How often do two of three fleas sit on one dog?

Question

Three fleas live on two dogs, Rex and Fido. Every minute, one of the three fleas is chosen at random and hops to the other dog. The state is the number of fleas on Rex, from 0 to 3. In the long run, what fraction of time are exactly two fleas on Rex?

Accepted Answer

This is the Ehrenfest model with N = 3. From a state with k fleas on Rex, the count rises with probability \frac{N-k}{N} (a Fido flea hops over) and falls with probability \frac{k}{N} (a Rex flea hops away). Detailed balance for this birth-death chain gives \frac{\pi{k+1}}{\pik} = \frac{N-k}{k+1}, the ratios of the binomial coefficients \binom{3}{k}. So the long-run shares are proportional to \binom{3}{0}, \binom{3}{1}, \binom{3}{2}, \binom{3}{3} = 1, 3, 3, 1, which add to 2^3 = 8. The stationary distribution is Binomial(3, 	frac12): \pik = \binom{3}{k}/8. For exactly two fleas on Rex, k = 2…

How often do two of three fleas sit on one dog?

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