[math-fun] Probability that all molecules of a gas are in one half of the container

A spherical vessel contains n molecules of gas.  What is the probability that all the molecules can be found in one hemisphere?  For a given hemisphere, it is 1/2^n.  For 6 hemispheres arranged like the faces of a cube, inclusion-exclusion gives 6/2^n - 12/4^n + 8/8^n.  But what is the probability if any hemisphere is allowed?  I'm stuck on this problem. In two dimensions, the probability that all n molecules lie in some semicircle is 2n/2^n. A related question is: Given n vectors x[1], ... , x[n], how do we test if they all lie in some half-space, i.e. does there exist a vector u such that u.x[i] > 0 for all i?  I'm stuck on this as well.   --  Gene