You win the final game of a tournament bracket iff you are the highest ranked player in that bracket. For example, in a tournament with ~8~ people, you win your first game iff you are the better player in your ~2~-player bracket, your second game iff you are the best player in your ~4~-player bracket, and your third game iff you are the best player among all ~8~ people. If we can find the probability that you win your ~i^\text{th}~ game using this, then we can use the linearity of expectation to sum these probabilities up to get our final answer.

So what is the probability that you win your ~i^\text{th}~ game? Well, there are ~2^k~ players in all, ~2^i~ players in this particular bracket, and ~r-1~ players better than you (or equivalently, ~2^k-r~ players worse than you). You win in the event that of the ~2^i-1~ other players in your bracket, all of them are chosen from the ~2^k-r~ players that are worse than you. Since all brackets are equally likely, this happens with probability:

$$\displaystyle \binom{2^k}{2^i-1} / \binom{2^k-r}{2^i-1}$$

It's dangerous to compute these two values directly and then divide them, since these are both very big numbers. We have two options for dealing with this:

1. We can work with logarithms of factorials instead of factorials directly.
2. We can do some simplification of the expression by cancelling common terms.