PL-OM 1987 Problem 2

A regular $n$ -gon is inscribed in a circle radius $1$ . Let $X$ be the set of all arcs $PQ$ , where $P, Q$ are distinct vertices of the $n$ -gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$ .