Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
TheProbabilityofComplementsandUnionsofEvents Theorem1:LetE beaneventinsamplespaceS.Theprobability oftheevent =S− E,thecomplementaryeventofE,isgiven by Proof:Usingthefactthat| | =|S| − |E|, Networking Platform 1 Extensible - CSE 240 – Logic and Discrete Mathematics 1 TheProbabilityofComplementsandUnionsofEvents Example:Asequenceof10 bitsischosenrandomly.Whatisthe probabilitythatatleastoneofthesebitsis0? Solution:LetE betheeventthatatleastoneofthe10bitsis0. Then istheeventthatallofthebitsare1s.Thesizeofthe samplespaceS is210.Hence, Networking Platform 2 Extensible - CSE 240 – Logic and Discrete Mathematics 2 TheProbabilityofComplementsandUnionsofEvents Theorem2:LetE1 andE2 beeventsinthesamplespaceS. Then Proof:Giventheinclusion-exclusionformulafromSection2.2,|A ∪B|=|A|+|B|−|A ∩B|,itfollowsthat Networking Platform 3 Extensible - CSE 240 – Logic and Discrete Mathematics 3 CombinationsofEvents Theorem:IfE1,E2,…isasequenceofpairwise disjointeventsinasamplespaceS,then •see Exercises 36 and 37 for the proof Networking Platform 4 Extensible - CSE 240 – Logic and Discrete Mathematics 4 Probability Whichismorelikely: a) Rollingan8when2dicearerolled? b) Rollingan8when3dicearerolled? c) Noclue. Networking Platform 5 Extensible - CSE 240 – Logic and Discrete Mathematics 5 Probability Whatistheprobabilityofatotalof8when2diceare rolled? Whatisthesizeofthesamplespace? 36 Howmanyrollssatisfyourconditionofinterest? 5 Sotheprobabilityis5/36~= 0.139 Networking Platform 6 Extensible - CSE 240 – Logic and Discrete Mathematics 6 Combinationswithrepetition • ThereareC(n+r-1,r)r-sizedcombinationsfromasetofn elementswhenrepetitionisallowed. • Example:Howmanysolutionsaretheretotheequation x1 + x 2 + x 3 + x 4 = 10 • Whenthevariablesarenonnegativeintegers? C(13,3) • Each#iscontribute1tothesumof10andsincewe € have4numbers,thenweneed3barstoseparatethe numbersandwecanplacethebarsnexttoeachother è wehave13spotstochoosewheretoplaceeach individualbar. •Onepossiblevalueis Networking Platform 7 Extensible - CSE 240 – Logic and Discrete Mathematics 1 + 3 + 6 + 0 = 10 7 Probability Whatistheprobabilityofatotalof8when3diceare rolled? Whatisthesizeofthesamplespace? 216 Howmanyrollssatisfyourconditionofinterest? C(7,2) Sotheprobabilityis21/216~=0.097 Networking Platform 8 Extensible - CSE 240 – Logic and Discrete Mathematics 8 ConditionalProbability LetEandFbeeventswithPr(F)>0.Theconditional probabilityofEgivenF,denotedbyPr(E|F)isdefinedto be: Pr(E|F)=Pr(EÇF)/Pr(F). E Networking Platform 9 Extensible - CSE 240 – Logic and Discrete Mathematics F 9 GroupProblem • Anurncontainsfourblueballsandfiveredballs. Whatistheprobabilitythataballchosenfrom theurnisblue? • Whatistheprobabilitythatwhentwodiceare rolled,thesumofthenumbersonthetwodiceis 7? Extensible Networking Platform 10 - CSE 240 – Logic and Discrete Mathematics 10 GroupProblem • Anurncontainsfourblueballsandfiveredballs.Whatisthe probabilitythataballchosenfromtheurnisblue? – Theprobabilitythattheballischosenis4/9sincethereare ninepossibleoutcomes,andfouroftheseproduceablue ball. • Whatistheprobabilitythatwhentwodicearerolled,thesum ofthenumbersonthetwodiceis7? – Bytheproductrulethereare62 =36possibleoutcomes.Sixof thesesumto7.Hence,theprobabilityofobtaininga7 is6/36= 1/6. Networking Platform 11Extensible - CSE 240 – Logic and Discrete Mathematics 11 GroupProblem • Anurncontainsfourblueballsandfiveredballs. Whatistheprobabilitythataballchosenfrom theurnisblue? • Whatistheprobabilitythatwhentwodiceare rolled,thesumofthenumbersonthetwodiceis 7? Extensible Networking Platform 12 - CSE 240 – Logic and Discrete Mathematics 12 GroupProblem • Anurncontainsfourblueballsandfiveredballs.Whatisthe