A table-tennis championship for $2^n$ players has $n$ rounds and is organized as a knock-out tournament, with players randomly matched together at the start. The last round is the final.

Before the tournament starts, I decide to pick two players to support at random from the list of entrants. What is the probability that they have to play against each other at some point in the tournament?

You may also be interested to try the more challenging problem FA Cup.