1. The 'cup' system for determining the champion amongst 2^n players consists of drawing lots to arrange the players in 2^n-1 pairs who are to play each other, then repeating this with the 2^n-1 winners of these matches, and so on. The winner and loser of the final match recieve the first and second prizes repectively.
Suppose that each of the players has a defined strength (all different), and the strongest player wins each match. then is it clear that the first prize goes to the strongest of all the players. what is the probability that the second prize goes to the second strongest player?
In the case n=4, what is the probability that the four strongest players all reach the semi-finals ?