Ehrenfest's urns is where you have two urns, and a number of balls divided between them. At every 'turn' a ball is selected at random and moved to the other urn.
Now, say we have ten balls.
Suppose that they are divided equally between the two urns, 5 in each. What is the expected length of time before the system returns to this state?
Or if you suppose the ten balls all start off in one urn - what is then the probability of returning to THAT state?
Clearly the first problem would be expected to take less time, since it is the 'equilibrium' as it were...