Given a triple (p(1), p(2), p(3)) of primes, the object of this paper is the study of the space Hom(T-p1,T-p2,T-p3, G) of homomorphisms from the triangle group T-p1,T-p2,T-p3 to a finite simple exceptional group G of Lie type B-2(2), (2)G(2), G(2) or D-3(4). With a few exceptions, we give precise asymptotic estimates for the size of Hom(T-p1,T-p2,T-p3, G) and determine the limiting probability that a randomly chosen homomorphism from T-p1,T-p2,T-p3 to G is surjective as |G| ---> infinity. (C) 2010 Elsevier Inc. All rights reserved.