probabilitythataballchosenfromtheurnisblue? – Theprobabilitythattheballischosenis4/9sincethereare ninepossibleoutcomes,andfouroftheseproduceablue ball. • Whatistheprobabilitythatwhentwodicearerolled,thesum ofthenumbersonthetwodiceis7? – Bytheproductrulethereare62 =36possibleoutcomes.Sixof thesesumto7.Hence,theprobabilityofobtaininga7 is6/36= 1/6. Extensible Networking Platform 13 - CSE 240 – Logic and Discrete Mathematics 13 GroupProblem Example:Whatistheprobabilitythatthenumbers 11,4,17,39, and23aredrawninthatorderfrom abinwith50 ballslabeledwiththenumbers1,2, …,50if a) Theballselectedisnotreturnedtothebin. b) Theballselectedisreturnedtothebinbeforethe nextballisselected. Extensible Networking Platform 14 - CSE 240 – Logic and Discrete Mathematics 14 GroupProblem Example:Whatistheprobabilitythatthenumbers11,4,17,39, and23aredrawninthatorderfromabinwith50 balls labeledwiththenumbers1,2,…,50if a) Theballselectedisnotreturnedtothebin. b) Theballselectedisreturnedtothebinbeforethenextball isselected. – Samplingwithoutreplacement:Theprobabilityis 1/254,251,200sincethereare50∙49∙47∙46 ∙45= 254,251,200waystochoosethefiveballs. – Samplingwithreplacement:Theprobabilityis 1/505 =1/312,500,000since505 =312,500,000. Extensible Networking Platform 15 - CSE 240 – Logic and Discrete Mathematics 15 Independence Definition:TheeventsE andF areindependentifandonlyif p(E⋂F) = p(E)p(F). Example:SupposeE istheeventthatarandomlygeneratedbitstringof lengthfourbeginswitha1andF istheeventthatthisbitstringcontains anevennumberof1s.AreE andF independentifthe16bitstringsof lengthfourareequallylikely? Solution:Thereareeightbitstringsoflengthfourthatbeginwitha1,and eightbitstringsoflengthfourthatcontainanevennumberof1s. – Sincethenumberofbitstringsoflength4is16, p(E) =p(F)=8/16=½. – SinceE⋂F={1111,1100,1010,1001},p(E⋂F)=4/16=1/4. WeconcludethatEandFareindependent,because p(E⋂F)=1/4=(½)(½)=p(E)p(F) Extensible Networking Platform 16 - CSE 240 – Logic and Discrete Mathematics 16 RandomVariables ForagivensamplespaceS,arandomvariable isanyreal valuedfunctiononS. S -2 • 0 2 Supposeourexperimentisarollof2dice.Sissetofpairs. X=sumoftwodice. Y=differencebetweentwodice. Z=maxoftwodice. Extensible Networking Platform 17 - CSE 240 – Logic and Discrete Mathematics X((2,3))=5 Y((2,3))=1 Z((2,3))=3 17 RandomVariables Example: Supposeweareplayingagamewithcardslabeled1to20, andwedraw3cards.Webetthatthemaximumcard hasvalue17orgreater.What’stheprobabilitywewin thebet? Letr.v.Xdenotethemaximumcardvalue.ThepossiblevaluesforXare3, 4,5,…,20. i 3 4 5 6 7 8 9 … Pr(X = i) ? ? ? ? ? ? ? 20 ? Fillinginthisboxwouldbeapain.Welookforageneralformula. Extensible Networking Platform 18 - CSE 240 – Logic and Discrete Mathematics 18 RandomVariables Xisvalueofthehighestcardamongthe3selected.20 cardsarelabeled1through20. WewantPr(X=i),i=3,…20. Denominatorfirst:Howmanywaysaretheretoselectthe3 cards? C(20,3) Howmanychoicesaretherethatresultinamaxcardwhose valueisi? C(i-1,2) Pr(X=i)=C(i-1,2)/C(20,3) Thesearethetablevalues. Wewinthebetifthemaxcard,Xis17orgreater.What’stheprobability wewin? Pr(X=17)+Pr(X=18)+Pr(X=19)+Pr(X=20) Extensible Networking Platform 19 - CSE 240 – Logic and Discrete Mathematics »0.51 19 PracticeProblem Aclasshas20womenand13men.Acommitteeoffiveis chosenatrandom.Find (a)p (thecommitteeconsistsofallwomen). (b)p (thecommitteeconsistsofallmen) (c)p (thecommitteeconsistsofallofthesamesex) Extensible Networking Platform 20 - CSE 240 – Logic and Discrete Mathematics 20 Solutions 1)Aclasshas20womenand13men.Acommitteeoffiveis chosenatrandom. (a) C(20,5)/C(33,5) (b) C(13,5)/C(33,5) (c) (C(20,5)+C(13,5))/C(33,5) Extensible Networking Platform 21 - CSE 240 – Logic and Discrete Mathematics 21