jmc

Consider the four random points before they are labeled $A$, $B$, $C$, or $D$. In the general case, they will be distinct, forming a convex quadrilateral. Suppose $A$ is labeled. If $B$ is labeled as the vertex opposite $A$, segments $AB$ and $CD$ will intersect; otherwise, they will not. Since there are 3 points to label as $B$, the probability these segments intersect is $\boxed{\frac{1}{3}}$. In this diagram, the green edges represent the labeling where $AB$ and $CD$ intersect, and the blue and red edges represent the equally likely labelings where $AB$ and $CD$ do not intersect.