General case

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.

Example problem

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).

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    $\begingroup$ This question is not yet well formulated: you need to specify the distribution of the number of marbles that are grabbed from the bag. You also need to specify how many of each type of marble is in the bag. And what does "grabbing exactly k unique marbles after the x number of selections" mean? Is it the total number of unique marbles seen in x selections, or would it be the number of new marbles seen in the next selection after the first x? Normally we would provide some help in formulating a question, but there are just too many unknowns here to know where you're going with this. $\endgroup$ – whuber Jan 1 '18 at 22:57
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    $\begingroup$ Thank you for the criticism. I've modified the question to be more clear. $\endgroup$ – terrigenus Jan 1 '18 at 23:28
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    $\begingroup$ Thank you. One seemingly minor but crucial detail remains, though: you must specify the probability distribution of the number of marbles you grab. The answer depends crucially on this and, unfortunately, there are myriad possibilities, so you cannot really leave it up to us to suggest what would be "correct." There's no way to know. If this question is inspired by an actual problem you have, then please consider describing that problem. $\endgroup$ – whuber Jan 1 '18 at 23:35
  • $\begingroup$ I think that is kind of what I am trying to determine myself. The initial assumption is that all marbles are equally probable to draw (I'll include this in the next edit). So the probability to grab any given marble is 1/n. To explain the particular problem would involve a fair amount of briefing on biology. So, I am trying to simplify as much possible to avoid that. Essentially, I am trying to provide a numerical argument that there exists a bias towards certain marbles and to quantify how strongly that bias is because I know their is one intuitively, like in my example. $\endgroup$ – terrigenus Jan 1 '18 at 23:48

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