Ehrenfest urn stationary distribution

HardProbability Stochastic Processes~7m

In an Ehrenfest model with $N$ balls split between two urns, a uniformly random ball switches urns each step. What is the stationary distribution of the number of balls in urn 1?

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Check detailed balance against a binomial guess.
Each ball is independently in urn 1 with probability $1/2$ at stationarity.

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$\text{Binomial}(N, 1/2)$

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Ehrenfest urn stationary distribution

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