11. Electoral Districts. In Anchuria, presidential elections are being prepared, in which President Miraflores wants to win. Exactly half of the numerous voters support Miraflores, and the other half support Dick Maloney. Miraflores is also a voter. By law, he has the right to divide all voters into two electoral districts at his discretion. Voting in each district is conducted as follows: each voter marks the name of their candidate on the ballot; all ballots are placed in an urn. Then one random ballot is drawn from the urn, and the candidate whose name is marked on it wins in that district. A candidate wins the election only if they win in both districts. If no winner is determined, a second round of voting is held under the same rules. How should Miraflores divide the voters to maximize the probability of his victory in the first round?
Solution. Suppose there are $2 n$ voters in Anchuria in total. It may happen that in each district, exactly half of the voters are supporters of Miraflores. The probability of this $p$ depends on the method of dividing into districts devised by Miraflores, but in any case, $p1, q=1$ and $p=0$.

If Miraflores includes only himself in the first district, then he will win in this district with the maximum possible probability $q=1$. Then in the opposition district, there will be $2 n-1$ voters, of which $n-1$ voters vote for Miraflores. Thus, the probability that Miraflores will win in the opposition district is $\frac{n-1}{2 n-1}$. Therefore, with such a division into districts, the probability of Miraflores' victory is

$$
1 \cdot \frac{n-1}{2 n-1}=\frac{n-1}{2 n-1}
$$

which is the maximum possible.

Answer: one district should consist of Miraflores alone, and the second district should consist of all other voters.

Comment: note that under such rules, Miraflores' victory is only a matter of time. He cannot lose, as each time the district consisting of only Miraflores votes against Dick Maloney. Thus, Maloney cannot win at any stage. Therefore, in the end, Miraflores will win. The solution to the problem can be slightly simplified by immediately accepting some circumstances as obvious.