I have two jars; initially one is empty while the other holds $n$ red and $n$ blue marbles. Every minute, I do the following procedure: take any pair of marbles of different colors but from the same jar, and then randomly divide them between the two jars, one in each. The procedure terminates once one jar contains all red marbles and the other contains all blue marbles.
I want to show this process terminates within $O(n^2)$ steps in expectation. How can I get this upper bound? I've tried reducing this from the drunkard's walk without success.
You're right that the result doesn't depend on how the pair is selected.
In each step, the difference between red and blue marbles in one of the jars is incremented or decremented with equal probability. It starts at $0$ and ends at $\pm n$. The expected number of steps for a simple symmetric one-dimensional random walk to reach $\pm n$ from $0$ is $n^2$. This can be shown by considering the recurrence relation
$$ a_k=1+\frac12\left(a_{k-1}+a_{k+1}\right) $$
for the expected value at position $k$, with boundary conditions $a_{\pm n}=0$, with solution $a_k=n^2-k^2$.