Let the urn $U_1$ with $m_1$ white balls and $n_1$ black balls, the urn $U_2$ with $m_2$ white balls and $n_2$ black balls. A balanced die is rolled and if an even number appears, a ball, chosen at random from $U_1$, is transferred to urn $U_2$. If an odd number appears, a ball, chosen at random from urn $U_2$, is transferred to urn $U_1$. What is the probability that, after the above experiment is performed twice, the number of white balls in the urn $U_2$ remains the same?
How to organize the events to apply Bayes Theorem? I don't know how to start solve the problem.