You are given a circle $C$ of unit radius and a point $M$ on it. Now, three points are chosen at random on the circle $C$ which divides it into three arcs. What is the expected length of the part that contains $M$?
My doubt is whether we should consider $M$ fixed or not.
We may consider $M$ fixed without loss of generality. $M$ cuts the circle into a line, which we take as having length 1 (scaling to the real circumference of $2\pi$ will happen at the end). The three random points on $C$ are then random variables $U_1,U_2,U_3\sim U(0,1)$ and $$E(\text{length of arc containing }M)=E(U_{(1)})+(1-E(U_{(3)}))=E(\mathrm B(1,3+1-1))+1-E(\mathrm B(3,3+1-3))=E(\mathrm B(1,3))+1-E(\mathrm B(3,1))=\frac14+1-\frac34=\frac12$$ where $U_{(i)}$ are order statistics and $\mathrm B$ is the beta function. Thus, on the unit circle, the expected length of the segment containing $M$ is $\frac12\cdot2\pi=\pi$.