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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.

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

$$\frac{\Sigma_{k = 0}^{d}C_{n - 1}^k}{2^{n - 1}}$$

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