= Binomial thinning of a geometric random variable
{title2=$p'=\frac{p}{p+r(1-p)}$}
Use the zero-based <geometric distribution>, with $\mathbb P(N=k)=p(1-p)^k$ for $k\geq0$ and $0<p<1$. After <binomial thinning> with retention <probability> $0\leq r\leq1$, the retained count is again zero-based <geometrically distributed>, with the displayed success <probability>. Indeed, writing $q=1-p$, its <probability generating function> is
$$
G_Y(s)=\frac{p}{1-q(1-r+rs)}=\frac{p'}{1-(1-p')s}.
$$
For $r=0$ this includes the degenerate success parameter $p'=1$ and retained count zero.
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