Elliptic curves with maximal Galois action on their torsion points

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation rho(E) : Gal((k) over bar/ k) -> GL(2)(Z). For a fixed number field k, we describe the image of rho(E) for a ` random' elliptic curve E over k. In particular, if k not equal Q is linearly disjoint from the cyclotomic extension of Q, then.E will be surjective for ` most' elliptic curves over k.