Suppose we have a set of $N$ numbers. At any given trial we can randomly choose $N^{1-a}$ of the numbers where $a\in(0,1)$. We replace the numbers back.

How many trials does it take in average case to have chosen all numbers?

If we use only $m$ trials what is the probability that we will leave out only $N^{b}$ of the set not covered where $b\in(0,1)$ holds?