Bayes Theorem - Statement, Proof, Formula, Derivation ...

Bayes' theorem describes the probability of occurrence of an event related to any condition. It is also considered for the case of conditional probability. CheckoutJEEMAINS2022QuestionPaperAnalysis: CheckoutJEEMAINS2022QuestionPaperAnalysis: × DownloadNow MathsMathArticleBayesTheorem Bayes'Theorem Bayes’theoremdescribestheprobabilityofoccurrenceofaneventrelatedtoanycondition.Itisalsoconsideredforthecaseofconditionalprobability.Bayestheoremisalsoknownastheformulafortheprobabilityof“causes”.Forexample:ifwehavetocalculatetheprobabilityoftakingablueballfromthesecondbagoutofthreedifferentbagsofballs,whereeachbagcontainsthreedifferentcolourballsviz.red,blue,black.Inthiscase,theprobabilityofoccurrenceofaneventiscalculateddependingonotherconditionsisknownasconditionalprobability. Inthisarticle,letusdiscussthestatementandproofforBayestheorem,itsderivation,formula,andmanysolvedexamples. TableofContents: Statement Proof Formula Derivation Examples Applications Practiceproblems FAQs BayesTheoremStatement LetE1,E2,…,En beasetofeventsassociatedwithasamplespaceS,wherealltheeventsE1,E2,…,En havenonzeroprobabilityofoccurrenceandtheyformapartitionofS.LetA beanyeventassociatedwithS,thenaccordingtoBayestheorem, \(\begin{array}{l}P(E_i│A)~=~\frac{P(E_i)P(A│E_i)}{\sum\limits_{k=1}^{n}P(E_k)P(A|E_k)}\end{array}\) foranyk=1,2,3,….,n BayesTheoremProof Accordingtotheconditionalprobabilityformula, \(\begin{array}{l}P(E_i│A)~=~\frac{P(E_i∩A)}{P(A)}…(1)\end{array}\) Usingthemultiplicationruleofprobability, \(\begin{array}{l}P(E_i∩A)~=~P(E_i)P(A│E_i)…(2)\end{array}\) Usingtotalprobabilitytheorem, \(\begin{array}{l}P(A)~=~\sum\limits_{k=1}^{n}~P(E_k)P(A|E_k)…(3)\end{array}\) Puttingthevaluesfromequations(2)and(3)inequation1,weget \(\begin{array}{l}P(E_i│A)~=~\frac{P(E_i)P(A│E_i)}{\sum\limits_{k=1}^n~P(E_k)P(A|E_k)}\end{array}\) Note: ThefollowingterminologiesarealsousedwhentheBayestheoremisapplied: Hypotheses:TheeventsE1,E2,…Eniscalledthehypotheses PrioriProbability:TheprobabilityP(Ei)isconsideredastheprioriprobabilityofhypothesisEi PosterioriProbability:TheprobabilityP(Ei|A)isconsideredastheposterioriprobabilityofhypothesisEi Bayes’theoremisalsocalledtheformulafortheprobabilityof“causes”.SincetheEi‘sareapartitionofthesamplespaceS,oneandonlyoneoftheeventsEioccurs(i.e.oneoftheeventsEimustoccurandtheonlyonecanoccur).Hence,theaboveformulagivesustheprobabilityofaparticularEi(i.e.a“Cause”),giventhattheeventAhasoccurred. BayesTheoremFormula IfAandBaretwoevents,thentheformulafortheBayestheoremisgivenby: \(\begin{array}{l}P(A|B)=\frac{P(B|A)P(A)}{P(B)}\:\:where\:\:P(B)\neq0\end{array}\) WhereP(A|B)istheprobabilityofconditionwheneventAisoccurringwhileeventBhasalreadyoccurred. Also,gettheBayesTheoremCalculatorhere. BayesTheoremDerivation BayesTheoremcanbederivedforeventsandrandomvariablesseparatelyusingthedefinitionofconditionalprobabilityanddensity. Fromthedefinitionofconditionalprobability,Bayestheoremcanbederivedforeventsasgivenbelow: