We have two urns. One contains four red balls and the other one four blue balls. Every round we randomly pick two balls one from each urn and then we placed them in the opposite urn. The experiments end when there are same number of red and blue balls in both urns, i.e., 2 red and 2 blue balls in each urn. What is the expected number of rounds needed to finish the experiment?
How to do this type of problem? Here there are so many possible cases (branching) . Is there any easy way to solve this? Please guide.
The state of the game is completely determined by the number of red balls in the first urn. Denote by $E_r$ the expected number of additional rounds when there are $r\in\{4,3,2\}$ balls in the first urn. Then $$E_4=1+E_3,\quad E_3=1+{1\over4}\cdot{1\over4}\>E_4+\left({1\over4}\cdot{3\over4}+{3\over4}\cdot{1\over4}\right)E_3+{3\over4}\cdot{3\over4}E_2,\quad E_2=0\ .$$ Solving this system gives $E_4={26\over9}$.