reader who realized that this figure seemedunreasonably large, the Tories admitted that the correct estimate was actually 5.4 percent. The errorwouldhavebeenforgivableiftheTorieshadnotfeltitnecessarytomakethefollowingpublicstatement:“Adecimalpointwasleftoutinacalculation.Itmakesnodifferenceatalltotheconclusionsofawide-rangingreportwhichshowsthatLabourhaveconsistentlyletdownthepoorestinBritain.”

This we’re giving you a number but who cares whether it’s right or wrong attitude ends upweakeningourcapacityformakingourownassessments,because,afterall,ifitdoesn’tmatter,thenwhybothergivingfiguresatall?

But it doesmatter. In the next case, not only were the statistical figures given in court multipliedtogetherincorrectlyasinMathErrorNumber1,butalsothefiguresthemselveswereinaccurateestimatesthrown out to the jury by an enterprising prosecutor. Once caught, these errors eventually led to theoverturningofaremarkableconviction—butnotbeforetheaccusedhadalreadycompletedtheirtermsinprison.

TheCaseofJanetCollins:HairstyleProbability

JuanitaBrookscrashedpainfullytotheground,hercaneunderneathher.Thegroceriesfromherwicker

shoppingbasketscatteredoverthepavement.Stunnedandinpain,ittooktheelderlywomanamomenttorealizethatshehadbeenviolentlypushedfrombehind,andanothertoliftherheadandscantheareaforherattacker.Whatshesawwasayoungblondewomantearingdownthealleyandroundingthecorneratthefarend.DanglingfromherhandwasJuanita’spurse.

JohnBass,wholivedonastreetofftheendofthealley,wasoutsidewateringhisfrontlawnwhenheheard Juanita scream.As he looked up fromhis hose, he saw a youngwoman, blonde ponytail flyingbehindher, runoutof thealleyandjumpintoabrightyellowcar thatwaswaitingat thecurb.Thecarrevvedupandtookoff,swingingwidelyaroundaparkedcarandpassingwithinsixfeetofBass.Tohissurprise,henoticedthatthedriverwasablackman.

Itwas1964.Interracialcoupleswereveryrare,andtheywerenottreatedwithindifference.Inpublic,inthestreet,theywerenoticed,singledout,andfrownedupon.

TheinvestigatorassignedtothecasewasOfficerKinseyoftheLosAngelesPolice.Kinseycollectedasmanydescriptivedetailsfromthetwowitnessesofthecrimeashecould.FromJuanitahelearnedthatthewomanshehadseenappearedtobegenerouslybuilt—Juanitaestimatedherweightat145pounds—with hair “between dark and light blonde.” She also described thewoman’s clothing as “dark.”Bassagreedabout thedarkclothingandguessed that thewomanhehadseenwasabout fivefeet tall,buthedescribedherbuildas“ordinary”andmentionedthatherdarkblondehairwastiedbackinaponytail,whichJuanitacouldnotremember.Healsostatedthatthedriverofthecarworeamustacheandbeard.

The police had no clues to the identity of the bag snatchers and no clear trail to follow. JuanitaBrooks’son,however,wasenragedbywhathadhappenedtohismother:notonlyhadshebeenattackedandrobbed,butalsoshehadsustainedadislocatedshoulderfromherfall.Hewasdeterminedtofindtheattackershimself.Havingcomeupwithasimpleplan,hevisitedeverygasstationintheneighborhoodwithadescriptionof thepairuntilhehitononewhosepersonnelconfirmed that indeedan interracialcouple came there regularly to fill up the tank of their yellow Lincoln. Brooks took this informationstraighttothepolice,whichexplainshow,fourdaysaftertherobbery,KinseywasringingatthedoorbellofthemodesthouseinhabitedbyMalcolmandJanetCollins.

WhenthepoliceofficerchosetofollowuptheinformationthatJuanita’ssonbroughthim—whenheagreedthattheCollinsesweresuspectssimplybecausetheymatchedthedescriptionofthethieves—hecouldnothaverealizedthathewasengaginginanunorthodoxidentificationprocedurethatwouldlaterresult in a serious legal puzzle. He assumed he would probably find evidence of their crime easilyenough;maybehewouldevenobtainaconfession.

AsKinseydrewupinfrontoftheCollinshouse,thefirstthinghesawwasayellowLincolnparkedinthestreet,andwhenJanetansweredthedoor,henotedwithsatisfactionthatshewaswearingaponytail.Itwasblonde,thoughhewouldhavedescribeditaslightratherthandark,andMalcolmwasnotwearingabeard,buttherewereenoughfeaturesincommonwiththethievesasdescribedbyJuanitaandBassforKinseytofeeljustifiedinaskingtheCollinsestoaccompanyhimtothepolicestation.Therehehadthemphotographed,andinterrogatedthemabouttheiractivitiesatthetimeoftherobbery.JanetexplainedthatMalcolmwasunemployed,butthatshehadbeenatworkatherjobasahousemaidinSanPedroonthemorningoftherobberystartingat8:50a.m.,andthatherhusbandhadcomewiththecartopickherupat1:00p.m.Accordingtobothofthem,theyhadthendriventothehomeofafriendinLosAngelesandhadspentthewholeafternoonthere.OfficerKinseyreleasedthemandhadthemdrivenhomeinapolicecarpendingfurtherinvestigation.

