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.
Best Answer
You can get an approximate answer. The coupon collector's problem tells you that the expected number of balls is $n*H_n$ or about 7485 for n=1000. So the average bin will have 7.485 balls. If you look at the Poisson distribution for 7.485 and find the number where it falls to 1/1000. I make that 17 or 18. The motivation is that you have 999 bins with an average occupancy of 7.485. The highest will be at a probability of about 1/999