Assuming I have two kind of balls, red and blue ones, and I pick randomly (with replacement) always a bunch of balls where the number of balls I pick is distributed according to the Poisson distribution. Now, I'd like to sample from the probability that I pick in total $z$ balls under the condition that I picked at least $k$ blue balls.
So, when denoted $Z$ as the random variable denoting the number of balls I pick in total, $X$ as the red balls I pick and $Y$ as the blue balls I pick. Then, I know that $Pr[ Z=X+Y ] \sim Poisson(\lambda) $ and I'd like to get samples of $Pr[Z\vert Y \geq k]$.
How can I do this?
I tried with rejection sampling, meaning throwing away all samples where $Y<k$, but this corresponds to samples of $Pr[Z,Y\geq k]$, right?