HisnextstepwastoshowthephotographsofMalcolmandJanettothevictimandtohisonlywitness.Theresult,however,wasdisappointing.JuanitacouldnotidentifyJanetatall,andBasswasabletosayonlythattheponytailofthewomanhehadseenrunningaway“lookedthesame”astheoneinthepicture.However,Kinseywasnotdeterred.HewasprettycertainthatMalcolmCollinswasashadycharacter.

Andhehadaplantoproveit.

ADAYor two later,OfficerKinseydrovearound theareawhereMalcolmandJanet liveduntilhesawthemarrivehomeintheiryellowLincoln.Hefollowedthem,parkedhiscarinapositionfromwhichhecould survey the rear of the house, and put in a call for backup at the Collins home. Following hisinstructions,theadditionalofficersrodeinamarkedpolicecarandpulledupostentatiouslyinfrontofthehouse.Thecar immediatelydisgorgednumerousuniformedofficers,whostormedup thefrontpathandloudlyrangthedoorbell.

Thistacticproducedexactlytheeffectthattheofficerhadhopedtoachieve.Fromhispositionattherear,KinseysawMalcolmgoracingoutofthebackdoor,scurryintoaneighbor’sgarden,andenterthehouse.WhenJanetopenedthefrontdoor,theotherofficerswentstraightintotheCollinshome,arrestedher,andledheroutside,wheretheypushedherintothepolicecar.Thentheywentnextdoor,enteredthehouse,andbeganaroom-to-roomsearchforMalcolm,whomtheydiscoveredtightlysqueezedinsideacloset. The couple was taken back to the police station, and this time theywere kept in custody andinterrogated formore than forty-eight hours.Yet oncemore, in spite of their shocking treatment of thesuspects,thepolicefailedtoobtainaconfessionorevenashredofsolidevidence,andforasecondtimeMalcolmandJanetwerereleasedwithnocharges.

Frustrated by the failure of his second premature attempt to extract a confession,Kinsey set to thetedious work of gathering evidence, questioning Janet’s employer, the friend whom the Collinses hadvisitedonthedayoftherobbery,andseveraloftheiracquaintancesandneighbors.Hecollectedasmuchinformation as he could—concerning theCollinses’ alibi, their financial situation, their hair color andstyle,andthepastandpresentstateofMalcolm’sbeard—andtwoweekslaterhearrestedMalcolmandJanetforthethirdtime.

AlthoughthefactsthatKinseycollectedthroughhisinvestigationwereactuallyratherimpreciseandconfusing,hedecided that this time theyweresufficient to justify thearrest,on thebasisof tworatherstrongpointsagainstthecouple.First,althoughJanethadclaimedthatherhusbandhadpickedherupfromworkat1:00,heremployerstatedthatthetimehadactuallybeenabout11:30.True,11:30wasthetimeatwhich Juanita said she had been robbed, which would have made it impossible for Janet to be theperpetrator. But neither Juanita nor Bass had been able to give the time of the robberywith any realprecision,andthedistancebetweenJanet’semployer’shomeandtherobberycouldhavebeencoveredinjustafewminutesbycar.

Second,onthedayfollowingtherobbery,Malcolmhadpaidofftwotrafficfinestotalingthirty-fivedollars.The police found the receipts in his pocket.Asked separately about the funds used to pay thefines,Malcolmexplainedthathehadusedmoneyhehadwonatagamblinghall,andJanetsaidhehadpaidthemoutofsomeofhersavings.ThetroublewasthatthestolenpursehadcontainedasumofmoneythatJuanitaestimatedasbetweenthirty-fiveandfortydollars.

Togetanideaofwhatthatsumrepresentedin1964,itisusefultohaveafewpointsofcomparison.TheaveragemonthlyrentintheUnitedStatesthatyearwas$115;aloafofbreadcostaround$0.20.JanetCollinsearned$12aweekforherpart-timeworkasahousemaid.Ontheonehand,her$50permonthmightseemhardlyenoughtosurviveon,butontheother,certainthingscostunimaginablylessthantheydotoday.JanetandMalcolmhadgottenmarriedonJune2,twoweeksbeforethedateoftherobbery,andhadtakenahoneymoontoTijuana,Mexico.ThetripcostJanetonly“apartofthat$12.”

Still,atJanet’spart-timesalary,thosetrafficfinesmusthaverepresentedaheavyfinancialburdenfortheCollinses.At the same time, themoneyobtained from thebag snatchingwouldhave seemed like anear-miraculouspieceofgoodfortune.Eveniftheyhadbeensodesperatethattheyhadplannedtosnatchsomeone’spurse,theycouldhardlyhaveimaginedfindingasumaslargeasthirty-fivedollars—exactly

what theyneeded to pay their fines.Remember, that amount ofmoneywould have beennearly Janet’smonthly salary; not many people walked around carrying somuch. Thematching of the sums seemedalmosttoogoodtobetrue.

WHILE IN custody and awaiting the preliminary hearing, Janet grew afraid, and requested a privateconversationwithOfficerKinsey.Duringthisconversation,JanettoldKinseyofherfearthatbecauseherhusband had a previous criminal record, a new conviction would net him a longer prison term. Sherepeatedlyexpressedheranxietyonthissubject,andfinallyshetoldOfficerKinseythatifeitherofthemweretobeconvicted,thenitshouldbeherandheralone;shewantedtotakealltheblame.Inexcerptsfromtheconversation,whichwasrecordedinitsentiretyandplayedbackduringthetrial,sheasksaboutthispossibilityagainandagain.

