Sample Midterm This midterm consists of 10 questions. The rst seven questions are multiple choice; the remaining three

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1 CS{74 Combiatorics & Discrete Probability, Fall 97 Sample Midterm :30{:00pm, 7 October Read these istructios carefully. This is a closed book exam. Calculators are permitted.. This midterm cosists of 0 questios. The rst seve questios are multiple choice; the remaiig three require writte aswers. 3. Aswer the multiple choice questios by circlig the correct aswer. You should be able to aswer all of these from memory, by ispectio, or with a very small calculatio. Icorrect aswers may attract a egative score, so if you do ot kow the aswer you should ot guess. 4. Write your aswers to the other questios i the spaces provided below them. Noe of these questios requires a log aswer, so you should have eough space; if ot, cotiue o the back of the page ad state clearly that you have doe so. Show all your workig. 5. The questios vary i diculty: if you get stuck o some part of a questio, leave it ad go o to the ext oe.. A fair 6-sided die is tossed repeatedly. The expected umber of tosses util two dieret outcomes are observed is A abset-mided secretary places letters i addressed evelopes i a completely radom maer. The probability that oe of the letters is placed i the right evelope is asymptotically e e e e 3. The largest umber of balls that ca be tossed ito bis, if the probability that ay cell cotais more tha oe ball is to be kept small, is o the order of a costat l p e l l 4. The radom variable X has the Biomial distributio with parameters ad p. (a) The expectatio of X is 0 p p p ot determied (b) The variace of X is p p( p) p p p p( p) ot determied

2 5. A o-egative radom variable X has expectatio E [X] = ad variace Var [X] = 3. Circle those three of the followig statemets that must be true about X : Pr[X 0] 0 Pr[X ] > 0 Pr[X ] = Pr[X ] Pr[X = ] > 0 E X = 4 Pr[X 3] = 0 6. balls are tossed at radom ito bis. (a) As!, the probability that the rst two bis are empty is 0 e e e = (b) As!, the probability that the rst bi cotais exactly oe ball ad the secod bi is empty is 0 e e e = 7. I a group of 0 people, 6 ca play the piao, 5 ca play the saxophoe, 4 ca play the violi, 4 ca play the piao ad the saxophoe, 3 ca play the piao ad the violi, ca play the saxophoe ad the violi, oly ca play all three istrumets. (a) How may people ca play at least oe istrumet? (b) How may ca play exactly two istrumets? [cotiued overleaf ]

3 8. Alice, Bob ad Charlie wat to choose oe of the umbers, ad 3 with equal probabilities. All they have is a biased coi that comes up heads with probability p, where 0 < p <. Alice suggests the followig method. Toss the coi twice. If the two ips are HH, output, if they are HT, output, if they are TH, output 3. If the ips are TT, repeat the experimet. Bob suggests the followig method. Toss the coi twice. Output the umber of heads obtaied, plus. Charlie suggets the followig method. Toss the coi three times. If heads is obtaied oly i the i-th toss, where i 3, output i. Otherwise, repeat the experimet. (a) For which values of p, if ay, does Alice's method produce the umbers, ad 3 with equal probabilities? (b) For which values of p, if ay, does Bob's method produce the umbers, ad 3 with equal probabilities? (c) For which values of p, if ay, does Charlie's method produce the umbers, ad 3 with equal probabilities? 3 [cotiued overleaf ]

4 (d) What is the expected umber of tosses used by Alice's method. (e) What is the variace of the umber of tosses used by Charlie's method. (f ) Suppose that you have a supply of biased cois, each with a ukow, ad possibly dieret, bias. You are allowed to use each coi oly twice. Ca you use the cois to select oe of the umbers, ad 3 with equal probabilities? Show how or explai why this is ot possible. 4 [cotiued overleaf ]

5 9. I the radom graph model G ;p, a radom graph o vertices is costructed by the followig experimet: start with vertex set V = f; ; : : :; g ad edge set E = ; for each of the possible edges e = fi; jg idepedetly, ip a coi with heads probability p; if the coi comes up heads, add e to E output G = (V; E) (a) What is the size of the sample space? (b) Let G be a graph i the sample space havig exactly m edges. What is the probability of G, as a fuctio of, p ad m? (c) Let the r.v. X deote the umber of edges i the graph G. What is E [X], as a fuctio of ad p? (d) A subset of vertices S V is called a clique if fi; jg E for all i; j S (i.e., all pairs of vertices i S are coected by a edge). For ay give set S of k vertices, what is the probability that S is a clique, as a fuctio of p ad k? (e) What is the expected umber of cliques of size k i G, as a fuctio of, p ad k? [cotiued overleaf ] 5

6 0. A fair -sided die is tossed three times. The three tosses are idepedet. Let X, X ad X 3 be the outcomes of these tosses (X, X ad X 3 each gets oe of the values ; ; : : :; with equal probabilities). Evaluate the followig probabilities: (a) Pr[X < X ] (b) Pr[X < X < X 3 ] (c) Pr[X < X ad X < X 3 ] (d) Pr[X < X j X < X 3 ] (e) Pr[X < X j X = X 3 ] 6