A note on the probability of generating alternating or symmetric groups

We improve on recent estimates for the probability of generating the alternating and symmetric groups $\mathrm{A}\mathrm{l}\mathrm{t}(n)$ and $\mathrm{S}\mathrm{y}\mathrm{m}(n)$. 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 $h(\mathrm{A}\mathrm{l}\mathrm{t}(n))$ such that a direct product of $h(\mathrm{A}\mathrm{l}\mathrm{t}(n))$ copies of $\mathrm{A}\mathrm{l}\mathrm{t}(n)$ can be generated by two elements.