Top two seeds meeting in a king-of-the-hill final

Fifty players numbered $1$ to $50$ share a single table; whenever two players meet, the higher number always wins. The first match is between two players chosen at random. After each match the loser is eliminated and the winner plays a new opponent drawn at random from those who have not yet played, until only one player is left ($49$ matches). What is the probability that the last match is between players $49$ and $50$?

$1/25$. The order in which players enter is a uniformly random permutation. Once $49$ or $50$ is in, they keep winning, so $49$ and $50$ meet in the final exactly when one of them is the very last player to enter. Each has probability $\tfrac1{50}$ of being last, giving $\tfrac2{50}=\tfrac1{25}$.