P(A|B)=P(A⋂B)/P(B),whereP(B)≠0 P(B|A)=P(B⋂A)/P(A),whereP(A)≠0 Here,thejointprobabilityP(A⋂B)ofbotheventsAandBbeingtruesuchthat, P(B⋂A)=P(A⋂B) P(A⋂B)=P(A|B)P(B)=P(B|A)P(A) P(A|B)=[P(B|A)P(A)]/P(B),whereP(B)≠0 Similarly,fromthedefinitionofconditionaldensity,BayestheoremcanbederivedfortwocontinuousrandomvariablesnamelyXandYasgivenbelow: \(\begin{array}{l}f_{X|Y=y}(x)=\frac{f_{X,Y(x,y)}}{f_Y(y)}\\f_{Y|X=x}(y)=\frac{f_{X,Y(x,y)}}{f_X(x)}\end{array}\) Therefore,  \(\begin{array}{l}f_{X|Y=y}(x)=\frac{f_{Y|X=x}(y)f_X(x)}{f_Y(y)}\end{array}\) ExamplesandSolutions Someillustrationswillimprovetheunderstandingoftheconcept. Example1: AbagIcontains4whiteand6blackballswhileanotherBagIIcontains4whiteand3blackballs.Oneballisdrawnatrandomfromoneofthebags,anditisfoundtobeblack.FindtheprobabilitythatitwasdrawnfromBagI. Solution: LetE1betheeventofchoosingbagI,E2theeventofchoosingbagII,andAbetheeventofdrawingablackball. Then,\(\begin{array}{l}P(E_1)~=~P(E_2)~=~\frac{1}{2}\end{array}\) Also,P(A|E1)=P(drawingablackballfromBagI)=6/10=3/5 P(A|E2)=P(drawingablackballfromBagII)=3/7 ByusingBayes’theorem,theprobabilityofdrawingablackballfrombagIoutoftwobags, \(\begin{array}{l}P(E_1|A)~=~\frac{P(E_1)P(A|E_1)}{P(E_1)P(A│E_1)+P(E_2)P(A|E_2)}\end{array}\) \(\begin{array}{l}=\large\frac{\frac{1}{2}~\times~\frac{3}{5}}{\frac{1}{2}~\times~\frac{3}{5}~+~\frac{1}{2}~×~\frac{3}{7}}\end{array}\) \(\begin{array}{l}=\frac{7}{12}\end{array}\) Example2: Amanisknowntospeakthetruth2outof3times.Hethrowsadieandreportsthatthenumberobtainedisafour.Findtheprobabilitythatthenumberobtainedisactuallyafour. Solution: LetAbetheeventthatthemanreportsthatnumberfourisobtained. LetE1betheeventthatfourisobtainedandE2beitscomplementaryevent. Then,P(E1)= Probabilitythatfouroccurs=1/6. P(E2)= Probabilitythatfourdoesnotoccur=1-P(E1)=1–(1/6)=5/6. Also,P(A|E1)=Probabilitythatmanreportsfouranditisactuallyafour=2/3 P(A|E2) =Probabilitythatmanreportsfouranditisnotafour=1/3. ByusingBayes’theorem,probabilitythatnumberobtainedisactuallyafour,P(E1|A) \(\begin{array}{l}=\large\frac{P(E_1)P(A|E_1)}{P(E_1)P(A│E_1)~+~P(E_2)P(A|E_2)}~ =~\frac{\frac{1}{6}~×~\frac{2}{3}}{\frac{1}{6}~×~\frac{2}{3}~+~\frac{5}{6}~×~\frac{1}{3}}\end{array}\) \(\begin{array}{l}=\frac{2}{7}\end{array}\) BayesTheoremApplications OneofthemanyapplicationsofBayes’theoremisBayesianinference,aparticularapproachtostatisticalinference.Bayesianinferencehasfoundapplicationinvariousactivities,includingmedicine,science,philosophy,engineering,sports,law,etc.Forexample,wecanuseBayes’theoremtodefinetheaccuracyofmedicaltestresultsbyconsideringhowlikelyanygivenpersonistohaveadiseaseandthetest’soverallaccuracy.Bayes’theoremreliesonconsolidatingpriorprobabilitydistributionstogenerateposteriorprobabilities.InBayesianstatisticalinference,priorprobabilityistheprobabilityofaneventbeforenewdataiscollected. PracticeProblems