Crossings in a random closed polygon in a circle

Choose points $P_1,P_2,\dots,P_n$ independently and uniformly on the circumference of a circle. Join $P_1$ to $P_2$, $P_2$ to $P_3$, and so on up to $P_{n-1}$ to $P_n$, and finally join $P_n$ back to $P_1$, giving $n$ chords. Derive the expected number of crossing points of these chords strictly inside the circle as a function of $n$, then evaluate it at $n=12$.

$\tfrac{n(n-3)}{6}$, which is $18$ for $n=12$. Of the $\binom n2$ pairs of chords, the $n$ consecutive pairs share an endpoint and cannot cross inside. Any other pair has four independent uniform endpoints and crosses with probability $\tfrac13$, so by linearity the expectation is $\tfrac13\left(\binom n2-n\right)$.