Choose two points on the top line (say T1 and T2), and two on the bottom line (say B1 and B2). Without loss of generality, assume T1 lies to the left of T2, and B1 to the left of B2. If this can be done in a uniform fashion (i.e., choosing any point is equally likely), the final probability is nothing but the probability of joining them in a way that the lines intersect.

Now, there are only two possibilities to form lines: Join T1 to B1 (so that T2 joins with B2), or join T1 to B2 (i.e., T2 joins B1). In the first case (T1B1 & T2B2), the lines don't intersect, and in the second (T1B2 & T2B1), they do. Thus, the desired probability is 1/2.