n(n 1) 2 C = total population total number of seats amount of increase original amount

Transcription

1 MTH 110 Quiz 2 Review Spring 2018 Quiz 2 will cover Chapter 13 and Section Justify all answers with neat and organized work. Clearly indicate your answers. The following formulas may or may not be useful. C = n(n 1) 2 Standard divisor = total population total number of seats Standard quota for a state = population of that state standard divisor Percent increase = amount of increase original amount 1. Suppose that the pairwise comparison method is used to determine the winner in an election. If there are 9 candidates, how many comparisons must be made? 2. Suppose that the pairwise comparison method is used to determine the winner in an election. If there are 5 candidates, A, B, C, D, and E, list all the comparisons that must be made. 3. The preference table for an election is shown. Number of Votes st choice D C C A 2nd choice A A B B 3rd choice B B D D 4th choice C D A C Suppose the Borda count method is used. Who is the winner? Is the majority criterion satisfied? Using one or more complete sentences, explain your answer. 1

2 4. The preference table for an election is shown. Number of Votes st choice A V V R 2nd choice R R A A 3rd choice V A R V Suppose the plurality method is used. Who is the winner? Is the head-to-head criterion satisfied? Using one or more complete sentences, explain your answer. 5. The preference table gives the results of a straw vote among three candidates A, B, and C. Number of Votes st choice C B A A 2nd choice A C B C 3rd choice B A C B (a) Suppose the plurality-with-elimination method is used. Who is the winner of the straw vote? (b) In the actual election, the four voters in the last column who voted ACB, in that order, change their votes to CAB. Using the plurality-with-elimination method, who wins the actual election? Is the monotonicity criterion satisfied? Using one or more complete sentences, explain your answer. 6. The preference table for an election is shown. Number of Votes st choice A B A B C C E B 2nd choice D A C A E B D A 3rd choice E D D C B A A E 4th choice C E E D D D C C 5th choice B C B E A E B D (a) Suppose the pairwise comparison method is used. Who is the winner? (b) Suppose that candidate C drops out, but the winner is still chosen using the pairwise comparison method. Who is the winner? Is the irrelevant alternatives criterion satisfied? Using one or more complete sentences, explain your answer. 2

3 7. A country is composed of four states, A, B, C, and D. The population of each state, in thousands, is given in the following table. State A B C D Population (in thousands) According to the country s constitution, the congress will have 40 seats, divided among the four states according to their respective populations. (a) Find the standard divisor, in thousands. How many people are there for each seat in congress? (b) Find each state s standard quota. (c) Find each state s lower quota. (d) Find each state s upper quota. (e) Use Hamilton s method to find each state s apportionment of congressional seats. (f) Use Jefferson s method with d = 48 to find each state s apportionment of congressional seats. (g) Use Adams s method with d = 52 to find each state s apportionment of congressional seats. (h) Use Webster s method with d = to find each state s apportionment of congressional seats. 3

4 8. A small country has 24 seats in the congress, divided among three states according to their respective populations. The table shows each state s population, in thousands, before and after the country s population increase. State A B C Total Original Population (in thousands) New Population (in thousands) (a) Use Hamilton s method to apportion the 24 congressional seats using the original population. (b) Find the percent increase, to the nearest tenth of a percent, in the population of each state. (c) Use Hamilton s method to apportion the 24 congressional seats using the new population. (d) What paradox occurs? Explain briefly. 4

5 9. The mathematics department has 30 teaching assistants to be divided among three courses, according to their respective enrollments. The table shows the courses and the number of students enrolled in each course. Course College Algebra Statistics Liberal Arts Math Total Enrollment (a) Apportion the teaching assistants using Hamilton s method. (b) Suppose the number of teaching assistants is increased from 30 to 31. Hamilton s method to reapportion the teaching assistants. Use (c) What paradox occurs? Explain briefly. 10. A corporation has two branches, A and B. Each year the company awards 100 promotions within its branches. The table shows the number of employees in each branch. Branch A B Total Employees ,000 (a) Use Hamilton s method to apportion the promotions. (b) Suppose that a third branch, C, with the number of employees shown in the table, is added to the corporation. Branch A B C Total Employees ,525 The company adds 5 new yearly promotions for branch C. Use Hamilton s method to reapportion the promotions. (c) What paradox occurs? Explain briefly. 5

