### Knock-out

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?

### Squash

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?

### Snooker

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

### 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.