We randomly place points $A, B, C$, and $D$ on the circumference of a circle, independently of each other. What is the probability that the chords $AB$ and $CD$ intersect?
Solution. The chords intersect if the points on the circumference are in the order $A C B D$ in one of the traversal directions. Place the points at arbitrary positions on the circumference. We can do this because the placement does not affect the intersection, only the names we give to the points. Let one of the placed points be $A$. Which one we choose still does not affect the intersection. If we now label one of the neighbors of $A$ as $B$, then the segments $A B$ and $C D$ do not intersect if we label the "opposite" one, otherwise there will be an intersection point. The selection of $B$ can be made randomly, so there is a $\frac{2}{3}$ probability of choosing a point adjacent to $A$. According to this, the probability that the chords intersect is $\frac{1}{3}$.