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Coverage After Sampling With Replacement $k$ Times

We have $n$ number of balls in a box, each has a unique label. $\{1, 2, ..., n\}$.

I want to randomly select a ball, record the label and place it back in the box. This action is repeated for $k$ times, and $k \geqslant n$. If the largest label in all previous samples is $v$, then the current label can be anything from $1$ to $v+1$. And we want the outcomes where the labels start from $1$ and have reached $n$ at least once.

For examples, the balls are $\{1, 2, 3\}$ and sampled $4$ times. Although there are $3^4$ outcomes, the desired outcomes are $(1, 1, 2, 3), (1, 2, 1, 3), (1, 2, 2, 3), (1, 2, 3, 1), (1, 2, 3, 2)$ and $(1, 2, 3, 3)$.

The questions are:

  1. what is the average value of $k$ so that after $k$ samples, we have at least one desired outcome?

  2. what is the probability that after $k$ samples, we have at least one desired outcome?

Mitch Huang
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  • Your question has a number of defects that will influence mathSE reviewers to react negatively rather than positively. If you are interested in reversing this reaction, see this article. – user2661923 Dec 11 '21 at 07:00
  • It looks as if the original balls' labels are not the labels you record (you can only label the first sample as $1$, and the second as $1$ if you see the same ball again or $2$ if you see a new one). A desired outcome is having seen each ball at least once, and the answer will be related to Stirling numbers of the second kind. This is has been asked many times before, sometimes more generally such as at https://math.stackexchange.com/questions/32800/probability-distribution-of-coverage-of-a-set-after-x-independently-randomly – Henry Dec 11 '21 at 10:21
  • If you first question is the expected number sampled when you first have a desired outcome, then this looks like the coupon collector's problem – Henry Dec 11 '21 at 17:55
  • Hi @Henry, I've read the coupon collector's problem and other related threads. Let me rephrase: – Mitch Huang Dec 12 '21 at 19:55
  • Let me rephrase: The first ball I draw is 1 and I place it back. I want to draw a 2 in the next attempt, otherwise I will draw again and again until I get 2. After a few attempts, I successfully draw a 2, in the next draw I want a 3 and I will keep trying until I get 3. Eventually, my desired result should at least includes one '4' ball. And the question becomes: averagely after how many attempts I can get a desired result. – Mitch Huang Dec 12 '21 at 20:07
  • The problem might look like a 'ordered' coupon collector's problem. I am not sure if that is the way to describe the problem but it is the closest thing I can think of. – Mitch Huang Dec 12 '21 at 20:08

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