“IfItoldyouthathedidn’tknowanythingaboutitandIdidit,wouldyoucuthimloose?”“Ijustwanthimout,that’sall,becauseIain’tneverbeeninnotrouble.Iwon’thavetodotoomuch

time,buthewill.”“What’sthemosttimeIcando?”“Would it be easier if Iwent aheadand said, if Iwasgoing to say anything, say it now insteadof

waitingtillcourttime?”

Atacertainpoint,OfficerKinseysummonedMalcolmtojoinintheconversation.Perhapshehopedtoextractsomecontradictionsbetweentheirstories—forexample,aboutexactlywhatmoneywasused topay the fines— and somehow parlay those contradictions into a confession. But that didn’t happen.Instead,likeJanet,Malcolmseemedinterestedonlyindiscussingstrategytoobtaintheleastpossibletotalpunishment. “I’m leaving it up to her,” he can be heard saying at one point in the recording. And atanother:“Thisisalittledelicateonmybehalf.”

Theconversationendedwhen thecouple finallydecided that theyneededmore time to think thingsover.ToOfficerKinseythetoneofthediscussionandtheconcernofthecouplewiththeirpossibleprisonsentencesweresignsofguiltybehavior,andhedescribedthemassuchduringthesubsequenttrial,saying,“They seemed to be conscious of their guilt, and looking to find the best solution to get out of thesituation.”

ButjustasMalcolmCollins’fleeingthepolicetohideinaneighbor’sclosetmaynotindicateactualguilt so much as a general fear of trouble with the police due to bitter past experience, so thisconversationcanbeunderstoodinquiteadifferentmanner.Marriedtoablackmanatatimeoframpantracism, nineteen-year-old Janet must have repeatedly experienced the censure of society. Ignorant,working-class,andpoor,shewouldnothavebeenused tostandingupforherrights.Sheprobablyfeltthat,guiltyor innocent, therewasessentiallynochanceatall thata trialwouldresult inacquittal.Hermarriagetoanunemployedblackmanwithapreviousrecordfitintoaperfectimageofacoupleofpettycriminals,andjailtimemusthavelookedlikeacertainbet.Underthosecircumstances,Janet’sdesiretoshoulder theblameinorder tosaveherhusbandfrommoreserioustroublemayseemindicativenotsomuchofguiltasoflove.

ASWASrevealedoverthecourseofthecouple’sjointseven-daytrial, theprosecution’sentirecasewasbuilt around the problem of identification. The defense stressed the impossibility of someone leavingworkat11:30andcommittingacrimesomestreetsaway,alsoat11:30.Theprosecutionrespondedbynotingthatnoneofthewitnesses,neitherJuanita,Bass,norJanet’semployer,werecertainoftheirtimestowithinafewminutes.Andasnotedabove,thatwasallitwouldhavetakenforthecoupletodriveto

thesceneoftherobbery.Therewas also the issue ofMalcolm and Janet’s alibi.Unfortunately for them, although the friend

whomtheyclaimedtohavevisited inLosAngelesonthedayof therobberyrememberedtheircomingovertoherplace,shewasnotabletorecalltheprecisetimeoreventheprecisedateoftheirvisit.

Still,bothof thesepiecesofevidencewereweak; theydidn’tproveanything.What theprosecutionwantedwastopinpointMalcolmandJanetCollinsastherobbersbyidentifyingthemthroughthephysicalevidencegiven inBass’description: theyellowcar,which theyundoubtedlydidpossess; Janet’s darkblondehair,ponytail,anddarkclothing;andMalcolm’smustacheandbeard.

If those details had beenmore precise and correspondedmore closely to the two defendants, theymightwellhavebeeninserioustrouble.Butinfact,theywerevagueanddidnotquitefit.Foronething,Malcolmwasnotnowwearingabeard.Bassnonetheless identifiedMalcolmat thetrialas themanhehadseeninthecar.Buttheeffectofhisdeclarationwasdestroyedbythedefense,whoprovidedevidenceprovingthatshortlybeforethetrialBasshadfailedtopickMalcolmoutofapolicelineup.

Asked whether he had worn a beard on June 18, the day of the robbery,Malcolm explained thatalthoughhedidoccasionallywearabeard,hehadnotbeenwearingone lately,having shaved itoff acoupleofweeksearlierforhisJune2weddingtoJanet.ThedefensecalledanumberofwitnesseswhowereacquaintedwithMalcolmandconfirmedhisclaim.However,thecourtclerkwhotookpaymentforMalcolm’strafficfinesonJune19testifiedthatherecalledhimashavingabeardonthatday.Intheend,thispointsimplycouldnotbedeterminedonewayortheother.

ThentherewastheproblemofJanet’sclothesandhair.EvidencewaspresentedthatonJune18shehad been wearing light-colored clothing, not dark. Furthermore, neither Juanita nor Bass was able toidentifyJanetinthecourtroom.Ontopofthat,whiletheybothagreedthatJanet’shairseemedsomewhatlighterincourtthanithadbeenonthedayoftherobbery,Janet’semployertestifiedthatshethoughtthatJanet’shairhadbecomedarkersincethatday.ThepossibilitythatJanethaddyed,bleached,orotherwisealteredherhair since June18wasdiscussed indetail,butcouldnotbe resolved. Janetherselfdeniedhavingdoneanythingspecialtoherhair.

MalcolmandJanetwerebothcalledtotestifyontheirownbehalf,andbothdeniedanyinvolvementinthecrime.Malcolm told the juryabouthowhehadpickeduphiswifeatworkanddrivenher to theirfriend’shomeinLosAngeles,wheretheyspenttheafternoon.Janetconfirmedthis.OfherconversationalonewithOfficerKinsey,sheaddedonthestandthat“inducementshadbeenheldouttoheronconditionthatsheconfessherparticipation,”andsheformallydeniedeverhavingmadeanyconfessionorintendingtodoso.

