Download The Probability of Complements and Unions of Events

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