You are given $2n$ boxes that are arranged circular (you can imagine all boxes are on the edge of a circular table). Then randomly, you put $k$ balls in the boxes such that each box is containing either 0 or 1 ball.

What you have to do is to pick $n$ consecutive boxes such that the number of balls you pick is as small as possible. What is the expected number of this number (the number of balls)?

On a special case, what if $k$ is equal to $O(\log n)$?