Is the probability that half the sets are half red, more than half? There are $2n$ sets, each of which contains an even number of integers.
Each integer is colored red with probability $1\over2$, independently of the others.
Say that a set is good if at least half its elements are red. 
Say that a coloring is good if at least $n$ out of the $2n$ sets are good
What is the probability $p$ that the random coloring is good?
It is clear that $p\geq {1\over2}$, since for every coloring in which $m$ sets are not good, in the opposite coloring at least $m$ sets are good. Therefore, for every not-good coloring, the opposite coloring is good.
Is it always true that $p>{1\over2}$ strictly?
 A: Yes, it's always strictly greater.
As T. Gunn says in the comments, it is sufficient to show that there is a very good colouring, that is to say a colouring which is good, and remains good after swapping all the colours.
Suppose we have a colouring which is good, but not very good, and let $k$ be the number of sets with a strict majority of red elements. Since these $k$ sets are precisely the sets which will have a strict majority of blue elements after we swap, these are the sets which are not good in the opposite colouring. And the original colouring was not very good, which means $k>n$.
Now pick any red element and colour it blue. Since we have $k>n$ sets with a strict majority of red in the original colouring, we have at least $k>n$ sets which are at least half red in the modified colouring. So the modified colouring is still good.
Finally, choose any good colouring with the smallest possible number of red integers. This must be very good, since if it wasn't we could use the above argument to get a good colouring with fewer red integers, contradicting our choice.
(This argument does not actually require that the number of sets is even, only that every set has even size.)
