Question
A chess tournament has $2^n$ players, each with a distinct rating. Assume that the player with the higher rating always wins against a lower rated opponent with probability $0.5 < p < 1$. The winner proceeds to the subsequent round. Since the tournament's structure resembles that of a knockout bracket, $n$ total rounds are played, including the final. The probability that the second-highest rated player defeats the highest rated player in the final round can be expressed as a function of $n$ and $p$. Determine this function evaluated at $n = 4, p = \frac{3}{4}$.