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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk December 14, 2010 Final
Name: This quiz is closed book, but you may have two 8.5" 11" sheets with notes in your own handwriting on both sides. Calculators are not allowed. You may assume all of the results presented in class. Please show your work. Partial credit cannot be given for a wrong answer if your work isn't shown. Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem. Please keep your entire answer to a prob lem on that problem's page. Be neat and write legibly. You will be graded not only on the correctness of your answers, but also on the clarity with which you express them. If you get stuck on a problem, move on to others. The problems are not arranged in order of difficulty. The exam ends at 4:30 PM. Problem Points Grade Grader Problem Points Grade Grader 1 20 8 20 2 12 9 11 3 10 10 10 4 10 11 10 5 15 12 32 6 10 13 10 7 10 total 180 Final Problem 1. [20 points] We define the following recurrence for n 0: Tn+2 = Tn+1 + 2Tn where T0 = T1 = 1. 2 (a) [8 pts] Prove by induction that Tn is odd for n 0. You do not need to solve the recurrence for this. Final 3 (b) [12 pts] Prove by induction that gcd( Tn+1 , Tn ) = 1 for n 0. You may assume that Tn is odd for all n. You do not need to solve the recurrence for this. Final Problem 2. [12 points] Find a closedform solution to the following recurrence: x0 = 4 x1 = 23 xn = 11xn1  30xn2 for n 2. 4 Final 5 Problem 3. [10 points] Note: in this question, you may use "choose" notation or factorials in your answers for both (a) and (b). In the card game of bridge, you are dealt a hand of 13 cards from the standard 52card deck. (a) [5 pts] A balanced hand is one in which a player has roughly the same number of cards in each suit. How many different hands are there where the player has 4 cards in one suit and 3 cards in each of the other suits? (b) [5 pts] Not surprisingly, a nonbalanced hand is one in which a player has more cards in some suits than others. Hands that are very desired are ones where over half the cards are in one suit. How many different hands are there where there are exactly 7 cards in one suit? Final 6 Problem 4. [10 points] Three pairs of twins are sharing a bucket of 24 pieces of chicken from Olive Oyl's. Through the magic of twinship, each person always eats exactly as many pieces of chicken as his or her twin. If each person eats at least one piece of chicken, how many ways can the chicken pieces be distributed amongst this family? For the pur poses of this problem, chicken pieces are indistinguishable, and every piece is eaten. Final 7 Problem 5. [15 points] A family of seven folk eat a meal at Kansas Flightless Chicken that comes with unlimited free biscuits (the restaurant owner has deep pockets). Being a car ing family, they only eat biscuits by sharing them with other family members. However, the family obeys the following two rules: A biscuit must be shared between exactly two people If two people share a biscuit, they cannot share another biscuit with each other. Prove that two members of this family must have shared the same number of biscuits over the course of dinner. Final 8 Problem 6. [10 points] Prove that for any m, n, k such that 0 k < m < n  k, the following identity holds: k n m nm = k i ki i =0 Final 9 Problem 7. [10 points] Let G be a bipartite graph with n nodes and k connected compo nents. You (mutually) independently color each of the nodes of G red or black with equal probabilities. What is the probability that your coloring is a valid 2coloring of G? (Hint: the answer does not depend on the number of edges.) Final 10 Problem 8. [20 points] In the faroff land of Spain, two soccer teams, Barcelona and Madrid have been battling each other for centuries to see who is more awesome. They play two games a year: one in the Spring and one in the Fall. The Madrid team has a tendency to fire their coaches very often to try to improve their results. The outcomes of the games are as follows: 1. If Madrid has not fired their coach since the previous game: Barcelona wins with probability Madrid wins with probability They tie with probability
1 5 2 5 2 5 2. If Madrid has fired their coach since the previous game (they haven't yet realized that it's a bad idea): Barcelona wins with probability Madrid wins with probability They tie with probability
