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$$k \approx\log n$?
Reference/Motivation. This is equalactually one of my work to attack this question: $O(\log n)$?Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game.