Suppose $G$ is a finite group. Let's define $n$-th generating fraction of $G$ as $gf_n(G) := \frac{|\{(a_1, ..., a_n) \in G^n| \langle a_1, ..., a_n \rangle = G\}|}{|G|^n}$.
What is the asymptotics of $gf_2(S_n)$?
The only thing that I managed to determine is that:
$$\frac{n! - 1}{(n!)^2} \leq gf_2(S_n) \leq \frac{3}{4}$$
The lower bound follows from $\frac{3}{2}$-generatedness of symmetric groups. The upper bound follows from the fact that $S_n$ has a subgroup of index $2$.
However, both those bounds are very crude and I would be interested to see something better.
An asymptotic result is that $$gf_2(S_n)\to\frac34$$ as $n\to\infty$, proven in [15] in the paper below.
More information can be found in this paper.