Given a function $G(x)= c \ln(1-{x \over 2})$ for $x \in [0,1]$ I first had to check for which $c$ the function $G$ is a probability generating function. I guess I need to check to property $G(1)=1$? If so, $G$ is a PGF for $c={1 \over \ln(0.5)}$.
Now for $X_c$ being the corresponding random variable, I need to calculate $Pr(X_c=1)$. I think I have to use the definition of the PGF and somehow get $G$ into a series expansion - but how to do so? Some help is much appreciated.
To recover probabilities from a PGF $G(x)$, use the relation $$\mathbb{P}(X = k) = \frac{G^{(k)}(0)}{k!}$$ where $G^{(k)}$ denotes the $k^{\textrm{th}}$ derivative of $G$.