3. N balls are placed in two urns. After each unit of time one ball is selected
at random and transfered from the urn it was in to the other one. Let the
number of balls in the first urn denote the state of the system.
• Write down the transition matrix $P = (p_{i,j})_{i,j=0,...,N}$ of this Markov
chain.
• Show that the unique invariant distribution of the system is $\mu_j = \binom{N}{j} 2^{-N}$. (Hint: use the detailed balance condition $\mu_i p_{i,j} = \mu_j p_{j,i}$.)
