[Math] the probability of two random line segments crossing in a unit square

Question

For the purposes of this question a random line segment is defined by connecting two random points inside the unit square, where a random point is found by generating two random numbers between 0 and 1, and taking them as the x and y co-ordinates. What is the probability of two such randomly generated line segments crossing?

Answer 1

First decide whether you mean the line segments only, or the whole lines through the two points. Then the following method should lead you to the answer:

Generate four points at random. There are three ways to pair those points together, and the probability that the lines/line segments intersect will be $0$ or $\frac13$ or $\frac23$ or $1$, depending on whether the four points form a convex quadrilateral or a triangle with a fourth point in its interior.

So the overall probability can be deduced once you know the probability that four randomly chosen points from the unit square form a convex quadrilateral (or a triangle with a point inside). Not a trivial probability problem but I think it is known.