  In this work we consider semi-classical Schr\"odinger operators with
potentials supported in a bounded strictly convex subset ${\cal O}$ of ${\bf
R}^n$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we
consider certain classes of small random perturbations and show that with
probability very close to 1, the number of resonances in rectangles
$[a,b]-i[0,ch^{2/3}[$, is equal to the number of eigenvalues in $[a,b]$ of the
Dirichlet realization of the unperturbed operator in ${\cal O}$ up to a small
remainder.
