M8 FINAL EXAMINATION DECEMBER, 26 Printed Name: Signature: Instructor: seat number: INSTRUCTIONS: This exam consists of 3 multiple-choice questions. Each question has one correct answer choice. Indicate your answer choice for each question by placing the appropriate CAPITAL letter in the correct space below. You may write in the test as much as needed, but no credit is given for anything written inside the exam itself. You may use only one-line or two-line scientific calculators. Please keep this cover page attached to your exam. GOOD LUCK! PLEASE USE CAPITAL LETTERS. ) ) 2) 2) 2) 22) 3) 3) 23) 4) 4) 24) 5) 5) 25) 6) 6) 26) 7) 7) 27) 8) 8) 28) 9) 9) 29) ) 2) 3) SCORE:
PROBLEM : A universal set U has two disjoint subsets, A and B, with n(u) =, n(a) = 42, and n(b) = 57. Which of the following is n( A B )? A) C) 68 E) B) 5 D) 2 N) none of the above PROBLEM 2: A die is weighted so that all of the even numbers are equally likely, and all of the odd numbers are equally likely. In addition, each even number is three times as likely as each odd number. If this die is rolled 4 times, how many times should we expect to roll a 2? A) /3 C) 5 E) 87.5 B) 2/3 D) N) none of the above PROBLEM 3: A system of equations A X=B is given by: 2 x 2 A = = 2, X = y, and B 4. Which of the following is the value of y? 2 5 z A) 3 C) 2 E) -6 B) -5 D) -/5 N) none of the above PROBLEM 4: A code is formed by selecting four letters from the set L = { A, B, C, D, E }, without replacement. For example, CDAB and DABC are two possible codes, but CAEC is not a possible code. How many such codes will contain the letter D? A) 2 C) 256 E) 96 B) 24 D) 24 N) none of the above
PROBLEM 5: Britney Spears new perfume, Uninhabited, comes in two strengths, regular and industrial strength. Each bottle of regular strength uses 3 ounces of skunk scent and 2 ounces of buttermilk, and each bottle of industrial strength uses 8 ounces of skunk scent and 2 ounces of buttermilk. Each day, Britney has 8 ounces of skunk scent and 42 ounces of buttermilk to be used for making perfume. If Britney makes the correct number of bottles of each strength to use up all of the available ingredients, how many bottles of industrial strength perfume will Britney make? A) 9 C) 65 E) 2 B) 45 D) N) none of the above PROBLEM 6: Which of the following represents the number of corner points of the solution set to the following system of inequalities? x 2 y 3 x + 2y 2x + y A) C) 4 E) 5 B) 3 D) 2 N) none of the above PROBLEM 7: A lunchbox contains six pieces of cheese and four hot dogs. Three of the items are randomly selected. What is the probability that exactly two hot dogs are selected? A) 3/ C) / E) /5 B) /2 D) 35/2 N) none of the above PROBLEM 8: Which of the following is the equation of the line that has slope m = -5/2 and passes through the point at which the lines 3x + 2y = 4 and x +2y = 2 intersect? A) -5x+2y=3 C) 5x+2y=3 E) -5x+2y=6 B) 5x+2y=6 D) 2x+5y=6 N) none of the above
PROBLEMS 9 AND REFER TO THE FOLLOWING SITUATION: The We Don t Really Karasotes Movie Theater sells two sizes of popcorn, a 5-gallon bucket and a 2-gallon bucket. Due to company policy, they must make at least twice as many 2-gallon buckets as 5-gallon buckets, In addition, the total number of buckets of popcorn made during a single workday cannot exceed 8 buckets. The company makes $6 profit for each 5-gallon bucket sold, and they make $5 profit for each 2-gallon bucket of popcorn sold. The company would like to maximize their daily profit. Let x = number of 5-gallon buckets made y = number of 2-gallon buckets made PROBLEM 9: Which ONE of the following is NOT a constraint for the above linear programming problem? A) x 2y C) x + y 8 E) y B) y 2x D) x N) none of the above PROBLEM : If the company makes the correct number of buckets of each size to maximize their daily profit, how many 2-gallon buckets will be made? A),2 C),8 E) 9 B) 6 D) N) none of the above PROBLEM : A corporate committee is to be formed by selecting four people from a set of 8 qualified applicants. The team will consist of a president, a vice-president, and two foremen (whose roles are not different from each other). How many different corporate committees are possible? A) 7 C) 68 E) 24 B) 84 D) 2 N) none of the above PROBLEM 2: A cow s weight gain is linearly related to the amount of food they eat per day. If they eat 4 pounds of food one day, they will gain 3.5 ounces that day. If they eat 6 pounds of food one day, they will gain 4.7 ounces that day. If, on a particular day, the cow gains 6.5 ounces, how many pounds of food did they eat that day? A) 8.5 C) 7.5 E) 9.5 B) 8 D) 9 N) none of the above
