I've got an idea for a simpler approach, but I've forgotten too much math to be able to actually try it.

The idea is to consider the set A of all circles that intersect the unit circle.

If you pick 3 random points inside the unit circle the probability that circle c ∈ A is the circle determined by those points should be proportional the length of the intersection of c's circumference with the unit circle.

The constant of proportionality should be such that the integral over all the circles is 1.

Then consider the set of all circles that are contained entirely in the unit circle. Integrate their circumferences times the aforementioned constant over all of these contained circles.

The ratio of these two integrals should I think be the desired probability.