  Given a finite group $G,$ let $e(G)$ be expected number of elements of $G$
which have to be drawn at random, with replacement, before a set of generators
is found. If all the Sylow subgroups of $G$ can be generated by $d$ elements,
then $e(G)\leq d+\kappa$ with $\kappa \sim 2.75239495.$ The number $\kappa$ is
explicitly described in terms of the Riemann zeta function and is best
possible. If $G$ is a permutation group of degree $n,$ then either $G=S_3$ and
$e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor+\kappa^*$ with $\kappa^* \sim
1.606695.$
