I'm a little rusty on my probability theory, so this might be an easy question (or not). I'm interested in this for a personal project, not homework. Consider an urn with $N$ balls, each distinguished by a number $1...N$. I pick out balls from the urn indefinitely, for each pick/time step $t$ I record the ball extracted, and put it back in the urn. Given some integer value $c$, what is the probability that at time/pick $t$, I have picked any single/distinct ball for the $c$'th time?
To be concrete - suppose there are $N=1000$ balls, and $c=4$. I'm interested in the probability that at time $t$ I've picked the same ball (any one of them, not a specific ball) $4$ times. Obviously after $N*(c-1)+1=3001$ picks this will have happened surely, but this is likely to happen before, what is the probability for every time step $t$?
Thanks a lot for the help!