# How many batches of balls have a leading ball when we fill $n$ bins by placing $k$ balls each turn

Suppose we have $$n$$ bins and $$k is some natural number. Each turn we select at random $$k$$ distinct bins out of the $$n$$ bins in which we place a ball. How many turns does it take on average until there is at least $$1$$ ball in each of the bins? This part has been solved in the comments (as it turned out to be a duplicate).

Related to this : If we call a collection of $$k$$ balls a winner if one of the balls is the first one to enter a bin, how many winners are there on average. So the first batch of ball has a 100$$\%$$ chance of being a winner and then the probability of being a winner decreases as more balls are added. For the coupon collector's problem mentioned in the comments (with $$k=1$$) it is obvious that there will always be exactly $$n$$ winning batches.

• When $k=1$ this is the coupon collector's problem, so you might start by researching that. – saulspatz Nov 13 '18 at 22:55
• Thank you for this reference, it gives me a good idea of how difficult the problem is and a good starting point to investigate further! – Darkwizie Nov 13 '18 at 23:07
• "Each turn we select at random k distinct bins out of the n bins in which we place a ball." Does it mean that $k$ balls are placed in (randomly chosen) $k$bins each turn? – user Nov 13 '18 at 23:14
• @user : Yes exactly – Darkwizie Nov 13 '18 at 23:21
• The first part is essentially the same as math.stackexchange.com/questions/2147576/… which has two good responses – Henry Nov 13 '18 at 23:42