Is a generic closed orientable hyperbolic 3-manifold Haken? 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.
 A: Firstly, the random Heegard splitting model and the random fibering model are discussed at length in my preprint. (I believe the OP is familiar with it). I believe the question is asked there, though not answered - it is pointed out that the smaller genus of an incompressible surface will grow, but that seems to be neither here nor there. A related question is whether a random fibered manifold has an incompressible surface which is not the fiber of the fibration you started with. Here, the answer is YES, for genus 1 (Floyd-Hatcher), and probably YES for higher genus, which would tend to suggest that a random Dunfield-Thurston manifold might be Haken, but nobody knows for sure.
As for Dylan's comment, it is certainly plausible. Any given configuration of tetrahedra (in particular one with a normal sphere which is not the link of a vertex), in fact, you might as well assume that it is topologically a thickened sphere, will appear with probability one, and it is highly unlikely that it will bound balls on both sides.
