So I was thinking about a local football tournament that is much like the English FA Cup. It's a knockout tournament combining teams from leagues of varying standards. To manufacture a bit of a "fairy tale" story, the draw has been rigged so that a team from a division lower than the top is always guaranteed a place in the last 4 (semi finals).
The question I was pondering, if the draw was made to be completely random (i.e. all teams are selected from the same pot) and we made the assumption that the teams from the top league always won. How would you go about calculating the probability that a team from the lower leagues make the semi final?
There are 10 teams in the top league and currently they join the competition at the Round of 32 (so 22 lower league teams). If they joined at the Round of 64 (10 top league and 54 lower league teams) would the chances significantly improve?
Note in the FA Cup a draw is made at the end of each round to decide the following round's matches so there is not a tournament tree like some tournaments.
A team from the lower leagues reaches the semi-final if and only if at least one of the four branches contains no team from the top league. Let $n$ be the number of teams per branch, where $n=8$ if the teams from the top league enter at the Round of $32$ and $n=16$ if they enter at the Round of $64$. Let $A_k$ be the event that branch $k$ contains no team from the top league. Then by inclusion-exclusion the probability for at least one of the four branches to contain no team from the top league is
\begin{align} P\left(\bigcup_{k=1}^4 A_k\right)&=\sum_{i=1}^4\binom4i(-1)^{i+1}P\left(\bigcap_{k=1}^iA_k\right)\\ &=\sum_{i=1}^4\binom4i(-1)^{i+1}\frac{\binom{(4-i)n}{10}}{\binom{4n}{10}}\\ &=\frac1{\binom{4n}{10}}\sum_{i=1}^4\binom4i(-1)^{i+1}\binom{(4-i)n}{10}\;.\\ \end{align}
For $n=8$, this is
$$ \frac1{\binom{32}{10}}\left(\binom41\binom{24}{10}-\binom42\binom{16}{10}\right)=\frac{162437}{1344005}\approx12\%\;, $$
and for $n=16$ it's
$$ \frac1{\binom{64}{10}}\left(\binom41\binom{48}{10}-\binom42\binom{32}{10}+\binom43\binom{16}{10}\right)=\frac{805494443}{4733537963}\approx17\%\;, $$
so the probability does increase a bit if the teams from the top league join at the Round of $64$.