Therewerenofurtherwitnesses,andthereforenofurthertestimonywasexpected.ItwasobviousthatthecaseagainstMalcolmandJanetwasweak.Butatthelastminute,asasuddenmovetobolsterupthefailingprocessofidentification,theprosecutorpresentedadramaticnewapproach.

RAY SINETAR, a thirty-year-old prosecutor with just two years of experience behind him, had beenwonderinghowtoexplaintothejurywhathesawintuitively:theCollinseshadtobethethievesforthesimplereasonthatthenumberofcouplesfittingtheirdescriptionwassosmallthat,itseemedtohim,theywerevirtuallycertaintobetheonlysuchcoupleintheneighborhood.Itwasfrustratingtoperceivethissoclearlyandyetnothavetheevidencetomakeitholdwaterincourt.ButSinetarhadalittleknowledgeofmathematicalmethods;hisbrother-in-lawwasEdThorp,amathematicianandblackjackgeniusfromNewMexicowhowouldtestifyintheroleofexpertwitnesstwomonthslateratthemurdertrialofJoeSneed(seechapter3).ItstruckSinetarthatmaybehecouldmathematicallyprovethattheCollinseshadtobetherightcouple,bycalculatingtheprobabilitythatanygivencoupleintheLosAngelesareacouldsharetheirdistinguishingfeatures,vagueastheywere:mixedrace,yellowcar,mustache,beard,blondeponytail.

Early in themorning of the second day of testimony, Sinetar dashed off a phone call to the localuniversity, California State University at Long Beach, and left a message that he urgently needed amathematiciantocomeintocourtandtestify.Themanwhoansweredthemessagewastwenty-six-year-oldprobabilitytheoristDanielMartinez,whohadcomeintoworkthatdaytoteachhisclassandthoughtitmightbe interesting to turnhisknowledge intoevidence inacourtcase.He later recalled,however,feelingabitunsettledwhenthecasebegantounfoldbeforehim,wonderingjustwhathehadgothimselfinto.

In court,Sinetar turned to the jury and explained to them that hewasgoing togive amathematicalproof. He would show that if a person searched for couples matching the physical characteristicsdescribed by the witnesses to the crime—couple in car, black man, beard, mustache, white woman,blonde,ponytail,yellowcar—thepossibilityofaprecisematchwassounlikelythatifanysuchcouplewereactuallytobefound,theoddswouldbeoverwhelmingthatitmustbethesamecoupleasthatseenbythewitnesses.

SinetarputMartinezonthestandandhadhimtestifytothevalidityoftheproductrule,which,aswesawinthepreviouschapter,statesthatiftwoeventsareindependent,thentheprobabilitythattheybothhappenisobtainedbymultiplyingtheprobabilitiesforeachonehappeningonitsown.

Next,Sinetargaveestimatesfortheseparateprobabilitiesassociatedwithfindingapersonwitheachofthecouple’sdistinguishingcharacteristics,asfollows:

• Blackmanwithabeard:1outof10• Manwithmustache:1outof4• Whitewomanwithblondehair:1outof3• Womanwithaponytail:1outof10• Interracialcoupleincar:1outof1,000• Yellowcar:1outof10

Sources differ as to whether Sinetar gaveMartinez the probability figures without mentioning thequalitiestheycorrespondedto,orwhetherhetoldMartinezthatjudgingthevalidityofthoseprobabilitieswasnothisconcern.Inatelephoneinterviewsomefortyyearslater,Sinetarrecalledthatheonlygavethefigures,butaquotedquestionandanswerfromtheoriginaltestimonyrecordedinthestatesupremecourtappealjudgmentindicatesthathesaidmore:

Now,letmeseeifyoucanbeofsomehelptouswithindependentfactors,andyouhavesomepaperyoumayuse.Yourspecialtydoesnotequipyou, Isuppose, togiveussomeprobabilityofsuch thingsasayellowcarascontrastedwithanyotherkindofcar,doesit?...Iappreciatethefactthatyoucan’tassignaprobabilityforacarbeingyellowascontrastedtosomeothercar,canyou?

Martinez’srecordedanswer:“No,Icouldn’t.”Inanycase,whatemergesclearlyisthatMartinezwasaskedtomultiplythenumberstogetherandwas

provided with paper and pencil for the purpose. He did so, obtaining the resultaprobabilityof1in12million.

Havingmadethiscalculation,themathematicianwasaskedtostanddown,andtheprosecutorgavehisinterpretationofthisresulttothejuryinanimpassionedspeech.Heexplainedthatthischanceof1in12millionrepresentedthelikelihoodofagivenpairofpersonsinLosAngelesfulfillingallofthecriteria

above—beingseentogetherinacar,havingablondeponytail,etc.—andthattherefore,havingfoundsuchacouple,onecouldbesurefarbeyondanyreasonabledoubtthatthismustbethecoupleinquestion.Infact, the prosecutor told the bemused jurors, this new type ofmathematical proofwas on the verge ofreplacing the traditional idea of proof beyond a reasonable doubt, a notion he described as “themosthackneyed,stereotyped,trite,misunderstoodconceptincriminallaw.”

