Source:P. Winkler's Puzzles book. (Chapter: Probability).
Highlight the part between the * symbols for the answer.
*This problem can be reformulated as the following problem.
Suppose I have a stack of black cards and one red card. Initially I take red card in my hand. Now I add black cards randomly between any two cards (so, initially its either above or below red). Note that the probability that I add the card above the red card, when x-1 is the number of cards above red and y-1 is the number of cards below red is x/(x+y). Let the problem be if red card is dividing the black cards into two sets, what is the expected number of black cards in the smaller section. So, we see that the two problems are equivalent.
Now this way, we are getting all possible combinations in which one red and n black cards can be mixed, we see that the probability that the red card is at height h is independent of h. So, the probability that the smallest urn contains n/2 balls or 1 ball (or any number of balls between 1 and n/2) are all same. So, expected number of balls in the smaller urn is asymptotically n/4. :)