I have an urn with N balls each with a different color and I am sampling them with replacement. I keep track of which colors I have seen. Given the number of unique colors observed I would like to characterize the distribution of the number of draws from the urn. For example, if I know only one color has been observed what is the probability that I have only drawn one ball from the urn? How about two draws, three draws, etc? What if two colors have been drawn? What are the tail probabilities in general, i.e., probability that m or more balls have been drawn given c unique colors have been drawn.
Now, consider I have k balls of each of the N colors (kN total balls). For each color, one of the balls is marked with a dot. I am again sampling with replacement. I will keep track of colors drawn if and only if the ball drawn has a dot. Again, I am interested in characterizing the total number of draws given the number of colors drawn (only those marked with a dot in this example). Also, how do I characterize the tail probabilities in this example?