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Voting in a democratic society does not always yield conclusive results. Consider a club with 55 voting members. Five of the members (Alice, Bill, Carol, Don, and Herkimer) decide to run for club president. Members are asked to rank the five candidates from 1 to 5, with 1 being the first choice, 2 being the second choice, and so on. The results of the ranking are shown below.

OK, who wins the election if

1234518 members

ALICE

BILL

CAROL

DON

HERKIMER

12 members

HERKIMER

CAROL

BILL

DON

ALICE

10 members

DON

HERKIMER

CAROL

BILL

ALICE

9 members

BILL

DON

CAROL

HERKIMER

ALICE

4 members

CAROL

HERKIMER

BILL

DON

ALICE

2 members

CAROL

DON

BILL

HERKIMER

ALICE

a) The winner is the candidate with the most first-place votes?

b) There is a run-off between the two candidates receiving the most first-place votes?

c) Five points are given for a first-place ranking, four points for a second-place ranking, three points for a third-place ranking, two points for a fourth-place ranking, and one point for a fifth-place ranking?

d) The winner is the person who beats each candidate in a two-person contest?

If you take the time to answer the questions, you'll see that the choice of voting process is not a trivial decision.

=========

Here's another voting situation. Suppose there are three candidates, Alice, Bill, and Carol.

PREMISE: It is known that 2/3 of the voters prefer Alice to Bill, and 2/3 of the voters prefer Bill to Carol.

QUESTION: Can we conclude that the voters prefer Alice to Carol?

If you think the answer is YES, look at the table below. The PREMISE is true (check it out).

Preference--->

1231/3 ALICE

BILL

CAROL

1/3 BILL

CAROL

ALICE

1/3 CAROL

ALICE

BILL

What portion of the voters prefer Carol to Alice?

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**Related Readings:**

Davis, Kenneth C. *Don't Know Much About History*. New York:
Avon Books, 1990.

Paulos, John Allen. *A Mathematician Reads the Newspaper*.
New York: Anchor Books, 1995.

Paulos, John Allen. * Beyond Numeracy: Ruminations of a Numbers
Man*. New York: Alfred A. Knopf, 1991.

Smith, Sanderson. *Agnesi to Zeno: Over 100 Vignettes from the
History of Math*. Berkeley, CA: Key Curriculum Press, 1996.

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