This and this are some related (but slightly different) questions. My question is more general.
I have a set of $N$ elements. I want to repeatedly draw $X$ items ($X \le N$) with replacement, i.e. I first choose $X$ items without replacement and then I replace them in the original set. I want each element to be drawn at least $K$ times. So how many times do I have to draw $X$ elements such that I have probability $P$ that each element has been drawn at least $K$ times?
Let's do an example. I have a deck of $N=10$ cards, labeled $1, 2, ..., 10$. At each step I draw $X=3$ cards, I look at them and then I put them back in the deck. How many times do I have to perform such step in order to draw each single card at least $K=5$ times with probability $P=0.95$?
