  We show that for all $n\leq X$ apart from $O(X\exp(-c(\log X)^{1/2}(\log \log
X)^{1/2}))$ exceptions, the alternating group $A_n$ is invariably generated by
two elements of prime order. This answers (in a quantitative form) a question
of Guralnick, Shareshian and Woodroofe.
