Let $S$ be a set of four points chosen independently, uniformly at random from 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 show that the probability that $S$ defines a pair of crossing edges of the same color is equal to $1/4$. This is connected to a recent result of Aichholzer et al. [GD 2019] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halfed. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation $\frac{1}{2}-\frac{7}{50}$ of the total number of crossings.

翻译：设$S$为从正方形中独立均匀随机选取的四个点构成的集合。连接$S$中每对点的直线段。若线段斜率为正则将其染为红色，否则染为蓝色。我们证明$S$定义同色交叉边的概率等于$1/4$。该结果与Aichholzer等人[GD 2019]的最新结论相关——他们证明了通过对几何图的边进行二染色并计算单色交叉而非全部交叉，交叉数量可减少超过一半。我们的研究表明，对于所描述的随机绘图，存在一种边染色方式，使得单色交叉数的期望值为总交叉数的$\frac{1}{2}-\frac{7}{50}$。