How often are six of eight balls on the left?

Question

Eight balls are split between a left and a right box. Every second, one of the eight is chosen at random and moved to the other box. The state is the number of balls in the left box, from 0 to 8. In the long run, what fraction of time does the left box hold exactly six balls?

Accepted Answer

This is the Ehrenfest model with N = 8. From a state with k balls on the left, the count rises with probability \frac{N-k}{N} (a right ball is picked) and falls with probability \frac{k}{N} (a left ball is picked). Detailed balance for this birth-death chain gives \frac{\pi{k+1}}{\pik} = \frac{N-k}{k+1}, so the long-run shares are proportional to \binom{8}{k}, and the stationary distribution is Binomial(8, 	frac12): \pik = \binom{8}{k}/2^8 = \binom{8}{k}/256. For exactly six balls on the left, k = 6, and \binom{8}{6} = 28: \pi6 = \frac{28}{256} = \frac{7}{64}. So the left box holds exactly s…

How often are six of eight balls on the left?

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