1 5 1 5 3 5 Now, Madrid does not fire their coach if they win or tie, but if they lose, they will fire their coach with 90% probability following the loss. For the following two questions, assume that Madrid did not fire their coach before the Spring 2010 game. Here, everything is mutually independent unless otherwise specified. You can use the following timeline to help you visualize the situation. (a) [10 pts] Given that Madrid lost the Fall 2010 game, what is the probability that they fired their coach between the Spring and Fall games of 2010? Final 11 (b) [10 pts] What is the probability that Madrid fired their coach between the Spring and Fall games in 2010 given that they lost BOTH of the games in 2010? Final 12 Problem 9. [11 points] Harvard's loan officer is evaluating her portfolio of student loans. There are 4 especially suspicious loans. Loan A is given to an average Harvard student who does average Harvard stuff. It has a probability of 1/10 of being paid back. Loan B was given to Bill Gates and has a 1/5 probability of being paid back. Loan C was given to David the evil TA and has a probability of 1/3 of being paid back. Loan D was given to a 6.042 student and has 1/2 probability of being paid back. In each of these parts, express any numbers in your answer as ratios of integers. (a) [3 pts] If the probabilities of being paid back are mutually independent, what is the probability that all four loans are paid back? (b) [4 pts] If the probabilities are pairwise independent, what is the most that you can say about the probability that all four loans get paid back? Final 13 (c) [4 pts] If you cannot assume any independence, what is the most you can say about the probability that all four loans are paid back? Final Problem 10. [10 points] 14 Consider a length n vector of integers, x = ( x1 , x2 , . . . , xn ) where the entries of the vector are integers in the set {1, 2, . . . , n}, and the xi are selected uniformly at random. Note that this means numbers can repeat; x could, for example, be a vector of all 1's. Let A be the number of entries xi in the vector for which xi i. Calculate Ex[ A] and give it in closed form. Final 15 Problem 11. [10 points] Let R be a positive random variable with Ex[ R3 ] = k. Prove that Pr ( R x ) k/x3 for any x > 0. Final Problem 12. [32 points] 16 In all parts of this problem, assume that we are using fair, regular dice (sixsided with values 1, 2, 3, 4, 5, 6 appearing equally likely). Furthermore, assume that all dice rolls are mutually independent events. (a) [4 pts] You roll two dice and look at the sum of the faces that come up. What is the expected value of this sum? Express your answer as a real number. (b) [7 pts] Assuming that the two dice are independent, calculate the variance of their sum. Express your answer as a real number. Final 17 (c) [7 pts] You repeatedly roll two fair dice and look at the sum. What is the probability that you will roll a sum of 4 before you roll a sum of 7? Express your answer as a real number. (d) [7 pts] What is the expected number of rolls until you get a sum of 4 or a sum of 7? (For example, if you get 7 on the first roll, the number of rolls is 1.) Express your answer as a real number. Final 18 (e) [7 pts] You roll 10 dice. Using the Chernoff Bound, give an upper bound for the probability that 8 or more of them rolled a 1 or a 2? You don't need to calculate the value with a calculator (since you do not have one), but please write it in simplest terms. Final Problem 13. [10 points] 19 An MIT student is walking along Mass Ave and is torn between making a trip to the pub and going to her dorm to finish her 6.042 homework. The student is n steps from the pub and T  n steps from the dorm where T >= 3 is the distance between the pub and the dorm, both located on Mass Ave (which we will assume to be a straight line). The student flips a coin at each step to decide which way to go. With probability p, the student takes 2 steps toward the pub and with probability 1  p, she takes 1 step toward her dorm. If the student is one or two steps away from the pub and takes two steps toward the pub, then she reaches the pub. If she is one step away from her dorm and takes a step toward her dorm, then she reaches the dorm.. Let Xn be the probability that the student reaches the pub before she reaches the dorm. Assume that all coin tosses are mutually independent. Find a recurrence for Xn . (You do not need to solve the recurrence, but your answer should contain sufficient information such that a 6.042 could solve it given enough time.) MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science

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