# Finding the probability that $n$ points which are randomly selected from $d-1$ dimensional spherical surface are 'semi-spherical'

Let $S^{d-1}=\{(x_1,\cdots,x_{d})\in {\mathbb R}^{d}|{x_1}^2+\cdots+{x_d}^2=1\}$, and let us call the intersection of any $d-1$ dimensional subset which passes through the origin and $S^{d-1}$ 'a great circle'. Also, let us call one of two components gotten by cutting $S^{d-1}$ by a great circle '$d-1$ dimensional semi-spherical surface'. Also, suppose that a point set on $S^{d-1}$ is semi-spherical when it is included in a $d-1$ dimensional semi-spherical surface.

Then, here is my question.

Question : Letting $d\ge 2$, find the probability $f(d,n)$ that $n$ points which are randomly selected from $S^{d-1}$ are semi-spherical.

Remark : Suppose that $n$ points are selected randomly with equable probability.

Example : We can easily get $f(2,n)=n\cdot 2^{1-n}.$

Motivation : I've known the $d=2$ case, but I'm facing difficulty for $d$ in general. Can anyone help?