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
________________________________ 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:
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