I have the following generating function for stochastic process
$$\sum_n z^n p_n=\left[1-\frac{(1-z)}{\Lambda(t)}\right]^{n_0},$$
and I want to extract the probability $p_0(t)$ but I am confused how I could do this. So far I've thought of
$$p_0(t)=\sum_n z^n p_n \delta_{n, 0},$$
to extract it but that doesn't seem to help solve this. Any help would be appreciated.
Evaluate at $z=0$ in your first expression. Notice that for $n>0$, one has $0^n=0$ and the only term remaining is the constant term $p_0$.