A knot in S^3 is small if its complement does not contain a closed incompressible surface. Is it a generic property for knots, meaning that among all knots with less than $n$ crossings, the proportion of small knots goes to 1 when $n$ goes to infinity?
In general, a knot is not small.
Abigail Thompson showed that if a thin position is not a bridge position, then there exists a closed incompressible, non-peripheral, surface in the knot complement (Corollary 3).
We need to show that in general, a knot in a thin position is not in a bridge position.
Even if it wasn't, Finkelstein--Moriah showed that there exists an essential meridional planar surface in the complement of a knot in a bridge position. Hence there exists a closed incompressible, non-peripheral, surface in the knot complement.
Following up on the first comment by Misha: Your question is very sensitive to the model you choose. I vaguely remember a talk by Ken Millet (http://math.ucsb.edu/~millett/) in which he gave a natural model of knot generation where the generic knot seemed to be a connect sum of $O(n)$ copies of the trefoil. If you tweaked the model, then the generic knot was the unknot.
And to reply to another comment above: one could condition on the knot being hyperbolic. Then the model is more interesting to analyze. However such a model is "unusable in practice" -- you can't actually generate knots this way because the waiting time is too long.