Chess Tournament

A knockout chess tournament has $K$ levels and $2^K$ players, numbered so that player $1$ is the strongest, player $2$ the second strongest, and so on down to player $2^K$. Before the first level, the players are assigned the slots $1$ to $2^K$ uniformly at random, and this bracket is then fixed: at the first level, players in adjacent slots play each other (slot 1 with slot 2, slot 3 with slot 4, and so on), and at each later level the winners of adjacent matches play each other. When two opponents play, the stronger player always wins. If the tournament has $8$ players, what is the probability that players $1$ and $2$ meet in the final level?