PROBLEM 3: A Markov chain has two states. If the chain is in state on a given observation, then it is three times as likely to be in state as to be in state 2 on the next observation. If the chain is in state 2 on a given observation, then it is twice as likely to be in state as to be in state 2 on the next observation. Which of the following represents the correct transition matrix for this Markov chain? 3/ 4 / 4 3/ 4 / 4 A) P = C) P = E) / 3 2 / 3 2 / 3 / 3 3/ 4 P = / 4 2 / 3 / 3 / 4 3/ 4 2 / 3 / 3 B) P = D) P = N) none of the above / 3 2 / 3 3/ 4 / 4 PROBLEM 4: A fair die is rolled twice. If it is known that the two numbers landing up are not the same number, what is the probability that the sum of the two numbers landing up is exactly eight? A) /6 C) 2/5 E) /5 B) 5/36 D) /9 N) none of the above PROBLEM 5: Suppose the matrix B is a 3x3 matrix such that B A = C, where A and C are given by: A = 2, 3 and 2 C = 2 3 2 Which of the following is the entry in the third row and second column of B? A) C) -/2 E) B) -3/2 D) /2 N) none of the above
PROBLEM 6: Using the following five matrices, which ONE of the following operations is NOT defined? 2 3 A = [ 2 3 ], B =,,, and =. 3 C = D = 5 E 3 4 2 2 A) B D+2B C) D A C E) A C B) C B C D) (C E) 2 N) none of the above. PROBLEM 7: Using the matrices in PROBLEM 6, which of the following is the element in the second row and first column of B -? A) -/3 C) /2 E) /6 B) -/2 D) /3 N) none of the above PROBLEM 8: Using the matrices in PROBLEM 6, which of the following is the element in the second row and second column of B E? A) -2 C) 4 E) 3 B) -3 D) N) none of the above PROBLEM 9: Suppose there is a 4% chance that Sid Slacker passes his finite math class. In addition, if Sid Slacker passes his finite math class, then there is only a 2% chance that he will get kicked out of school. On the other hand, if Sid does not pass his finite math class, then there is a 7% chance he will get kicked out of school. If Sid Slacker gets kicked out of school, what is the probability that he passed his finite math class? A).25 C).6 E).2 B).8 D) 4/2 N) none of the above
PROBLEM 2: Which of the following is the value of x in the solution to the following system of equations? w 2x + y + 6z = 2w x + 8y + 9z = 8 A) x= -4-2y+z, y & z arbitrary C) x= 2-5y-4z, y & z arbitrary E) x = 3, y & z arbitrary B) x= -4+2y-z, y & z arbitrary D) x= 4-2y+z, y & z arbitrary N) none of the above PROBLEM 2: MTV plans to visit three of the following five cities during spring break: Atlanta, Boston, Cincinnati, Dallas, and Ellettsville. An itinerary for their trip is a list of the three cities chosen in the order they are to be visited. If the itinerary used by MTV is randomly selected from among all possible itineraries, what is the probability that Dallas is visited immediately before Ellettsville? A) 3/ C) / E) 3/2 B) /2 D) 3/5 N) none of the above PROBLEM 22: The transition matrices of three Markov chains are shown below. Which of these represent Markov chains that are regular? P =.2 Q =.5.8.5 R =.2.3.7..7 A) P only C) Q and R only E) Q only B) P and Q only D) R only N) none of the above PROBLEM 23: A fair six-sided die is tossed five times. What is the probability that a number less than 3 lands up exactly four times? A) 2/243 C) 5/243 E) 5/32 B) 3/6 D) /243 N) none of the above
PROBLEM 24: Which of the following represents the minimum value of the function 8x 3y on the solution set shaded below? A) 7 C) there is no minimum E) 27.5 B) -9 D).5 N) none of the above NOTE: The solution set is unbounded, and the points labeled with dotted lines are NOT corner points. y (, 3) (2, 3) U N B O U N D E D (, -3) (4,.5) x (, - 3.5) PROBLEM 25: There are five Poodles and four Chihuahuas at an animal shelter. If two of the animals are randomly selected to be shown at the mall, what is the expected number of Poodles selected? A).2 C) 9/9 E). B) /9 D).9 N) none of the above PROBLEM 26: A Markov chain has the transition matrix P below. If the chain is in state on the 3 rd observation, what is the probability that it will be in state 2 on the 6 th observation? P = / 2 / 2 A) /4 C) 3/4 E) /2 B) D) /8 N) none of the above
PROBLEM 27: There are three football jerseys in a laundry hamper, with the numbers 8, 88, and 93 on the back, respectively. A total of three selections are made from the hamper, each time noting the number on the back of the selected jersey. If the number 8 jersey is selected on any reach into the hamper, it is returned before the next selection. Otherwise, the selected jersey is not replaced before the next reach into the hamper. How many different outcomes are possible? A) 2 C) 4 E) 3 B) D) 5 N) none of the above PROBLEM 28: K-Fed s concerts behave like a Markov chain. If the current concert gets cancelled, then there is an 8% chance that the next concert will be cancelled also. However, if the current concert does not get cancelled, then there is only a 6% chance that the next concert will be cancelled. What is the long-run probability that a concert will be cancelled? A) 2/3 C) /4 E) 3/4 B) 7/ D) 4/5 N) none of the above PROBLEM 29: Using the Markov chain of the previous problem, if the current concert is four times as likely to be cancelled as to be not cancelled, what is the probability that the next concert will also be cancelled? A).76 C).8 E).36 B).64 D).24 N) none of the above PROBLEM 3: A sample space S has two independent events, A and B, with Pr[A]=.3, and Pr[B]=.4. Which of the following is Pr [ A B ]? A).8 C).28 E).3 B).2 D).42 N) none of the above