May 19, 2010Math 132: Foundations of Mathematics 12.5 Homework Solutions 27. (a) 28. (b) 29. (d) 30. (e) 53. Positive Correlation, Weak 54. Negative Correlation, Moderate 55. No Correlation 56. Negative Correlation, Weak 57.The sign of the correlation shows whether the correlation is pos/neg; the closer to 1, the stronger the correlation.

May 19, 2010Math 132: Foundations of Mathematics Amy Lewis Math Specialist IU1 Center for STEM Education

May 19, 2010Math 132: Foundations of Mathematics 14.1 Voting Methods Understand and use preference tables. Use the following methods to determine an election’s winner: –Plurality –Borda count –Plurality-with-elimination –Pairwise comparison

Preference Tables Preference ballots: ballots in which a voter is asked to rank all of the candidates in order of preference. Preference table: a table that shows how often each particular outcome occurred. Refer to the preference table on page 773. May 19, 2010Math 132: Foundations of Mathematics

Preference Tables Preference Table for the Election of Student Body President Number of Votes First ChoiceSASB Second ChoiceASAS Third ChoiceBBCA Fourth ChoiceCCBC May 19, 2010Math 132: Foundations of Mathematics How many students voted in the election? How many students selected the candidates in this order: B, S, A, C? How many students selected Samir (S) as their first choice for student body president?

Popular Voting Methods The plurality method The Borda count method The plurality-with-elimination method The pairwise comparison method May 19, 2010Math 132: Foundations of Mathematics

The Plurality Method The candidate (or candidates, if there is more than one) with the most first-place votes is the winner. A plurality occurs when no single candidate receives a majority of first-place votes (more than 50% of the votes). May 19, 2010Math 132: Foundations of Mathematics

The Plurality Method Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD May 19, 2010Math 132: Foundations of Mathematics Who is declared the winner using the plurality method? Four candidates are running for mayor of Smallville: Antonio (A), Bob (B), Carmen (C), and Donna (D). The voters were asked to rank all the candidates in order of preference.

The Borda Count Method Developed by the French mathematical and naval captain Jean-Charles de Borda. Assigns points to each candidate based on how they are ranked by the voters: –Last-place: 1 pt. –Second-to-last-place: 2 pts. –Third-from-last-place: 3 pts. –Etc. The points are totaled for each candidate separately. The candidate with the most points is the winner. May 19, 2010Math 132: Foundations of Mathematics

The Borda Count Method Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD May 19, 2010Math 132: Foundations of Mathematics Who is declared the winner using the Borda Count method?

The Borda Count Method Preference Table for the Smallville Mayoral Election Number of Votes First Choice: 4 pts. A: 130*4 = 520 D: 120*4 = 480 D: 100*4 = 400 C: 150*4 = 600 Second Choice: 3 pts. B: 130*3 = 390 B: 120*3 = 360 B: 100*3 = 300 B: 150*3 = 450 Third Choice: 2 pts. C: 130*2 = 260 C: 120*2 = 240 A: 100*2 = 200 A: 150*2 = 300 Fourth Choice: 1 pt. D: 130*1 = 130 A: 120*1 = 120 C: 100*1 = 100 D: 150*1 = 150 May 19, 2010Math 132: Foundations of Mathematics A gets = 1140 points B gets = 1500 points C gets = 1200 points D gets = 1160 points Bob wins!

The Plurality-with-Elimination Method The candidate with the majority of first-place votes wins. –If no candidate receives a majority of first-place votes, eliminate the candidate with the fewest first-place votes. Move the candidates in each column below the eliminated candidate up one place. –The candidate with the majority of first-place votes in the new preference table wins. –Repeat the process until a candidate receives a majority. May 19, 2010Math 132: Foundations of Mathematics

The Plurality-with-Elimination Method Does any candidate have the majority? Who do we eliminate? What does the new preference table look like? May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD

The Plurality-with-Elimination Method Does any candidate have the majority now? Who do we eliminate? What does the new preference table look like? May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceBDDC Second ChoiceCBBB Third ChoiceDCCD

The Plurality-with-Elimination Method Does any candidate have the majority now? Who wins? Carmen wins! May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceCDDC Second ChoiceDCCD

Pairwise Comparison Method The preference table is used to make a series of comparisons in which each candidate is compared to each of the other candidates. For each pair of candidates, X and Y, use the table to determine how many voters prefer X to Y and vice versa. If a majority prefer X to Y, then X receives 1 point. If a majority prefer Y to X, then Y receives 1 point. If the candidates tie, then each receives ½ point. After all comparisons have been made, the candidate receiving the most points is the winner. May 19, 2010Math 132: Foundations of Mathematics

Pairwise Comparison Method How many comparisons do we need to make? –Antonio vs. Bob –Antonio vs. Carmen –Antonio vs. Donna –Bob vs. Carmen –Bob vs. Donna –Carmen vs. Donna May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD

Pairwise Comparison Method Bob gets 1 point. May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Antonio vs. Bob

Pairwise Comparison Method Antonio gets ½ pt. Carmen gets ½ pt. May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Antonio vs. Carmen

Pairwise Comparison Method Antonio gets ½ pt. Donna gets ½ pt. May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Antonio vs. Donna

Pairwise Comparison Method Bob gets 1 point May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Bob vs. Carmen

Pairwise Comparison Method Bob gets ½ pt. Donna gets ½ pt. May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Bob vs. Donna

Pairwise Comparison Method Carmen gets ½ pt. Donna gets ½ pt. May 19, 2010Math 132: Foundations of Mathematics Preference Table for the Smallville Mayoral Election Number of Votes First ChoiceADDC Second ChoiceBBBB Third ChoiceCCAA Fourth ChoiceDACD Carmen vs. Donna

Pairwise Comparison Method Who wins? –Antonio: 1 point –Bob: 2½ points –Carmen: 1½ points –Donna: 1 point Bob wins! Again! May 19, 2010Math 132: Foundations of Mathematics

Who were our winners? Plurality: Donna Borda count: Bob Plurality-with-elimination: Carmen Pairwise comparison: Bob Who should be mayor of Smallville?!? May 19, 2010Math 132: Foundations of Mathematics

May 19, 2010Math 132: Foundations of Mathematics Homework Page 782: #7 Apply all 4 voting methods to determine the kind of play the theater society will perform next semester. Next Session: Thursday, May 20