And now for a related piece of trivia, which may or may not help with this problem: If you distribute points uniformly on the surface of a unit sphere, their x-coordinates are uniformly distributed in the interval [-1,1]. This is specific to the 2-sphere in 3-space, and was almost certainly first noted by Archimedes. Sincerely, Adam P. Goucher http://cp4space.wordpress.com

----- Original Message ----- From: Eugene Salamin Sent: 09/17/13 08:43 PM To: math-fun Subject: Re: [math-fun] Probability that all molecules of a gas are in one half of the container

And more generally, the particles can have any spherically symmetric distribution.  If preparing to integrate over space, a useful distribution is the Gaussian, exp(-r^2) = exp(-x^2) exp(-y^2) exp(-z^2).

  --  Gene

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________________________________ From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Tuesday, September 17, 2013 12:28 PM Subject: Re: [math-fun] Probability that all molecules of a gas are in one half of the container

On 17/09/2013 20:16, Mike Stay wrote:

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For each possible point on the sphere, the probability that all the particles are in the hemisphere centered on that point is 2^{-n}. Pull the constant out and integrate over the sphere to get 2^{-n} * 1.   Or am I missing something?

I think so. The cases overlap, so the probabilities don't just add.

I have the feeling I've seen this problem, or one awfully similar to it, before. If so, I don't remember the answer.

One trivial remark: The answer is the same whether you choose particles uniformly *on the surface of the sphere* or *in the ball bounded by the sphere*.

-- g

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