For the little it might be worth, here are the results of some simulations:
(source: landsburg.com)
The columns correspond to values of $|A|$ and the rows to values of $|B|$. The four-tuple in each cell is (Probability winner is in $A$, Probability winner is in $B$, Probability a given member of $A$ is the winner, Probability a given member of $B$ is the winner).
I simulated each of these 10,000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors).
I was struck by the non-monotonicity in the third entry as you go down the third column, so I repeated these trials and got the same result.
