Comparing the expected number of random elements from the symmetric and the alternating groups needed to generate a transitive subgroup

Given a transitive permutation group of degree n , we denote by e T ( G ) the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators of a transitive subgroup of G is found. We compare e T (Sym( n )) and e T (Alt( n )) .