[math-fun] Thought up while drawing polygons ...

In the unit square pick 4 points A,B,C,D at random (x,y each uniformly distributed). Draw lines AB, BC, CD, DA. What is the probability that two of these lines will cross (that is, we get a reflexive quadrilateral)? With 5 points there are several possible topologies: no crossings, one, two, three, or five. Is 4 crossings possible? What are the probabilities of each configuration? With a given number of crossings, how many distinct topologies are possible? (E.g. with 3 crossings, in one configuration one region is not adjacent to the outside, and in another all are adjacent to the outside.) Generalize to N points. I don't have answers or know if this is an interesting problem.