On 9/17/2013 12:02 PM, Eugene Salamin wrote:

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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?

They overlap.

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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

Yes, unless they're antipodal (with probability zero), we can take a line (say, the diameter orthogonal to their perpendicular bisector) partitioning the circle into two semicircles, one of which contains both points. Sincerely, Adam P. Goucher