My question is as follows: "Is a generic closed orientable hyperbolic 3-manifold Haken?" Of course the word 'generic' can be interpreted in many ways, and the answer might depend on the way how one interprets 'generic'. As an example, let's consider the related question "Is a generic closed (aspherical) 3-manifold hyperbolic?" In this case there are various notions of 'generic', in each case the answer is a resounding yes. For example, almost all Dehn fillings on a hyperbolic knot result again in a hyperbolic 3-manifold. Furthermore Maher (Random Heegaard splittings) showed in a precise sense that a 'random gluing of two handlebodies of genus >1' gives rise to a hyperbolic 3-manifold. Similarly, in a precise sense, a 'random' fibered 3-manifold is hyperbolic.

My hunch would have been that a generic hyperbolic 3-manifold is non-Haken (why should it have an incompressible surface?), but I just had lunch with another 3-manifold topologist and his guess was that a generic hyperbolic 3-manifold should be Haken.