12.5 homework solutions

Math 132: Foundations of MathematicsMay 19, 2010

12.5 Homework Solutions27. (a)28. (b)29. (d)30. (e)53. Positive Correlation, Weak54. Negative Correlation, Moderate55. No Correlation56. Negative Correlation, Weak57. The sign of the correlation shows whether the

correlation is pos/neg; the closer to 1, the stronger the correlation.


Math 132:Foundations of Mathematics

Amy LewisMath Specialist

IU1 Center for STEM Education


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

Math 132: Foundations of Mathematics

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, 2010


Preference TablesPreference Table for the Election of Student Body President

Number of Votes 2100 1305 765 40

First Choice S A S B

Second Choice A S A S

Third Choice B B C A

Fourth Choice C C B C

May 19, 2010

• 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, 2010


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, 2010


The Plurality Method

Preference Table for the Smallville Mayoral Election


First Choice A D D C

Second Choice B B B B

Third Choice C C A A

Fourth Choice D A C D

May 19, 2010

• 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, 2010


The Borda Count MethodPreference Table for the Smallville Mayoral Election






May 19, 2010

• Who is declared the winner using the Borda Count method?


The Borda Count MethodPreference Table for the Smallville Mayoral Election


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, 2010

• A gets 520 + 120 + 200 + 300 = 1140 points• B gets 390 + 360 + 300 + 450 = 1500 points• C gets 260 + 240 + 100 + 600 = 1200 points• D gets 130 + 480 + 400 + 150 = 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, 2010



• Does any candidate have the majority?• Who do we eliminate?• What does the new preference table look like?

May 19, 2010









• Does any candidate have the majority now?• Who do we eliminate?• What does the new preference table look like?

May 19, 2010



First Choice B D D C

Second Choice C B B B

Third Choice D C C D



• Does any candidate have the majority now?• Who wins?• Carmen wins!May 19, 2010



First Choice C D D C

Second Choice D C C D


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, 2010


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, 2010









• Bob gets 1 point.

May 19, 2010







Antonio vs. Bob



• Antonio gets ½ pt.• Carmen gets ½ pt.

May 19, 2010







Antonio vs. Carmen



• Antonio gets ½ pt.• Donna gets ½ pt.

May 19, 2010







Antonio vs. Donna



• Bob gets 1 point

May 19, 2010







Bob vs. Carmen



• Bob gets ½ pt.• Donna gets ½ pt.

May 19, 2010







Bob vs. Donna



• Carmen gets ½ pt.• Donna gets ½ pt.

May 19, 2010







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, 2010


Who were our winners?

• Plurality: Donna• Borda count: Bob• Plurality-with-elimination: Carmen• Pairwise comparison: Bob• Who should be mayor of Smallville?!?

May 19, 2010


Homework

Page 782: #7Apply all 4 voting methods to determine the kind of play the theater society will perform next semester.

Next Session: Thursday, May 20

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