  We improve on recent estimates for the probability of generating the
alternating and symmetric groups $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$. In
particular we find the sharp lower bound, if the probability is given by a
quadratic in $n^{-1}$. This leads to improved bounds on the largest number
$h(\mathrm{Alt}(n))$ such that a direct product of $h(\mathrm{Alt}(n))$ copies
of $\mathrm{Alt}(n)$ can be generated by two elements.
