The recent paper "Positroids, knots, and $q,t$-Catalan numbers" by Galashin and Lam (https://arxiv.org/abs/2012.09745) gives geometric meaning to the $q$-Catalan numbers (both kinds of $q$-Catalan numbers, in fact!).

Check out Corollary 1.3 for a specific probabilistic statement in line with your inquiry.

We have the following elegant but baffling corollary. $$ $$ Corollary 1.3. The probability that a uniformly random $k$-dimensional subspace of $\mathbb{F}_q^n$ belongs to $\Pi_{k,n}^{\circ}(\mathbb{F}_q)$ is given by $$\mathrm{Prob}(V \in \Pi_{k,n}^{\circ}(\mathbb{F}_q)) =\frac{(q-1)^n}{q^n-1}$$ The probability $\frac{(q-1)^n}{q^n-1}$ does not depend on $k$. We do not have a direct explanation for this phenomenon.

Here $\Pi_{k,n}^{\circ}(\mathbb{F})$ is the open positroid variety inside the Grassmannian $\mathrm{Gr}_{k,n}(\mathbb{F})$.

The recent paper "Positroids, knots, and $q,t$-Catalan numbers" by Galashin and Lam (https://arxiv.org/abs/2012.09745) gives geometric meaning to the $q$-Catalan numbers (both kinds of $q$-Catalan numbers, in fact!).

Check out Corollary 1.3 for a specific probabilistic statement in line with your inquiry.

We have the following elegant but baffling corollary. 

Corollary 1.3. The probability that a uniformly random $k$-dimensional subspace of $\mathbb{F}_q^n$ belongs to $\Pi_{k,n}^{\circ}(\mathbb{F}_q)$ is given by $$\operatorname{Prob}(V \in \Pi_{k,n}^{\circ}(\mathbb{F}_q)) =\frac{(q-1)^n}{q^n-1}.$$ The probability $\frac{(q-1)^n}{q^n-1}$ does not depend on $k$. We do not have a direct explanation for this phenomenon.

Here $\Pi_{k,n}^{\circ}(\mathbb{F})$ is the open positroid variety inside the Grassmannian $\operatorname{Gr}_{k,n}(\mathbb{F})$.