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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 generatorsis 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 $\kappa$ isexplicitly described in terms of the Riemann zeta function and is bestpossible. 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 $\kappa^* \sim1.606695.$

The expected number of elements to generate a finite group with $d$-generated Sylow subgroups