If I have a bag of blank paper slips of number N.

I choose one at random

If it is blank I colour it red and replace.

If it is already coloured I just replace it.

How form does the number of red slips (n) as a function of number of picks (as it approaches N)?

Obviously it grows fastest at the start when all the slips are blank and slows down as you try to get the few remaining.

It seems similar to sampling without replacement but in some sense there is a 'replacement'. or alteration.

Eventually I want to generalize to include three colours per round with an additional rule that I only colour with a certain probability which is different for the three colours.

Thanks for any help

Peter