Considerascenariowhereyouareinchargeofanalysingthedataandmodellingapandemic. Weconsideragivendisease(let'scallit'VIRUS'),whichhasanunknownprevalencein thepopulation(wewillassumethat[0,1]istheproportionofthepopulationthathas thedisease).Wewillwritetheprobabilitythatapersonisdiseasedasp(D)=.

Letusconsiderthatyouareexperimentingwithavaccineagainstthedisease.Youhave 1000subjectsingroupAwhotakethevaccineand1000ingroupBwhotakeaplacebo.Let usassumethatyoutestthesubjectsinbothgroupsdaily,andafteronemonthyouobtainthe followingresults:2subjectsfromgroupAtestedpositiveatsomepointduringthemonth, and40subjectsfromgroupB.Inthispartwewillassumethatweareusingatestwiththe followingstatistics:

theprobabilityofhavingthediseaseiftestedpositiveisp(D|T)=0.7

theprobabilityofhavingthediseaseiftestednegativeisp(D|T)=0.01.

(i)Accountingforthelimitationsofthetest,howmanysubjectsingroupAandBdidpossiblycatchthediseaseduringthismonth?

(ii)Theefficacyofavaccineistypicallycalculatedas

p(D|V ) p(D|V )

Ev =

p(D|V)

Useyourresultsfromabovetocalculatetheefficacyofthevaccine.Discusswhat wouldhappenifyourtestwerelessaccurate:Whatwouldhappenifp(D|T)wouldbe lower?Ifp(D|T)wouldbehigher?