We consider the probability that a pair of random permutations from the symmetric group generates the alternating group or the symmetric group. The best known results (Babai, Dixon) require the classification of finite simple groups (CFSG). We give an elementary CFSG-free proof. We use character theory and character estimates in our proof. This method carries over to other mathematical problems: We establish a new lower bound on the number of $T_2$- systems of the alternating group and we develop an asymptotic formula for the multiplicities of the root number functions in the symmetric group.<eng>