Recognizing that some peoplemight be disturbed by the replacement of an actual search for solidproofofguiltbyapurelytheoreticalandnumericaloperation,andthatthismightinfactevenbeasourceofjudicialerror,headmittedthat“onsomerareoccasion...aninnocentpersonmaybeconvicted.”Butthatcanhappenanyway,hesaid,andifitcametoachoicebetweenusingthe“newmath”toconvicttheoccasionalinnocentorusingtheoldsystemandlettingtheguiltygofree, thensurelythenewmathwaspreferable, for otherwise, “Life would be intolerable . . . because there would be immunity for theCollinses, for peoplewho choose not to be employed to go and push old ladies down and take theirmoney and be immune, because how could we ever be sure they are the ones who did it?” Sinetarconcludedhisspeechbynotingthat theestimateshehadgivenwereactuallyconservative, that therealvalues were probably even smaller, so that “in reality, the chances of anyone else besides thesedefendantsbeingthere...issomethinglikeoneinabillion.”

Ittookthejuryeighthoursandfiveballotstodeliveraverdictofguilt.Surprisingly,acaseoflittleimportance in theannalsofcrime turnedout tohave tremendoussignificance in theannalsof law. Inaway,whatthejurywasdoingthatdaywasnotdeterminingwhethertheCollinscouplewasguiltyornot.Rather, they were making a judgment about whether mathematical calculation could replace hardevidence.TherewasnoseriousevidenceagainsttheCollinses:nothingabouttheirappearancecouldbeidentified clearly, their alibi was not firm but could not be proved false, and as for their poverty,miserable social situation, and fearful behavior aroundpolice, not onlywere these factors insufficientproofsofguilt,butthethirdonemaywellhavebeencausedbytheothertwo.

The jury found the Collinses guilty of second-degree robbery, and they were sentenced to prisonterms.ThepublicreactionstoSinetar’slegalexploitfollowedquickly.“JusticebyComputer”and“LawofProbabilityHelpsConvictCouple”weresamplesof theheadlines thatappearedsoonafter the trial.Attention started to build up around the event, and within a month a feature on the case ran in Timemagazine.InitsissueofJanuary8,1965,undertheheading“Trials:TheLawofProbability,”Time toldthenation:

Ajuryhasconvictedthe[Collins]coupleofsecond-degreerobberybecauseProsecutorRaySinetar,30,cannilyinvokedatotallynewtestofcircumstantialevidence—thelawsofstatisticalprobability.

Convictedbymath,MalcolmCollins receivedasentenceofoneyear to life.JanetCollinsgot“notlessthanoneyear.”

JANETCOLLINSdidnotappealhersentence,preferringtoseeitthrough,stayundertheradar,andkeepoutofthewayofthelawinfuture.ButMalcolm,perhapsmorerebelliousbynature,andpossiblyencouragedbyalawyerwhosensedanopportunityforlegalinnovation,didappeal.Andwhenhelost,heappealedagain,andthecasewentuptothesupremecourtofCalifornia.There,Sinetar’stechniquemetitsmatch—in the person of twenty-five-year-old law clerk Laurence Tribe,whowas assisting one of the court’sjudges.

ItsohappenedthatTribehadmajoredandexcelledinmathematicsatHarvardbeforeturningtolawschool.Thus,inhismemoforthejudge,hewasabletosystematicallycalloutalltheerrorsthatSinetarhadmanaged tomake in the courseofhismisleadingly simple argument.Readersof the state supreme

court’s judgmentwill findTribe’smemo,unsigned,at theend,andanymathematicianwill recognizeatonce that it was written by an unusually knowledgeable hand. Tribe’s arguments are flawless andconvincing.

ThefirsttwoerrorsheraisedarenoneotherthanMathErrorNumbers1and2.Theprosecutorgavearbitraryvaluesoutofhisownheadfortheprobabilitiesofsuchfeaturesascarsbeingyellow—nottomentiontheabsurdnamingofa“probability”thatagirlwillbewearingaponytail,giventhathairstyle,unlikecarcolor,canbealteredinstantaneouslyandatwill.Asfortheassumptionthatthethieveswereactuallyamarriedcouple,itseemedtohavenojustificationwhatsoever;therewasnothingtoprovethatthepeopleintheyellowcaratthesceneoftherobberyweremarried.Inshort,Sinetar’sassumptions,andhisnumbers,werebasedonnostatisticalresearch(oralmostnone—Sinetardoesrecallhavingaskedthelawofficesecretaries for theirguessesbeforemakinghischart!)andcertainlynohardevidence. Suchestimationscanbevaluableineverydaylife;theabilitytomakeaneducatedguessisoneoftheweaponspeoplecanusetofightagainsttheabuseofnumbersinthepublicdomain.Butvagueestimationshavenoplace in a court of law—no person’s freedom should depend on them. As though to underline theirinaccuracy, the prosecutor went so far as to tell the jury that he considered his estimates to be quite“conservative”and,aspointedoutintheTimearticle,invitedthemtomakeuptheirown,suggestingthatpracticallyanynumberswoulddo.

Thesupremecourtjudgmentreads:

The prosecution produced no evidence whatsoever showing, or from which it could be in any wayinferred,thatonlyoneoutofeverytencarswhichmighthavebeenatthesceneoftherobberywaspartlyyellow,thatonlyoneoutofeveryfourmenwhomighthavebeenthereworeamustache,thatonlyoneoutof every ten girls who might have been there wore a ponytail, or that any of the other individualprobabilityfactorslistedwereevenroughlyaccurate.

