Probability of generation by random permutations of given cycle type

Suppose $\pi$ and $\pi'$ are two random elements of $S_n$ with constrained cycle types such that $\pi$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $\pi'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle \pi, \pi' \rangle$ is transitive and $G \geq A_n$ both have probability approximately $$(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),$$ provided $(x, x')$ is not close to $(0, \infty)$ or $(\infty, 0)$. This formula is derived from some preliminary results in a recent paper (arXiv:1904.12180) of the authors. As an application, we show that two uniformly random elements of uniformly random conjugacy classes of $S_n$ generate the group with probability about 51%.