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 $\exp (-c(n\log n{)}^{1\mathrm{/}2})$. As a corollary we get Dixon’s asymptotic expansion $$1-1\mathrm{/}n-1\mathrm{/}{n}^2-4\mathrm{/}{n}^3-23\mathrm{/}{n}^4-\cdots$$ for the probability that two random elements of $S_n$ (or $A_n$) generate a subgroup containing $A_n$.