MATH 11008 Explorations in Modern Mathematics Fall 2013 Circle one: MW7:45 / MWF1:10 Dr. Kracht Name: Exam Score: /100. (110 pts available) Exam 1: Version C Academic Honesty Pledge Your signature at the bottom indicates your agreement to abide by the following rules. 1. All purses, bags, books, notes, and other papers are placed in the designated area of the classroom. 2. All electronic devices (including cell phones, ipods, etc.) except for hand-held calculators are placed in the designated area of the classroom. 3. I will not share my calculator with another student during the exam. 4. I will not communicate with other students during the exam. 5. I will not seek help from or give help to others during the exam. 6. I will turn my exam in and will not take it from the classroom. 7. I will not discuss the exam outside of class with another student who has not yet taken the exam. 8. I will not cheat in any other way. 9. I will follow any other instructions from my professor. Signature:. Good Luck! 1
Part I: Long Answer. No credit for answers without sufficient justification. Use standard mathematical notation correctly. 1. (25 points) The students in the Math Club are electing a treasurer. The candidates are Adam, Brandon, Constance, Darlene, Edith, and Frank. Here is the preference schedule for the election. # voters 10 9 7 4 6 1 1st place Frank Constance Brandon Adam Constance Darlene 2nd place Adam Adam Constance Darlene Brandon Edith 3rd place Constance Brandon Darlene Constance Adam Frank 4th place Brandon Darlene Adam Edith Edith Constance 5th place Edith Frank Edith Brandon Frank Brandon 6th place Darlene Edith Frank Frank Darlene Adam (a) (2 pts) How many students voted in the election? (b) (3 pts) How many votes are needed for a majority? (c) (5 pts) Find the winner using the Plurality Method. (Show your work!) (d) (5 pts) Compute the Borda points earned by Adam. (Show your work!) (e) (5 pts) Who wins the Pairwise Comparison between Edith and Frank? (Show your work!) (f) (5 pts) In the Plurality-with-Elimination method, is there a winner declared in the first round? If so, who? If not, who is eliminated in the first round? 2
2. (25 points) Consider a four-sided die (with sides labeled 1, 2, 3, and 4 ) and a seven-sided die (with sides labeled 1, 2,..., 7 ). Assume that both dice are fair (honest). The dice are tossed and we observe the number that comes up on each die. (a) Consider the sample space S for this random experiment, where each outcome is an ordered pair of the form (f, s) where f is the number that comes up on the four-sided die and s is the number that comes up on the seven-sided die. i. Write out the sample space S for this random experiment completely. Use set notation. (For your sanity and mine, write the set out in some systematic order.) ii. Find S = N. (b) Let E 1 be the event roll doubles or a sum of six. i. Write E 1 as a set. ii. Find E 1. iii. Find Pr(E 1 ). (c) Let E 2 be the event roll a sum of fourteen. i. Write E 2 as a set. ii. Find E 2. iii. Find Pr(E 2 ). 3
Part II: Multiple Choice (5 points each) Circle the letter of the best answer. 3. Burger Bar 154 offers a build-your-own burger sandwich with choice of burger, bun, topping, and sauce. There are 4 different burgers, 3 buns, 10 toppings, and 5 sauces to choose from. How many different burger sandwiches are there if you choose exactly one from each category? (a) 600 (b) 1200 (c) 22 (d) 9600 4. Thirty contestants are entered in the So You Think You Can Do Math Contest. In how many ways can one choose the top four finishers regardless of order? (a) 30 4 (b) 30 P 4 (c) 30 C 4 (d) 4! 5. Thirty contestants are entered in the So You Think You Can Do Math Contest. In how many ways can one choose a winner and the first-, second-, and third-runners-up? (a) 30 4 (b) 30 P 4 (c) 30 C 4 (d) 4! 6. Consider the sample space S = {o 1, o 2, o 3, o 4 }. Suppose Pr(o 1 ) = 0.27 and Pr(o 2 ) = 0.31. If o 4 is six times as likely as o 3, find Pr(o 4 ). (a) 0.35 (b) 0.36 (c) 0.42 (d) 0.06 4
7. Find the probability of an event E if the odds against E are 7 to 1. (a) 7 8 (b) 1 8 (c) 6 7 (d) 1 7 8. An honest coin is tossed 37 times in a row. The probability that exactly 36 of the tosses come up heads is (a) 36 37 (b) (c) 37 2 37 36 2 37 (d) 37 P 36 9. In an election involving 15 candidates, what is the maximum number of columns possible in the preference schedule? (a) 2 15 (b) 15 P 3 (c) 15 C 10 (d) 15! 10. Consider an election with 501 voters. If there are 10 candidates, at least x votes are needed to have a plurality of the votes. Find x. (a) 251 (b) 50 (c) 51 (d) 250 5
11. In a round robin corn-hole tournament, every player plays against every other player. If 18 players are entered in a round robin corn-hole tournament, how many matches will be played? 18 17 (a) 2 (b) 18 17 (c) 18 2 (d) 2 18 12. An election with four candidates (A,B, C, and D) is decided using the method of Pairwise Comparisons. Suppose that A loses one pairwise comparison and ties one, B loses two and ties none, and C loses two and ties one. Find how many pairwise comparisons D wins. (a) 2 (b) 1 (c) 3 (d) 0 13. A candidate who beats each other candidate in a head-to-head comparison is called a(n) (a) Condorcet candidate (b) Arrow candidate (c) Borda candidate (d) Impossible candidate 14. Arrow s Impossibility Theorem implies (a) that in every election, no matter what voting method we use, at least one of the four fairness criteria will be violated. (b) that it is impossible to have a voting method that satisfies all four of the fairness criteria. (c) that in every election, each of the different voting methods must produce a different winner. (d) that every voting method can potentially violate each of the four fairness criteria. 6