I'm interested in the following question:
Given $g,h \in S_n$, what is the probability that $ \langle g,h \rangle$ is a primitive group of given O'Nan-Scott-type?
Can someone point me towards interesting literature concerning this problem? I know there is a lot of work on the expected number of elements to generate a group (i.e. formulas using the Moebius function on the subgroup lattice), but I can't seem to find anything concerning the above question.