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.

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.