Probability that two randomly chosen permutations will generate $S_n$.

It can be shown the odds of two random permutations generating $S_n$ or $A_n$ is $1 - \frac{1}{n} - O(\frac{1}{n^2})$.

• $\mathbb{P} = \frac{1}{n}$ both permutations fix the same element, so $\langle S \rangle \subset S_{n-1}$
• $\mathbb{P} = \frac{1}{4}$ both permutations have the same parity (odd or even), $\langle S \rangle \subset A_n$

This result due to László Babai, on The Probability of Generating the Symmetric Group.

However, if you don't want to use the classification of finite simple groups you can show this probability is greater than $1 - \frac{2}{\log \log n}$. This is proven by John Dixon around 1970.

More recently, the same author gets further terms for the odds of generating the symmetric group: $$1 - \frac{1}{n} - \frac{1}{n^2} - \frac{4}{n^3} - \frac{23}{n^4} - \frac{171}{n^5} - \dots $$ You can count the coincidences by hand. The odds both elements fix or transpose the same two elements is of order $\frac{1}{n^2}$, etc.