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6.042/18.062JMathematicsforComputerScience November08,2010TomLeightonandMartenvanDijk

ProblemSet10

Problem1. [15points] SupposePr{}:S [0,1] isaprobabilityfunction onasamplespace,S,andletBbeaneventsuchthatPr{B}>0. DefineafunctionPrB{}onoutcomesw S bytherule:

PrB{w}= Pr{w}/Pr{B} ifwB, (1)0 ifw /B.

(a) [7pts] Prove that PrB{} is also a probability function on S according to Definition14.4.2.(b) [8pts]Provethat

PrB{A}= Pr{AB}Pr{B}

forallA S.Problem2. [20points](a) [10pts] Here are some handy rules for reasoning about probabilities that all follow

directly from the Disjoint Sum Rule. Use Venn Diagrams, or another method, to provethem.

Pr{AB}=Pr{A} Pr{AB} (DifferenceRule)Pr A = 1Pr{A} (ComplementRule)

Pr{AB}=Pr{A}+Pr{B} Pr{AB} (Inclusion-Exclusion)

Pr{AB} Pr{A}+Pr{B}. (2-eventUnionBound)

IfAB, then Pr{A} Pr{B}. (Monotonicity)

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2 ProblemSet10(b) [10pts] Prove the following probabilistic identity, referred to as theUnion Bound.

Youmayassumethetheoremthattheprobabilityofaunionofdisjoint sets isthesumoftheirprobabilities.Theorem.LetA1, . . . , An beacollectionofeventsonsomesamplespace. Then

n

Pr (A1 A2 An) Pr (Ai).i=1

(Hint: Induction)Problem3. [15points] Recallthestrangedicefromlecture:

Inthebookweprovedthat ifwerolleachdieonce,thendieAbeatsB moreoften,dieBbeatsdieC moreoften,anddieC beatsdieAmoreoften. Thus,contrarytoourintuition,thebeatsrelation>isnottransitive. Thatis,wehaveA > B > C > A.Wethenlookedatwhathappensifwerolleachdietwice,andaddtheresult. Inthisproblemwewillshowthatthebeatsrelationreversesinthisgame,thatis,A < B < C < A,whichisverycounterintuitive!(a) [5pts]ShowthatrollingdieC twiceismorelikelytowinthanrollingdieB twice.(b) [5pts]ShowthatrollingdieAtwiceismorelikelytowinthatrollingdieC twice.(c) [5pts]ShowthatrollingdieB twiceismorelikelytowinthanrollingdieAtwice.

Problem4. [14points] Letsplayagame! Werepeatedlyflipafaircoin. Youhavethesequence HHT, and I have thesequence HT T. Ifyour sequencecomes up first, then youwin. Ifmysequencecomesupfirst,thenIwin. Forexample,ifthesequenceoftossesis:

T T HT HT HHT

2

6 7

1

5 9

3

4 8

A B C

Imageby MIT OpenCourseWare.

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3ProblemSet10then you win. What is the probability that you win? It may come as a surprise that theanswerisverydifferentfrom1/2.This problem is tricky, because the game could go on for an arbitrarily long time. Drawenough of the tree diagram to see a pattern, and then sum up the probabilities of the(infinitelymany)outcomesinwhichyouwin.Itturnsoutthat foranysequenceofthreeflips, there isanothersequencethat is likelytocomeupbeforeit. Sothereisnosequenceofthreeflipswhichturnsupearliest! ...andgivenanysequenceofthreeflips,knowinghowtopickanothersequencethatcomesupsoonermorethanhalfthetimegivesyouanicechancetofoolpeoplegamblingatabar:-)Problem5. [15points] Wereinterestedintheprobabilitythatarandomlychosenpokerhand(5cardsfromastandard52-carddeck)containscardsfromatmosttwosuits.(a) [7pts] What is an appropriate sample space to use for this problem? What are the

outcomesintheevent,E,weareinterestedin? Whataretheprobabilitiesoftheindividualoutcomesinthissamplespace?(b) [8pts]WhatisPr(E)?Problem6. [21points]Ihaveadeckof52regularplayingcards,26red,26black,randomlyshuffled. Theyall liefacedowninthedecksothatyoucantseethem. Iwilldrawacardoffthetopofthedeckand turn it face up so that you can see it and then put it aside. I will continue to turnupcards likethisbutatsomepointwhiletherearestillcards left inthedeck,youhavetodeclare

that

you

want

the

next

card

in

the

deck

to

be

turned

up.

If

that

next

card

turns

up

blackyouwinandotherwiseyoulose. Eitherway,thegameisthenover.(a) [4pts] Show that if you take the first card before you have seen any cards, you then

haveprobability1/2ofwinningthegame.(b) [4pts]Supposeyoudonttakethefirstcardand itturnsupred. Showthatyouhave

thenhaveaprobabilityofwinningthegamethatisgreaterthan1/2.(c) [4pts]Iftherearerredcardsleftinthedeckandbblackcards,showthattheprobability

ofwinninginyoutakethenextcardisb/(r+b).(d) [9pts]Either,

1. comeupwithastrategyforthisgamethatgivesyouaprobabilityofwinningstrictlygreaterthan1/2andprovethatthestrategyworks,or,

2. comeupwithaproofthatnosuchstrategycanexist.

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6.042J / 18.062J Mathematics for Computer Science

Fall2010

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