An immersed surface S in a Riemannian manifold M induces a probability measure on the space G_2(M) of two-planes in the tangent bundle of M. If M is a hyperbolic 3-manifold and S_n is a sequence of surfaces with principal curvatures going to zero, then any weak* limit of their induced measures is a convex combination of the Liouville (equidistributed) measure on G_2(M), and measures that come from immersed totally geodesic surfaces in M. We consider two ways of generating a “random” nearly geodesic surface in M, one by bounding the genus, and the other by bounding the area. We show that limits of the measures in the former case must come exclusively from the totally geodesic surfaces (if there are any in M), while limits in the latter case must have some portion that is equidistributed. This is joint work with V. Markovic and I. Smilga.