Knockout bracket where the two best teams collide last

A single-elimination tournament has $2^n$ teams placed into the bracket uniformly at random. The stronger team always wins each match. What is the probability that the two strongest teams play each other in the final?

$\dfrac{2^{n-1}}{2^n-1}$. They meet in the final exactly when they start in opposite halves of the bracket. Given the best team's slot, $2^{n-1}$ of the other $2^n-1$ slots lie in the opposite half.