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 $99\%$.
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).