Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.

Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a link diagram is a choice for each crossing of either an A-resolution or a B-resolution. The resulting state graph is the graph whose vertices correspond to the circles, and whose edges correspond to dotted lines. A link diagram is said to be semi-adequate if either the state-graph resulting from A-resolving all crossings, or the state-graph resulting from B-resolving all crossings (or both) have no single-edge loops.

In Section 1.3 of Guts of surfaces and the colored Jones polynomial, the authors state that, "The condition that a link be semi-adequate seems to be rather mild". As evidence, they point out several low-crossing-number computations, including a computation of Stoimenow that shows that at least 249,649 of the 253,293 prime knots with 15 crossings are semi-adequate.

The question which lept to my mind (and I have no idea whether this is trivial, well-known, hard, or open) is whether this is an instance of the Strong Law of Small Numbers (which perhaps in this context should be called the `strong law of small links'):

Ilya's comment is very much on point. In general, part of what makes this question difficult is that the answer is likely to depend in a huge way on your model of "random knots". Choosing uniformly at random from a list of all $n$-crossing diagrams and choosing uniformly at random from a list of $n$-crossing knots are only two of many different models. Some others: stick knots, knot complements built out of $n$ tetrahedra (Champanerkar-Kofman-Patterson started an enumeration), etc. I don't believe the answer to your question is known for any of these... – Dave Futer Jul 17 '13 at 16:19