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))