We place $N$ red balls and $N$ black balls in $N$ boxes. No empty boxes. Someone picks one ball from one box.
Red ball = success, black ball = fail.
How to place balls so that we have highest probability of success (not fail). If we put all red balls in all boxes ( $1$ red in each) and all black in one, probability of picking red will be close to $1$ as $N$ grows, but how I can prove that that is best distribution (if it is) (since it will be also close to $1$ as $N$ grows if we for example put $N-1$ black balls in one and one in some other?