The prosecutor’s next error was to ask his expert witness to apply the product rule to hisprobabilities,withoutverifyingorallowingthewitnesstoverifywhetherall theeventsinhislistwereindependent.Thiswasaseriousmistake,sincemanyofthoseeventsareactuallynotindependentatall.Theprobabilityofamanhavingbothamustacheandabeardismostcertainlynotequaltotheprobabilityofamanhavingamustachemultipliedbythatofamanhavingabeard,giventhatbeardsnotaccompaniedbymustachesarequiterare.

As a matter of fact, this flaw had not gone unnoticed. Rex deGeorge, the lawyer who defendedMalcolm at his appeal, had already raised the same point, albeit unconvincingly. Who could takeseriouslyhis saying, “There is adependencybetweenNegrodrivers andyellowcars; thereareby farmanymoreNegroesthanCaucasiansdrivingyellowcars,”or,“Thereisadependencybetweenblondesandintermarriage;blondesandredheadstendtobemoreadventuresome,moredaring,andmorelikelytochoosetobewithaNegro,”orhisclaimthat“thereisadependencybetweenthewayawomanwouldnormallywearherhairandhowshewouldfixitwhenshegoestocarryoutarobbery”?

DeGeorge’sappealfailed,buthismainpointisjustified,andwasmademorerigorouslybyTribeinthesupremecourtbrief.

There was another glaring defect in the prosecution’s technique, namely an inadequate proof of thestatisticalindependenceofthesixfactors.Noproofwaspresentedthatthecharacteristicsselectedweremutuallyindependent,eventhoughthewitnesshimselfacknowledgedthatsuchconditionwasessentialtotheproperapplicationof the“productrule”or“multiplicationrule.” . . .Totheextent that thetraitsor

characteristics were not mutually independent (e.g., Negroes with beards and men with mustachesobviously represent overlapping categories) the “product rule” would inevitably yield a whollyerroneous and exaggerated result even if all of the individual components had been determined withprecision.

OthererrorsTriberaiseddealwithmoresubtledifficulties.Forexample,heshowedthatevenifoneaccepts thatonlyonecouple in12millionshares the traitsof the thievingcouple, this figurecannotbeconfused with the probability that having found one such couple, it must be the right one. This is adifferentcalculationandsignificantlymorecomplex;wewillfaceaverysimilarprobleminthecaseofDianaSylvester(seechapter5).Indeed,Tribecalculatedthattheprobabilityofanothercoupleintheareaexistingwith the samemain identifying featuresas theCollinses (yellowcar,blondehair, etc.)was inexcessof40percent.

Duetothemisuseofmathematicsanditsexcessiveeffectonthejury’sdecisionmaking,theoriginaljudgmentwasreversed;Malcolm’sconvictionwasquashed,andhewasreleasedtojoinhiswife,whobythattimehadbeenoutofjailforsomethreeyearsalready.

Thebagsnatchingofwhich theCollinseswereaccusednormallywouldhavesunkwithoutarippleinto the infiniteseaofminor legalcases.But thenoveluseofmathematics in their trialshot it into thenewspapers,andfromthereintotheannalsoflegalhistory.LaurenceTribewentontobecomeamajornameinAmericanlaw,defendingAlGoreagainstGeorgeBushinfrontoftheUSSupremeCourtafterthe2000presidentialelections,teachingaseriesofbrilliantstudents—includingPresidentBarackObama,inwhose administration Tribe later served—and, perhaps most importantly, writing a series of seminalpapers in the early 1970s rejecting the use of mathematics at trial. It is practically impossible tooverestimate the influenceTribe’spapershavehad in the criminal justice system,bringing the fieldofresearchintotheproperandcorrectuseofmathematicsattrialtoavirtualhaltfordecades.Itwouldtakea new generation, and all the mathematical difficulties associated with the modern science of DNAanalysis,torevivetheurgencyofthatmovementandbringitbacktolife.

MATHERRORNUMBER3»TRYINGTOGETSOMETHINGFROMNOTHING

SUPPOSEYOUAREPROOFREADINGaone-thousand-pagemanuscript,andyourworkissupposedtobegoodenough tooverlooknomore thanamaximumof twentyerrors throughout thebook.Youhave readfiftypagesattentively,andsofaryouhavenotfoundasingletypo.Whatcouldyouconcludeaboutthenumberoferrorsthatyouarelikelytofindthroughoutthemanuscript?Therearezeroinyoursample:howmanymighttherebeinthebook?

Wouldyoufeelinclinedtoassumethatthepartyoureadisrepresentativeofthewholeandestimatethenumberoferrorsasbeingfewerthanoneperfiftypages,orfewerthantwentythroughoutthewholebook?Wouldyousnapthepagesshut,gotoyourboss,andassurehimthatthejobisdone?

Orareyouthekindofpersonwhowouldthinkofthe101reasonswhymistakesmightstilloccurintherestofthebook?Afterall,itcouldbethatawholegroupofthemappeartogetherabitfurtheralonginthetext. The authormight have had a bad day or an inattentivemoment. Hemight havewritten an entirechapteronatopicthathappenedtobetheonlywordhedidn’tknowhowtospell.Ormaybehetypedthelatertextduringaparticularlybumpytrainride.Ifyou’rethelatterkindofperson,youwouldn’trestuntilyouhadthoroughlycheckedeverysinglepage;otherwiseyou’dneverbecertainthatyouhaddoneagoodjob.

