medium Probability

Triangular Selection

A release-checked medium problem for training Combinations and Permutations, Geometric Probability.

Question

Fix $n \geq 4$. Three distinct vertices are selected uniformly at random from a regular $n-$gon. These points are then connected pairwise via line segments to form a triangle. Find the probability that at least one of the sides of the triangle is also a side of the $n-$gon when $n = 10$.

Practice focus

This Probability problem is tagged Combinations and Permutations, Geometric Probability. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.