Problem 67: PIN guessing ( ) In order to get money from a cash dispenser I have to punch in a Personal Identification Number. I have forgotten my PIN, but I do know that it is equally likely to be any one of the integers 1, 2, .... n. I plan to punch in integers in ascending order until I get the right one. I can do this at the rate of r integers per minute. As soon as I punch in the first wrong number, the police will be alerted. The probability that they will arrive within a time t minutes is 1 - e-At, where A is a positive constant. If I follow my plan, show that the probability of the police arriving before I get my money is 1 - -)(k-1)/ Simplify the sum. On past experience, I know that I will be so flustered that I will just punch in possible integers at random, without noticing which I have already tried. Show that the probability of the police arriving before I get my money is 1 - n - (a- l)e->/r .Problem 65: A knock-out tournament ( V ) A tennis tournament is arranged for 2" players. It is organised as a knockout tournament, so that only the winners in any given round proceed to the next round. Opponents in each round except the final are drawn at random, and in any match either player has a probability ; of winning. Two players are chosen at random before the start of the first round. Find the probabilities that they play each other: (i) in the first round; (ii) in the final round; (iii) in the tournament.1.2 Problems Problem 1: Determine if each of the following objects is a member of Z: (5). (3, -1), 7.12, v5, a = the 2,00th decimal digit in the base-10 expres- sion for *. Problem 2: Let A be the set of digits in the base-10 expression of the rational number . Let B be the same for me. Prove that A = B. Problem 3: Prove or disprove: the set C of digits in the base-10 expression of 40363 63637 3 33000 00000 equals the set A of the previous problem. Problem 4: Define sets A = (1. (4). (2).3, 4. 5) B = ({ {1,4,5.3, 1)}}. C = (1. (3).2.1). D = (1,1,3), E = (1.4, (5).(3} }. F = (1,8, (1.2,3.4)). and, aj = 1, as = (2), as = (2, 1). as = (2, 1, 3, 4), as = (3, 1, 5). For each of aj,..., as, determine if it is a member of the sets A.... , F respectively. Present your answer in the following table: ABCDEF Use an "E" to stand for membership and a blank to stand for nonmemn- bership. Problem 5: Define A, B. C. D. E. and F as in Problem 4. Calculate the following sets. Anc BOF DUC CRE CU(DNF) AnE. @ Chung-Chih Li. Kishan Mehrotra 14 1. Sets Problem 6: Define U = (3,1,3, 2 ). V = (1.3. (1.3). (1,2.3}}. ISUE V? LA U C V? Problem 7: Let A and B be sets. If A C B, what does that tell you about AnB ad AUB? Problem 8: Let A. B and S be sets. If A C S and B C S. what can you say about A U B? Problem 9: Find the cardinality of the set S = (p/q |p.q c N+, p.q. $ 10 ). Problem 10: List all the subsets of (1.2. (6)). Problem 11: List all the elements of (b.c,d) x (e,o). Problem 12: Describe all sets that have no proper subsets. Problem 13: Let A, B,C be sets. Suppose that A is a subset of B, and C is a proper subset of B. Is A a proper subset of C'?18. The following data lists eight different investment amounts (X) and the amount of interest they earned (Y): X $1000 $2000 $3000 $10000 $500 $5000 Y $50 $100 $150 $500 $25 $250 What is the best estimate for the regression coefficient r between X and Y? a) r will be positive and close to zero b) r will be positive and close to one c) r will be exactly one dir will be larger than one e) r will be negative 19. In a study of possible correlation between the height in cm (X) and weight in kg (Y) of chimpanzees, a sample of 40 animals produces a correlation coefficient of r=+0.813 and a regression line with equation Y = 034 X + 19 5. What is the expected weight of an 80 cm tall chimpanzee? a) 46.7 kg b) 177.9 kg c) 24.0 kg d) 34.8 kg e) 57.1 kg 20. The following table lists the number of days that five houses had been up for sale, as well as their selling price. X (days) 45 12 3 17 32 Y (1000$) 275 401 420 212 365 Calculate the correlation coefficient r between the number of days (X) and the selling price (Y) for this sample. a) - 0.465 b) - 0.219 c) +0.219 d) + 0.512 e) +0.897 21. The Vitamin C content of a particular brand of vitamin supplement pills is normally distributed with mean 490 mg and standard deviation 12 mg. What is the probability that a randomly selected pill contains at least 500 mg of Vitamin C? a) 0.7967 b) 0.8333 c) 0.0525 d) 0.1123 e) 0.2033Problem 28: Let A. B. U. and V be any sets such that A C U and B C V. Is the following correct? Explain. ( A x B) C ( U x V ). Problem 29: Let A. B. C be sets. Prove A x (BUC) = (A x B) U (A xC). Problem 30: Let A. B be sets. and C:= AU B. Use the result in Problem 20 to prove that A x B C A x C CC xC. Problem 31: Let A be a set. Define diagonal of A x A := {(a. a)la E A). Suppose A C B. Prove or disprove the following identities. 1. (A x B)n (B x A) = B x B. 2. (the diagonal of A x A)n (A x B) = the diagonal of B x B. Problem 32: Let X = {1.2.3. a). List the elements of P(X). Problem 33: Let X be the set (1. 2, 3. 4.5.6. (1}}. Find Y such that Y = XU(XnP(X)). Problem 34: List the elements of the following sets. 1. P(0) 2. P(10)) 3. P(P(O) 4. (0) x P(0) 5. 4 x P(0) 6. P(W) x P(0) Problem 35: List the elements of A x P( A), where A = (a. 1). Problem 36: Let A. B be sets. Prove P(AnB) = P(A)nP( B). Problem 37: Prove or disprove that if ACP(B) & Be P(A). then A = B. Problem 38: Let A be the set of nonnegative integers and B the set of nonnegative odd integers. Prove that the cardinality of A is equal to the cardinality of B. [See Theorem 1.10 on page 12] Problem 39: Prove that for any set A. |4|