In fact, thebestattitudedependson theexactworkyouaredoing. Ifyouarecheckingaproduct inwhich you feel certain that any errors will be distributed equally throughout—for example, anautomaticallyproducedproduct—youwouldprobablybesafewiththefirstmethod.Butifyouhavenoreason to believe that the errors will be spread out evenly, then you might easily miss somethingembarrassing.Inthelattercase,it’sdifficulttodrawanyconclusionaboutthetotalnumberoferrorsinthewholebookbasedon thezero inyoursample.Guessing that thewholemanuscript isperfectwouldberisky—andmakingsuchaguessinacourtoflawisoutrightwrong.

The prosecution in the next case we describe combined the previous two math errors, givingunjustifiedstatisticalestimatesandmultiplyingthemtogetherfornon-independentestimates,andtotopitoff,themostimportantoftheseestimateswasmadebasedonafindingofzerointhesampleexamined.

TheCaseofJoeSneed:AbsentfromthePhoneBook

ThemurderoccurredonAugust17,1964,intheswelteringheatofaNewMexicohighsummer.SilverCity,whichhadcomeintosuddenexistenceasatentcitywhenalargesilverminewasdiscoveredtherein1870,hadsurvivedtheplagueofabandonmentthatturnedsomanyofthenearbyminingcommunitiesintoghost townswhen theveinsoforerandry.Thiswasperhaps thanks to thesplendidsolitudeof itssituationinthemidstofasoutherndesertparadise.Overtheensuingyears,thetowngrewanddevelopedinto a ranching community, policed by sheriffs and, unofficially, by posses, and home to legendarycriminalfiguresofthefarWest,suchasButchCassidyandBillytheKid.

ThetownwasalreadyblanketedbythecoolingdarknessofnighttimewhenPaulineHicks,fromthecomfortofherhomein theupper-middle-classsectionofSilverHeights,heardsomething thatsoundedlikesharpshotsringingoutintotheshadows.Disturbed,shesteppedoutintohergardenandwalkedovertoward her neighbors’ house to take a quick look around. All seemed quiet, however, soMrs. Hicksreturnedindoorsandwenttobed.“Ithoughtnothingmoreaboutit,”shetestifiedlater.

SilverCityMines

ButchCassidy

BillytheKid

Butthenextmorning,August18,ayoungmancamerunningtoherhouse,knockingandpoundingonthedoor inastateof fearandshock.“Help!”heshouted.“Myparentshavebeenmurdered!They’vebeenshot!”

Mrs.Hicksrecognizedtwenty-year-oldJoeSneed,herneighbors’son,althoughshehadnotseenhimforsometime.HehadgraduatedfromSilverCityHighSchooltwoyearsearlier,andhadsubsequentlymovedtoCalifornia,wherehehadbeenlivingforaboutayear.

SheandJoecalled thepolice,whoarrivedalmostatonceandentered thehouseby thebackdoor,whichJoehadleftopen.Theywereshockedatwhattheyfound.

EllaMaeSneed,Joe’sforty-eight-year-oldmother,waslyinginherbed,dead,herheadrestingonherpillow.Shehadbeenshotthreetimes:intheleftear,intheleftside,andintheback.Itwasobviousthatshehadbeenkilledinhersleep.

Herhusband,fifty-year-oldJoeAlvieSneed,waslyinginhispajamasinasmallentrywaybetweenthebedroomandthebathroom,withabulletwoundtothesideandtwointheback.ButthepossibilitythatJoehadbeenkilledbecausehewasupandoutofbed—rousedbyaburglar,perhaps,orintheheatofaquarrel—wasquicklydiscounted.Bloodytracksaswellasthepositionofhiswoundsprovedthathe,likehiswife,hadbeenshotinbed,probablywhileasleep.Thepoormanhadsucceededinstaggeringtothedoorofthebedroombeforecollapsingonthefloor,dead.

Therewasnot theslightestsignofarobbery.Notevenaforcedentry.Thebulletscamefroma .22caliberpistol,whichwasnotfoundatthescene.

The police brought young Joe down to the station to ask him for details about his discovery. Thequestioningwascordial;someoftheofficerswereJoe’sfriends,andseveralhadcometoknowhimwhenhehadworkedinthestreetsofSilverCityasanewspaperdeliveryboyafewyearsearlier.HetoldthemthathehadbeenreturningtoSilverCitytovisithisparentsafterhislongabsence.HehadmadethetripfromCaliforniabycar,andhadspentthefinalnightofhisjourneyinamotelinLasCruces,arrivingathisparents’homeearlyinthemorningtocatchthematbreakfast.Discoveringthecrimesceneonentering,hehadrushedimmediatelytotheneighbors’houseforhelpwithouttouchingasinglething,heexplained.

Askedifhewaswillingtotakealiedetectortesttocheckhisstory,herepliedthathewas.Thetestwas administered at a specialized center in El Paso, and Joe’s answers and reactions were closelymonitored,butheremainedperfectlycoherentandquiteunflappable,and the test indicated thathewastellingthetruth.

Asadoubleprecaution,policealsosubjectedhimtoaparaffintest,whichexaminestheskinofthehandsformicroscopicparticles thatbecomeembeddedtherewhenthehandshaverecentlyfiredagun.But Joe’s hands were absolutely clean. The police let him leave, and he went to stay with hisgrandparents in the nearby town of Central. On the following morning, August 19, a coroner’s juryreturnedaverdictofdeathbygunshotwoundsat thehandsofapersonorpersonsunknown.“Mysterymounts indoubleslayingofaprominentSilverCitycouple founddead in thebedroomof theirhome,”reportedthedailynewspapers,“asliedetectorandparaffintestsprovenegative.”

