Expected length of arc in a randomly divided circle

The length of the arc from $(1,0)$ to the next point is the minimum $M$ of three (or $n$) independent uniform random numbers $X_1,\ldots, X_n$ from $[0,2\pi)$. Since $P(M>x)=\prod P(X_i>x)=\left(1-\frac x{2\pi}\right)^n$, we find $$E(M) = \int_0^{2\pi}P(M>x)dx =\left. -\frac{2\pi}{n+1}\left(1-\frac x{2\pi}\right)^{n+1}\right|_0^{2\pi}=\frac{2\pi}{n+1}.$$ The same holds for the arc length to the the other arc end, hence the expected length of the full arc containing $(0,1)$ is $$\frac{4\pi}{n+1}.$$