I have a bag of n marbles. I select a handful of marbles out of the bag x number of times. Each marble is equally probable to draw. Each time I select a handful, I mark them with a black dot and I put them all back.
Here's the hard part for me: The size of the handfuls will vary between selections. (For example, if I make 3 selections from a bag of 100 marbles, the sizes of my handfuls may be: 5 the first time, 19 the second, 8 the third - 5,19,8 are just random numbers i chose).
The quantity of interest is the probability of having exactly k marbles that are marked after my selections.
I will then use this to get the probability of having less that k marbles that are marked.
I understand my explanation is probably confusing so here is an example for more clarity.
There are 100 marbles in the bag. (n = 100). I reach in and grab 5 marbles all at once. I mark them with a black dot and put the 5 marbles back. I then reach in and grab 19 marbles all at one. I mark them with a black dot and put the 19 marbles back. ...etc for 8 marbles on the third selection...
I then count the number of marbles that are marked with a black dot. I see that there are only 22 marbles with a black dot (intuitively unlikely). I need to figure out the probability that I would have drawn <23 distinct marbles?
I've had some guidance with a simplified version of the problem here but I have struggled with understanding how to incorporate varying size selections (handfuls).