  We prove $\chi_s'(G)\leq 1.93 \Delta(G)^2$ for graphs of sufficiently large
maximum degree where $\chi_s'(G)$ is the strong chromatic index of $G$. This
improves an old bound of Molloy and Reed. As a by-product, we present a
Talagrand-type inequality where it is allowed to exclude unlikely bad outcomes
that would otherwise render the inequality unusable.
