You have a bag with marbles and draw without replacement. The marbles have $k$ distinct colors and there are exactly $n$ marbles of each color. All marbles are equally likely to be drawn. You continue drawing until the bag runs out of a color so you have drawn all $n$ marbles of that color. What is the expected number of marbles you need to draw and what is the probability distribution to model this process?

You clearly have do draw at least $n$ marbles and at most $k(n-1) +1$ by the pigeon hole principle. This looks similar to the [negative hypergeometric distribution](https://en.wikipedia.org/wiki/Negative_hypergeometric_distribution) but it doesn't quite fit because I want to stop drawing the moment I run out of any color not just one particular color.

Simplest example: There are $2$ red and $2$ green marbles in the bag. Without loss of generality let the first marble drawn be red. There is now a $1/3$ probability that the second marble is also red and I stop. If the second marble is green, the third marble will be either red or green and the bag will be out of that color so I stop no matter what. So with probability $2/3$ I will draw $3$ marbles. Hence the expected number of marbles drawn is $2\cdot (1/3) + 3\cdot (2/3)=2.67$.