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Before a knockout tournament with 2^n players I pick two players. What is the probability that they have to play against each other at some point in the tournament?

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If the score is 8-8 do I have more chance of winning if the winner is the first to reach 9 points or the first to reach 10 points?

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A player has probability 0.4 of winning a single game. What is his probability of winning a 'best of 15 games' tournament?

Voting Paradox

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

It is an exercise in simple probability and combinatorics that provides an intriguing and paradoxical situation for investigation.

Possible approach

The class could name 3 candidates to rank in order. Then everyone could write down their order of choice. You could then take 3 at a time and the class could discuss whether those three are transitive or not. After discussing several sets of 3 rankings they should be able to make conjectures about when the set will be transitive and when it will be intransitive.

Key question

How many possible sets of choice can be made in total by the voters?
How many of these sets are intransitive?

Possible support

See the problems A Dicey Paradox and Winning Team and the article Transitivity.