We have two rooms and ten people. Two initial states are Room A- 10 People, Room B -0 and Room A-9 People, Room B-1 person. Every minute, a person is chosen at random and (s)he must swap thier room. What is the probablity that as time tends to infinity, both rooms have 5 people each.
This is clearly a first order Markov chain which i solved using Matrix method and a bit of Mathematica.
Is there any other, more intuituve way for this?
The answer is about $0.2461$ for anyone wondering