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Let $S$ be a set of four points chosen independently, uniformly at randomfrom a square. Join every pair of points of $S$ with a straight line segment.Color these edges red if they have positive slope and blue, otherwise. We showthat the probability that $S$ defines a pair of crossing edges of the samecolor is equal to $1/4$. This is connected to a recent result of Aichholzer etal. [GD 2019] who showed that by 2-colouring the edges of a geometric graph andcounting monochromatic crossings instead of crossings, the number of crossingscan be more than halfed. Our result shows that for the described randomdrawings, there is a coloring of the edges such that the number ofmonochromatic crossings is in expectation $\frac{1}{2}-\frac{7}{50}$ of thetotal number of crossings.

A Note on the 2-Colored Rectilinear Crossing Number of Random Point Sets in the Unit Square