圆周折线穿越计数

You select a uniformly random starting point on the circumference of a circle, say $P_1$. Then, you select another uniformly random point on the circumference, $P_2$, and draw a line segment between $P_1$ and $P_2$. You then continue to select uniformly random points on the circumference of the circle, say $P_3, \dots, P_n$, and draw a line segment between $P_i$ and $P_{i+1}$ for each $1 \leq i \leq n-1$. Lastly, you draw a line segment from $P_n$ back to $P_1$. Find the expected number of intersections between line segments on the interior of the circle as a function of $n$. Evaluate this for $n = 12$.