SolvethefollowingproblemsusingBayesTheorem. Abagcontains5redand5blackballs.Aballisdrawnatrandom,itscolourisnoted,andagaintheballisreturnedtothebag.Also,2additionalballsofthecolourdrawnareputinthebag.Afterthat,theballisdrawnatrandomfromthebag.Whatistheprobabilitythatthesecondballdrawnfromthebagisred? Ofthestudentsinthecollege,60%ofthestudentsresideinthehosteland40%ofthestudentsaredayscholars.Previousyearresultsreportthat30%ofallstudentswhostayinthehostelscoredAGradeand20%ofdayscholarsscoredAgrade.Attheendoftheyear,onestudentischosenatrandomandfoundthathe/shehasanAgrade.Whatistheprobabilitythatthestudentisahosteler?  Fromthepackof52cards,onecardislost.Fromtheremainingcardsofapack,twocardsaredrawnandbotharefoundtobediamondcards.Whatistheprobabilitythatthelostcardisadiamond? FrequentlyAskedQuestionsonBayesTheoremWhatismeantbyBayestheoreminprobability? InProbability,Bayestheoremisamathematicalformula,whichisusedtodeterminetheconditionalprobabilityofthegivenevent.Conditionalprobabilityisdefinedasthelikelihoodthataneventwilloccur,basedontheoccurrenceofapreviousoutcome. HowisBayestheoremdifferentfromconditionalprobability? Asweknow,Bayestheoremdefinestheprobabilityofaneventbasedonthepriorknowledgeoftheconditionsrelatedtotheevent.Incase,ifweknowtheconditionalprobability,wecaneasilyfindthereverseprobabilitiesusingtheBayestheorem. WhencanweuseBayestheorem? Bayestheoremisusedtofindthereverseprobabilitiesifweknowtheconditionalprobabilityofanevent. WhatistheformulaforBayestheorem? TheformulaforBayestheoremis: P(A|B)=[P(B|A).P(A)]/P(B) WhereP(A)andP(B)aretheprobabilitiesofeventsAandB. P(A|B)istheprobabilityofeventAgivenB P(B|A)istheprobabilityofeventBgivenA. WherecanweuseBayestheorem? Bayesrulecanbeusedintheconditionwhileansweringtheprobabilisticqueriesconditionedonthepieceofevidence. Students,areyoustrugglingtofindasolutiontoaspecificquestionfromBayestheorem?Wewillmakeiteasyforyou.ForadetaileddiscussionontheconceptofBayes’theorem,downloadBYJU’S–TheLearningApp. TestyourknowledgeonBayes'Theorem Q5 PutyourunderstandingofthisconcepttotestbyansweringafewMCQs.Click‘StartQuiz’tobegin! Selectthecorrectanswerandclickonthe“Finish”buttonCheckyourscoreandanswersattheendofthequiz StartQuiz Congrats! VisitBYJU’SforallMathsrelatedqueriesandstudymaterials Yourresultisasbelow 0outof0arewrong 0outof0arecorrect 0outof0areUnattempted ViewQuizAnswersandAnalysis MATHSRelatedLinks ProtectorInMaths AdditionAndSubtractionOfDecimals LessThanSymbolInMaths MultiplicationAndDivisionOfIntegers Class10MathsIndex SectionOfSolids Symmetry-AxisOfSymmetryFigures TriangleInequality SurfaceAreaOfSphere FactorsOf70 5Comments Subhalaxmipani August30,2019at10:12am Superbexplanation.Thankstobujy’s🙂 Reply Nitin August12,2020at9:48am verynicelyexplained. Reply SanchitBhardwaj December5,2020at1:18pm Wellexplainedtheory. Reply unknown December15,2020at9:21pm thankyouforexplainingreallygood Reply MuhammadArman December20,2020at4:41pm wellexplained Reply LeaveaCommentCancelreplyYourMobilenumberandEmailidwillnotbepublished.Requiredfieldsaremarked* * SendOTP DidnotreceiveOTP? 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