Question
$ 50$ people are competing in a ping pong tournament where there is only one ping pong table. The competitors are numbered $1$ through $50$. Suppose that if two competitors meet, the one with the larger number wins. Two competitors are chosen at random, and the loser is removed from the tournament. The winner moves on to the next round, where their opponent is chosen at random. This process is repeated until one person is left (a total $49$ rounds will be played). Compute the probability that competitor $49$ is still in the tournament after the first $10$ rounds have been played.