Probabilistic generation of finite groups with a unique minimal normal subgroup

Let L be a finite group with a unique minimal normal subgroup, say N. We study the conditional probability P_{L,N} (d) that d randomly chosen elements of L generate L given that they generated L modulo N. In particular we prove that if d ≥ d(L) then P_{L,N} (d) ≥ 1/2. Several applications to general questions on the generation of finite and profinite groups are described