  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%.
