Consider a set $D$ of $n$ uniformly random binary strings of length $m$.
Fix $l \leq m$.
We now sample an $m$-bit string uniformly at random from $\{0,1\}^m$. Let $\alpha$ be the $l$-bit prefix of this string. I'm interested in the probability that $\alpha$ is the longest common prefix between two strings in $D$. How can this be calculated? Is there a closed-form formula?