Consider the scheme of random placing balls into $N=1000$ cells. We continue the procedure of placing balls as long as a last cell remains empty. The process terminates when a ball is placed into this cell. At this moment several cells (or a certain cell) contain(s) a maximum number of balls among all cells. What is the expectation of this maximum?
As an application of this scheme consider $N$ people who enter a lottery game. Each raffle is equivalent to the random placing of a ball into $N$ cells. We take a look at this process as long as each person has won at least once.