Dixon's asymptotic without CFSG

Without using the classification of finite simple groups, we show that the probability that two random elements of $S_n$ generate a primitive group smaller than $A_n$ is at most $\exp(-c(n \log n)^{1/2})$. As a corollary we get Dixon's asymptotic expansion $$1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - \cdots$$ for the probability that two random elements of $S_n$ (or $A_n$) generate a subgroup containing $A_n$.