So I was thinking about how many digits of a random irrational number it would take on average to hit every digit 0-9, then I wondered what if I only cared about hitting certain digits (say 0-4). Or what if I was working in a different base and so on. Then I realized this question is equivalent to any other situation where you care about randomly picking all the elements or some of the elements in a set. So to put my question into marbles in a bag:
Suppose you had n marbles in a bag and you randomly pick one of the marbles from the bag then put it back and draw again. 1) Whats the expectation value for the number of draws it would take to get every marble? 2) Also What is the probability of drawing all n marbles after x drawings (assuming x>=n). To make the question more general, suppose m rocks were also placed in the bag with the marbles such that any marble would have a 1/(n+m) chance of being drawn. Again what is the expectation value for the number of draws it would take to get all n marbles and what is the probability of drawing all n marbles after x drawings (assuming x>=n)?