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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

Age 14 to 18 Challenge Level:

How many different possible ways can each of the three voters vote?
For a paradox we are looking for situations in which

(A beats B beats C beats A) or (B beats A beats C beats B) etc.

Try counting the number of ways in which such situations are possible. To reduce the number of cases to analyse, you might consider what happens if two voters agree on their first choice.