JoeAlvieSneedandhiswifewereprominentcitizensofSilverCity.Thecouplewasresponsibleforthe circulation sectionof theSilverCityDailyPress. They had twomarried daughterswhono longerlivedathome;youngJoewas their thirdchildandonlyson.Describedin the localnewspapersas“anaverage American youth,” nothing emerged from his history in Silver City that might indicate anypropensity toviolent and shockingacts.Yetgiven thathewas thediscovererof thebodies, thepoliceweredutyboundtotreathimasasuspect.Itseems,however,thatinthosefirstdaystheydidnottakethisobligationasseriouslyastheymighthave;veryprobablytheydidnotbelievehewasthemurderer.

Unfortunately,thiscasualattitudeledthemtomakeamistake—thefirstofaseries—whichnearlyledtoagravejudicialerror.

ONTHEdayfollowingthediscoveryofthebodies,SergeantRichardIngramoftheSilverCitypoliceforcecalledJoe’sgrandparentsinCentralandaskedthemtobringJoeintotheSilverCitypolicestationforfurtherquestioning.Theyoungmandroveupbyhimself,usinghisgrandfather’scar.Advisedthathedidnothavetoanswerquestions,heappearedsurprised,sayingthathereallywantedtoanswerthem,thatheintendedtocooperatewithpoliceandtry tobeuseful,andthatheabsolutelywantedtoknowwhohad

committed thecrime.At somepointhementioned to theofficers thathewould like togethisowncarback,sinceithadremainedathisparents’housewhenthepolicehaddrivenhimtothepolicestationthedaybefore.SergeantIngram,whowasheadingtheinvestigativeteam,toldSneedthathewouldgofetchthecar.HeclaimedthatJoerespondedbywillinglyhandingoverthekeys.

The police fetched the car, but that was not all they did. They also searched inside—without awarrant, although they could have obtained one within the hour—and discovered something rathersuspicious there that gave them something specific to look for as they gathered their evidence in LasCruces.On theverynextday,August20, JoeSneedwasarrested for themurderofhisparents.At thepreliminary hearing, Sneed and his lawyer, J. Wayne Woodbury, contested the production of thedocuments found in the car on the grounds that the officers had obtained themby unlawful search andseizure.Duringthehearing,SergeantIngramwasaskedexactlywhathadgivenhimtherighttoconductsucha searchwithoutobtainingawarrant.Heclaimed thathehadaskedpermission fromJoehimself,whohadvoluntarilygivenit.

INGRAM:Joe,thedefendant,wasworriedabouthiscar,andItoldhimthatwewouldtrytogetitdowntohimassoonaswecould...IwentandaskedJoeifwecouldhavethekeystohiscar,thatwewantedtosearchitandthenwewouldbringitdowntotheCityHall.

Question:Whatdidhesay?INGRAM:Hehandedmethekeys.

Question:WasMr.Sneedunderarrestatthattime?INGRAM:No.

Question:Forwhatpurposedidyouwishtoquestionhimatthattime?INGRAM:IhadsomequestionsIwouldliketoaskhimabouthistrips,etcetera.

Question:Whatpromptedyoutoseekthesequestions,theseanswersfromMr.Sneed?INGRAM:Iwastryingtofindaclue.

Question:Isee,inotherwordsyouweretryingtofindaclueagainstMr.Sneed?INGRAM:No,justacluetoshedalittlelightonthecase...Ididn’tinterrogatehim,Ijustaskedhimsomequestions.

Question:Makesomedistinctionforme,itisyours,what’sthedifferencebetweenaskingquestionsandinterrogating?INGRAM:Iwasn’taccusinghimofanything.

Question:Washenotthenunderarrest?INGRAM:No,sir.

Question:Norwasheasuspect?Nomorethananybodyelsemightbe?INGRAM:Iguessmoresothanalotofpeople.

Question:Moresothanalotofpeople?INGRAM:Yes.

Question:Butsofarasyouwereconcerned,thiswasafriendlyassistanceJoewaslendingtothepolicedepartment.INGRAM:Yes.

“Unlawful search and seizure” is the act of conducting a searchwith neither the permission of theownerofthesearchedpropertynoraproperlyissuedwarrant.Inhistestimony,IngramindicatedthatJoeSneedhadgivenanoral assent to thepropositionof searching thecar,but it sounds somewhat forced,giventhatSneedwasnotasuspectat the timeandprobablywouldhavereactedwithsurprise ifaskedspecificallywhetherhiscarcouldbesearched.Joehimselfdeniedhavingbeentoldanysuchthing,andhecertainlyneverwaivedhisconstitutionalrightsagainstsearchandseizureinanyformalmanner,aswouldbenecessaryforasearchwithoutwarranttobeconductedlegally.Hetestifiedatthehearingthathehadbeenunawarethatthepolice,whowerehisfriendsandwhohadkindlyofferedtobringhimhiscar,meanttouseittoconstructsufficientevidenceagainsthimtochargehimwiththecrime—that,inshort,hehadbeentricked.

Inhindsight,itissurprisingthatJoegavethepolicehiskeysatall.Perhapshedidn’trealizewhatitwas that he had given them the chance to see. Two insignificant little bits of paper—he had possiblyforgottenheevenhadthem.Andindeed,theydidn’tmeanmuchuntilthepolicefollowedupthetrailtheyindicated.

OneofthemwasareceiptfromtheHolidayInnatYuma,Arizona,datedAugust12,fivedaysbeforethemurder.Thiswouldn’thavebeenunusualinitself,sinceYumawasonthewayfromCaliforniatoNewMexico.But the receiptwasnotmadeout in JoeSneed’sname.Thenameon the receiptwas“RobertCros