# Picking points on a sphere at random

Suppose we pick up $N$ points uniformly at random on a sphere. The probability that these points lie within a 'fixed' hemisphere is easily calculated to be $1/2^N$. But what is the probability that all the points lie within any hemisphere on the sphere? I am actually interested in this question for $d$-dimensional hypersphere as well, so method of computation that extends to higher dimensions will be much appreciated.

• This was investigated by the inimitable Kevin S. Brown here – deinst Mar 27 '16 at 3:54
• Thank you! Its a brilliant solution. – anurag anshu Mar 27 '16 at 13:38
• That was indeed wonderful! – Sum-Meister Feb 6 '17 at 14:31
• Some visualization here: youtube.com/watch?v=OkmNXy7er84 – user558317 Jun 1 at 18:35

Kevin S. Brown proved in "N points on sphere all in one hemisphere", that the probability that $$N$$ points, chosen uniformly at random on a $$d$$-dimensional sphere, are lying all in one hemisphere is:
$$\frac{\Sigma_{k = 0}^{d}C_{n - 1}^k}{2^{n - 1}}$$