Odds of choosing n members of a x sized subset of a y sized set.

Odds of choosing n members of a x sized subset of a y sized set.

Lets say I have a bag of 8 rocks. 3 rocks are red, the rest are black. I
choose, randomly, with replacement, 3 rocks. What are the odds that at
least one that I choose is red?
It seems in first pass that the odds if I draw just one time, is 3/8. If I
draw twice, then, would the odds be 6/8? Doesn't seem so, since drawing 3
times would by that reasoning given 9/8 > 1.0, so somewhere I've gone
wrong.
To expand to the general case, how do I determine the odds of choosing n
members of a x sized subset of a y sized set, if I draw z times with
replacement. The real world problem I'm really trying to solve involves
figures on the order of thousands.
Apologies if some of my terminology is off. Been a few years since I took
discrete...