On 9/17/2013 12:02 PM, Eugene Salamin wrote:
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.
I don't understand the arrangement. Are the hemispheres stuck onto the faces of a cube?
In two dimensions, the probability that all n molecules lie in some semicircle is 2n/2^n.
That doesn't look right. n=2 -> P=1
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.
If the boundary of the half-space is a plane, can't you just take the inner product with a vector normal to that plane? Brent
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