Probability of this event 
Let $S$ be a set of 27 pairwise different numbers. Let $T$ be a set of 9 elements chosen from $S$ uniformly at random (no repetitions). Let $u$ be an element sampled uniformly at random from $T$. Let $S'= \{s_1, s_2, \dots, s_{27}\}$ be the sequence of elements in $S$ but in increasing order. Given that $u$ is in the middle third of $T$, what is the probability that the $i$-th element from $S'$ was selected.

This is my attempt. On the one hand, I’m not sure if $\mathbb P(u = s_i) = 1/21$, where $i = 4, 5, \dots, 24$, and 0 otherwise. On the other hand, I don’t know if the answer should be $$\mathbb P( u = s_i) = {27\choose 9}^{-1}\times\sum^{8}_{j=0} {{i-1}\choose{9+j}}\cdot{{27-i}\choose{18-j-1}},$$ where $i = 4, \dots, 24$ and 0 otherwise.
 A: Without loss of generality, assume $S=\{1,...,27\}$.

Let $T$ be a random sample from $S$ with $|T|=9$, and let $u$ be a randomly selected element of $T$.

Write $T=\{t_1,...,t_9\}$, where $t_1 < \cdots < t_9$.

For arbitrary $i\in S$, our goal is to compute $P\Bigl(u=i{\,{\large{\mid}}\,}u\in \{t_4,t_5,t_6\}\Bigr)$.

Then we get
\begin{align*}
&
P\Bigl(\bigl(u=i\bigr){\,{\large{\land}}\,}\bigl(u\in \{t_4,t_5,t_6\}\bigr)\Bigr)
\\[4pt]
=\;&
\sum_{k=4}^6 P\Bigl(\bigl(u=i\bigr){\,{\large{\land}}\,}\bigl(u=t_k\bigr)\Bigr)
\\[4pt]
=\;&
\sum_{k=4}^6
P\Bigl(u=i{\,{\large{\mid}}\,}u=t_k\Bigr)
P\bigl(u=t_k\bigr)
\\[4pt]
=\;&
{\small{\frac{1}{9}}}
\sum_{k=4}^6
P\Bigl(u=i{\,{\large{\mid}}\,}u=t_k\Bigr)
\\[4pt]
=\;&
{\small{\frac{1}{9}}}
\sum_{k=4}^6
\frac
{
{\large{
{i-1\choose 3+(k-4)}
{27-i\choose 3+(6-k)}
}}
}
{
{\large{
{27\choose 9}
}}
}
\\[4pt]
=\;&
\frac
{1}
{
{\small{9}}
{27\choose 9}
}
\sum_{k=4}^6
{\small{
{i-1\choose 3+(k-4)}
{27-i\choose 3+(6-k)}
}}
\\[4pt]
\end{align*}
hence we get
\begin{align*}
&
P\Bigl(u=i{\,{\large{\mid}}\,}u\in \{t_4,t_5,t_6\}\Bigr)
\\[4pt]
=\;&
\frac
{
P\Bigl(\bigl(u=i\bigr){\,{\large{\land}}\,}\bigl(u\in \{t_4,t_5,t_6\}\bigr)\Bigr)
}
{
P\Bigl(u\in \{t_4,t_5,t_6\}\Bigr)
}
\\[4pt]
=\;&
\frac
{
P\Bigl(\bigl(u=i\bigr){\,{\large{\land}}\,}\bigl(u\in \{t_4,t_5,t_6\}\bigr)\Bigr)
}
{
\Bigl(
{\small{\frac{1}{3}}}
\Bigr)
}
\\[4pt]
=\;&
\frac
{1}
{
{\small{3}}
{27\choose 9}
}
\sum_{k=4}^6
{\small{
{i-1\choose 3+(k-4)}
{27-i\choose 3+(6-k)}
}}
\\[4pt]
\end{align*}
For example, for $i=12$ we get
\begin{align*}
&
P\Bigl(u=12{\,{\large{\mid}}\,}u\in \{t_4,t_5,t_6\}\Bigr)
\\[4pt]
=\;&
\frac
{
{\large{
{11\choose 3}
{15\choose 5}
+
{11\choose 4}
{15\choose 4}
+
{11\choose 5}
{15\choose 3}
}}
}
{
3
{\large{
{27\choose 9}
}}
}
\\[4pt]
=\;&
\frac{539}{6555}
\approx
0.08222730740
\\[4pt]
\end{align*}
