Let $B_{n}$ denote the $n$-strand braid group. Let $B_{[m,n]}$ denote the subgroup of $B_{n}$ generated by the strands $\sigma_{m},...,\sigma_{n-1}$. If $b$ is a braid, then let $|b|$ denote the length of the shortest braid word that represents $b$.

Suppose now that $n,r,s$ are natural numbers with $1<s<r<n$. If $b\in B_{n}$ is a braid chosen at random, then what are some bounds for the expected value of $\textrm{Min}\{|abc|:a\in B_{r},c\in B_{[s,n]}\}$. I am especially interested in the cases where $s$ and $n-r$ are small compared to $n$. I am especially interested in the case when $s=2$. I will let the answerer choose the distributions to use. Heuristic and experimental answers are welcome.