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Please help me with the following probability problem and give some reference how to deal with such problems. Thank you.

In a pool, we have $200$ balls with numbers from $1$ to $200$. Each time, we randomly pick $20$ balls from the pool, write down their numbers, and return the balls to the pool. Assuming that the probability of selecting all the balls is equal, how many times do we have to repeat this sampling to be $99\%$ sure that the numbers of all the balls are written down?

The same problem but $90\%$ instead of $99\%$.

RobPratt
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Fermat
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  • Please [edit] the question to show us your work, and where you are stuck. Can you answer the question with much smaller numbers? – Ethan Bolker Jun 06 '21 at 18:35
  • I think if we assume that in the first experiment we select 20 balls with numbers $a_{1},...,a_{20}$, for the second experiment we need to select at least one of the balls that is not selected in the previous experiment and proceed in this way. But I am not sure this approach leads to the correct answer or not, as I have not studied probability for a long time. – Fermat Jun 06 '21 at 19:01
  • In my opinion, this is a very challenging problem. What is the source of the problem; is it from a book/class, or is it (for example) something that you made up or saw on the internet? – user2661923 Jun 06 '21 at 20:06
  • @user2661923 Someone asked me a few days ago but I don't know the original source of the problem. Another approach comes to mind is, if we assume that the number of experiments is $r$, and in i th experiment we randomly select the set $E_i$ of the balls( $E_i$s are not necessarily disjoint), we should determine $r$ in such a way that the $E_i$s union is equal to the set of all balls, considering that the probability is at least $99%$ . – Fermat Jun 06 '21 at 20:17
  • I am out of my depth here. – user2661923 Jun 06 '21 at 20:20
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    This is a variation of the Coupon Collector's Problem – awkward Jun 07 '21 at 18:19
  • Thank you very much @awkward. – Fermat Jun 08 '21 at 19:34

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As derived at Sample all elements from a set at least once, with replacement and Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen, if you have $n=200$ balls and you draw $m=20$ each time, the probability that you’ve seen all balls after $t$ draws is

$$ \sum_{j=0}^n(-1)^j\binom nj\left(\frac{\binom{n-j}m}{\binom nm}\right)^t\;. $$

With a bit of trial and error, you can use this to find that you need $72$ draws for the probability to be at least $90\%$ (Wolfram|Alpha computation) and $94$ draws for the probability to be at least $99\%$ (Wolfram|Alpha computation).

joriki
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