While trying to answer this question I realized that the probability for two points to lie on the same side of the line joining two other points is directly related to the probability for four points to form a convex quadrilateral. Since results are known for the latter for various distributions but I couldn't find any results for the former, I thought it might be useful to record the connection and transfer the known results for easy reference in the form of an answer here.
So let some distribution on the plane be given, for instance a uniform distribution over some region. If we know the probability that four points form a convex quadrilateral, how can we obtain the probability that two points lie on the same side of the line joining two other points (where all points are independently drawn from the given distribution)?
(Note that I'm asking for two particular points to lie on the same side of the line joining two particular points, not for any two out of four to lie on the same side of the line joing the other two.)