6 11. You can purchase a particular model of a new car with a choice of ten colors, with or without automatic transmission, with or without four-wheel drive, with or without air conditioning, and with two, three, or four radio-cd speakers. In how many different ways can you order the car? 12. An ice cream store sells two drinks (sodas or milk shakes), in three sizes (small, medium, or large), and two flavors (vanilla or chocolate). In how many ways can a customer order a drink? 13. You are taking a multiple-choice test that has five questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions? 14. License plates in a particular state display two letters followed by three numbers, such as AT-887 or BB-013. How many different license plates can be manufactured for this state? 6

7 Voting Method Plurality Method Bourda Count Method Voters rank all candidates from the most favorable to the least favorable. Each last-place vote receives 1 point, each next-to-last-place vote 2 points, and so on. The candidate with the most points is the winner. Plurality-with- Elimination Method Pairwise Comparison Method How the Winning Candidate Is Determined The candidate with the most first-place votes is the winner. The candidate with the majority (over 50%) of first-place votes is the winner. If no candidate receives a majority, eliminate the candidate with the fewest first-place votes. Either hold another election or adjust the preference table. Continue this process until a candidate receives a majority of first-place votes. That candidate is the winner. Voters rank all the candidates. A series of comparisons is made in which each candidate is compared to each of the other candidates. The preferred candidate in each comparison receives 1 point; in case of a tie, each receives 1 point. The candidate with the most points 2 is the winner. Fairness Criterion Majority Criterion Head-to-Head Criterion Monotonicity Criterion Irrelevant Alternatives Criterion Description If a candidate receives a majority of first-place votes in an election, then that candidate should win the election. If a candidate is favored when compared head-to-head with every other candidate, then that candidate should win the election. If a candidate wins an election and, in a reelection, the only changes are changes that favor the candidate, then that candidate should win the reelection. If a candidate wins an election and, in a recount, the only changes are that one or more of the other candidates are removed from the ballot, then that candidate should still win the election. 7

8 Method Divisor Apportionment Hamilton s Standard divisor = total population total number of seats Jefferson s The modified divisor is less than the standard divisor. Adams s Webster s The modified divisor is greater than the standard divisor. The modified divisor may be less than, greater than, or equal to the standard divisor. Round each standard quota down to the nearest whole number. Initially give each group its lower quota. Give surplus items, one at a time, to the groups with the largest decimal parts. Round each group s modified quota down to the nearest whole number. Apportion to each group its modified lower quota. Round each group s modified quota up to the nearest whole number. Apportion to each group its modified upper quota. Round each group s modified quota to the nearest whole number. Apportion to each group its modified rounded quota. Paradox Alabama Paradox Population Paradox New-States Paradox Description An increase in the total number of seats to be apportioned results in the loss of a seat for a state. State A loses seats to State B, even though the population of State A grew at a faster rate than that of State B. A new state is added along with its fair share of seats, but this results in an old state losing a seat to another old state. 8

9 MTH 110 Quiz 2 Review Answers Spring AB, AC, AD, AE, BC, BD, BE, CD, CE, DE 3. A. No, because C got a majority of first place votes but did not win. 4. V. No, because A is favored when compared head-to-head with every other candidate but does not win. 5. (a) C (b) B. No. C won the straw poll, then the only changes were changes that favored C, but then C lost. 6. (a) A (b) B. No. A won, and then in a recount the only change was that another candidate was removed from the ballot, but then A lost. 7. (a) (50050 people), (b) 5.49, 7.83, 12.21, (c) 5, 7, 12, 14 (d) 6, 8, 13, 15 (e) 6, 8, 12, 14 (f) 5, 8, 12, 15 (g) 6, 8, 12, 14 (h) 6, 8, 12, (a) 4, 6, 14 (b) 28.3%, 26.3%, 14.7% (c) 3, 7, 14 (d) The population paradox occurs, because A has a faster growing population than B, but A loses a seat and B gains a seat. 9. (a) 16, 8, 6 (b) 17, 9, 5 (c) The Alabama paradox occurs, because when the total number of TA s is increased, Liberal Arts Math loses a TA. 10. (a) 10, 90 (b) 11, 89, 5 (c) The new states paradox occurs, because branch C is added and enough new promotions are added for C, but then B loses a promotion to A ,000