Math 130 Final Exam Study Guide. 1. Voting

Transcription

1 1 Math 130 Final Exam Study Guide 1. Voting (a) Be able to interpret a top choice ballot, preference ballot and preference schedule (b) Given a preference schedule, be able to: i. find the winner of an election using the A. plurality method B. borda count method C. plurality-with-elimination method D. method of pairwise comparisons ii. find the ranking of the candidates using the A. extended plurality method B. extended borda count method C. extended plurality-with-elimination method D. extended method of pairwise comparisons iii. find the ranking of the candidates using the A. recursive plurality method B. recursive borda count method C. recursive plurality-with-elimination method (c) Be able to recognize the following criteria, and identify any violations: i. The Majority Criterion ii. The Condorcet Criterion iii. The Independence of Irrelevant Alternatives Criterion iv. The Monotonicity Criterion (d) Be able to evaluate an election for fairness based on the four fairness criteria 2. Weighted Voting (a) Understand the notation, vocabulary, and definitions for weighted voting: i. Quota ii. Dictator iii. Dummy iv. Veto Power v. Coalitions and sequential coalitions A. how many coalitions B. coalition weights C. winning and losing coalitions vi. Be able to identify critical and pivotal players vii. Be able to compute the power distribution using the Banzhaf definition of power viii. Be able to compute the power distribution using the Shapley-Shubik definition of power

2 2 3. Fair Division (a) Method of Markers i. Be able to identify fair shares for any player ii. Be able to identify shares that would not be fair iii. Be able to use the Method of Markers to determine a fair allocation for reach player iv. Be able to identify left-over items v. Understand why this method results in a fair division (b) Method of Sealed Bids i. Be able to identify which person wins each item ii. Be able to calculate fair shares for each person iii. Be able to calculate the amounts paid to and from the estate iv. Be able to calculate the surplus v. Be able to calculate the end result and describe what each person receives and/or pays 4. Apportionment (a) Be able to calculate and interpret the standard divisor (b) Calculate the standard quotas, lower quotas and upper quotas (c) Calculate the relative fractions (d) Calculate the number of surplus seats if the lower quotas are allocated (e) Calculate the number of surplus seats if the upper quotas are allocated (f) Find the final apportionment using the following methods: i. Hamilton s Method ii. Lowndes Method (relative fractions) iii. Jefferson s Method (modified divisor) iv. Adams s Method (modified divisor) v. Webster s Method (modified divisor) vi. Understand the paradoxes and when a violation has occurred A. Alabama Paradox B. Population Paradox C. New States Paradox vii. Understand the Quota Rule and when a violation has occurred viii. For each allocation method, know which paradox may apply and if the method can violate the quota rule

3 3 5. Graphs (Euler) (a) Understand the following terms, and be able to identify them on a graph: i. Graph ii. Vertex iii. Edge iv. Loop v. Degree vi. Bridge vii. Connected/disconnected viii. Path ix. Circuit x. Euler Path xi. Euler Circuit (b) Be able to write out paths and circuits using vertex names (c) Be able to draw a graph to model a given scenario (d) Be able to tell if a graph has an Euler Circuit or an Euler Path or neither (e) Use Fleury s algorithm to eulerize a graph (f) Understand Euler s Theorems (g) Apply Euler s Theorems to answer questions about a given scenario (h) Be able to tell how many deadhead edges were added to a graph and how many total edges there are after eulerization 6. Graphs (Hamilton) i. definition of a complete graph on n vertices ii. definition of a Hamilton Circuit iii. relationships between the number of edges, vertices and the degree of each vertex iv. efficient, inefficient, optimal and approximate algorithms v. no algorithm is yet known that is both efficient and optimal (b) Be able to name Hamilton Circuits using the vertex names (c) Be able to calculate the weight of a Hamilton Circuit (d) Be able to calculate the total number of Hamilton Circuits on a graph with n vertices (e) Be able to apply the following algorithms, and be able to interpret the results and answer questions about a given scenario i. Brute Force Algorithm ii. Nearest-Neighbor Algorithm iii. Repetitive Nearest-Neighbor Algorithm iv. Cheapest-Link Algorithm

4 4 7. Trees i. definition of tree ii. minimum spanning tree iii. number of edges in a minimum spanning tree with n vertices iv. Kruskal s Algorithm (b) Be able to apply Kruskal s Algorithm to find the minimum spanning tree given a graph (c) Calculate the weight of the minimum spanning tree (d) Be able to use the results of Kruskal s Algorithm to answer questions about a given scenario 8. Scheduling (a) Understand the following i. digraph ii. priority list iii. precedence relations iv. task v. processor vi. Decreasing Time Algorithm vii. critical time viii. Backflow Algorithm ix. critical path x. Critical Time Algorithm xi. scheduling (b) Be able to create a digraph modeling a given situation (c) Be able to interpret a digraph (d) Be able to find a priority list using the Decreasing Time Algorithm (e) Be able to list the critical times for each task (Backflow Algorithm) (f) Be able to identify critical paths/interpret critical paths (g) Be able to find a priority list using the Critical Time Algorithm (h) Be able to schedule tasks on two and three processors given a digraph and a priority list

5 5 9. Descriptive Statistics (a) Be able to compute markups and markdowns (b) Understand variables, continuous vs. discrete and numerical vs. categorical (c) Interpret frequency tables, bar graphs and pie charts (d) Be able to compute mean, median, quartiles from frequency tables, bar graphs and pie charts (e) Mean- be able to define, compute and interpret (f) Median- be able to define (pg 537 in the text), compute and interpret (g) Understand outliers (h) Standard deviation-be able to define, compute and interpret (i) Percentiles- be able to define, compute and interpret (j) Understand and compute quartiles (k) Be able to compute the Five Number Summary (l) Be able to create and interpret a Box and Whisker Plot (Box Plot) 10. Probability i. counting number of outcomes- The Multiplication Rule ii. Calculating probability, including PR(E), Pr(E or F), Pr(E and F), and Pr(not E) iii. Odds iv. Odds to probability conversion v. Probability to odds conversion 11. Normal Distributions (a) Understand the relationship between mean and median when the data are normally distributed (b) Understand mean and standard deviation (c) Know the Empirical Rule (d) Given the mean and standard deviation, be able to understand and interpret a graph of normally distributed data (e) Be able to identify the median, mean and standard deviation from a graph of normally distributed data (f) Be able to answer questions about normally distributed data 12. Weighted Averages and Expected Values (a) Be able to compute and interpret a weighted average given a written description (b) Be able to compute and interpret the expected value