Dixon's theorem

Statement
Let $$S_n$$ denote the symmetric group of degree $$n$$, and $$A_n$$ denote the subgroup that is the alternating group. Pick two elements independently of each other, both uniformly at random, from $$S_n$$. Denote by $$f_1(n)$$ the probability that they generate all of $$S_n$$, and by $$f_2(n)$$ the probability that they generate $$A_n$$.

Then:

$$\lim_{n \to \infty} f_1(n) = \frac{3}{4}$$

and:

$$\lim_{n \to \infty} f_2(n) = \frac{1}{4}$$

Thus:

$$\lim_{n \to \infty} f_1(n) + f_2(n) = 1$$

Another way of putting this is as follows:


 * Define by $$g_1(n)$$ the probability that two permutations generate the whole group conditional to the elements not both being even permutations. Then, clearly $$f_1(n) = (3/4)g_1(n)$$. The first part of Dixon's theorem states that $$\lim_{n \to \infty} g_1(n) = 1$$.
 * Define by $$g_2(n)$$ the probability that two permutations generate the whole group conditional to the elements both being even permutations. Then, clearly $$f_2(n) = (1/4)g_2(n)$$. The second part of Dixon's theorem states that $$\lim_{n \to \infty} g_2(n) = 1$$.

Related facts

 * Symmetric group on a finite set is 2-generated
 * Alternating group on a finite set is 2-generated