Top seed meeting the third-best player in the final

In a single-elimination tournament, $128$ players with all-different ratings are placed into a bracket uniformly at random, and every game is won by the higher-rated player. The event runs for $7$ rounds, the last being the final. What is the probability that the final is played between the best player and the third-best player?

$\tfrac{32}{127}$. The final is between the two half-bracket winners, so we need the second-best player in the top player's half of 64 and the third-best in the other half. That happens with probability $\tfrac{63}{127}\cdot\tfrac{64}{126}=\tfrac{32}{127}$.