Q:

Probability that either $S_1$ or $S_2$ will reach semi-final.

$2^n$ players of equal strength are playing a knock-out tournament, where for each match players are paired randomly. Given two players $S_1$ and $S_2$, what is the probability of exactly one of them reaching the semi-finals?

For there to be semi-finals, we obviously need to assume $n\geq2$. 
If $n=3$, then there are $8$ players; in this case the $2$ particular players can't face each other in the first round.
How can I comprehend the cases for general $n$?

A:

Since every possibility is equally likely, forget all the matches and just choose $4$ players for the semifinals.
That makes the probability $$\dfrac{\displaystyle{2\choose 1}{2^n-2 \choose 3}}{\displaystyle{2^n \choose 4}}$$

