# More on multiply transitive permutation groups

This question is a sort of a follow-up to these two: reference on classfication of multiply transitive permutation groups and Multiply transitive groups, continued

The question is simply: is it true that the set of $n,$ for which any doubly transitive subgroup of $S_n$ is one $S_n$ or $A_n$ is of density one (since any doubly transitive group is either affine or almost simple, this seems clear, but I might be missing something silly).

the stronger statement is proved that, for almost all $n$, the only primitive permutation groups of degree $n$ are $A_n$ and $S_n$. (By a fortunate coincidence I saw a reference to this in a paper I was reading a day or two ago.)