Base sizes for S-actions of finite classical groups

Let G be a permutation group on a set Omega. A subset B of Omega is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher's collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Omega form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Omega are bases.