Given a circle of circumfence $1$ and on that circle, $n$ uniformly randomly placed points $X_1, \ldots, X_n$. We can identify each point $X_i$ with some real value in the interval $[0, 1)$, i.e., the $X_i$ are chosen from $U(0,1)$. Adjacent points on the circle form arcs. I'm confused about the expected lengths of those $n$ arcs.
On the one hand, the problem is completely symmetric, so the expected lengths of all arcs should be equal, thus $1/n$. None of the arcs should be "special".
On the other hand, one of the points is the smallest value and one is the largest. The expected value of the minimum is $1/(n+1)$ and the expected value of the maximum is $n/(n+1)$. This means that the expected length of the arc crossing $0$ is $2/(n+1)$, but for all other arcs $1/(n+1)$. This makes the 0-crossing arc "special".
I'm having trouble reconsiling those to answers. Help please?