What are the odds of hitting at least M cards in a deck in Y showings of Z cards on each showing? Say that you have a deck of cards that contains O cards. You want to be able to take at least M cards out of a deck that fit a certain criteria (i.e. they need to be aces or they need to be spades). Let's say that there are N such cards that fit the criteria (where N >= M) in the deck.
During a showing of cards, someone takes Z cards from the deck and places them face up. If you see a card that you want during this showing, you will take it into your possession and remove it from the deck. After the showing, the unclaimed cards will be shuffled back into the deck. Any removed cards are permanently gone from the deck for future showings.
If there are Y of these showings of Z cards, what are the odds that you will have at least X cards that fit the criteria at the end?
 A: For it to be always possible for $\ Z\ $ cards to be shown, we have to assume that $\ X\le O-Z+1\ $, so that the number of cards collected before the final showing can never exceed $\ O-Z\ $.
If, at any stage, you have collected $\ c< X\ $ of the cards satisfying the given criteria, then the probability that you will collect $\ k\le N-c\ $ more on the next showing is
$$
\frac{{N-c\choose k}{O-N\choose Z-k}}{O-c\choose Z}\ .
$$
You can thus treat $\ c\ $ as the state of a time-homogeneous Markov chain whose initial state is $\ c=0\  $, and the entries of whose $\ (N+1)\times(N+1)\ $ transition matrix $\ P\ $ are given by
$$
p_{cd}=\cases{0& if $\ 0\le d<c<X\ $, $\ d>c+Z\ $,\\
& or $\ c\ge X\ $ and $\ d\ne c$\\
\frac{{N-c\choose d-c}{O-N\choose Z-d+c}}{O-c\choose Z} & if $\ 0\le c\le d\le c+Z\ $ and $\ c<X\ $\\
1& if $\ c=d\ge X\ $,}
$$
with states $ X,X+1, \dots,N\ $ being absorbing.
After $\ Y\ $ showings, the state of the chain will be
$$
f=\pmatrix{1&0&\dots&0}P^Y
$$
and the probability that at least $\ X\ $ of the desired cards have been collected is $\ \displaystyle\sum_{k=X}^Nf_k\ $, so the odds in favour are
$$
\frac{\displaystyle\sum_{k=X}^Nf_k}{1-\displaystyle\sum_{k=X}^Nf_k}\ .
$$
