We place $n$ points on the unit circle uniformly and independently . What is the probability that their convex hull does not contain the origin?
I came up with this:
If the convex hull does not contain the origin then there is one point $x$ such that the next point $y$ in clockwise order forms an angle $xOy$ larger than $\pi$. Therefore we just need to calculate the probaility that this happens at least once, but notice that this cannot happen more than once . Suppose that $x_1,x_2,\dots x_n$ are the points that are to be placed independently, then the probability that the next point after $x_i$ forms an angle larger than $\pi$ is $\frac{1}{2^{n-1}}$. Therefore the probability the hull does not contain the origin is $\frac{n}{2^{n-1}}$
I would greatly appreciate a proof verification, alternative proofs and insight as to whether this can be generalized to higher dimensions, when there are $d+2$ points it is possible, as is shown here.