At time 0, a cell culture starts with one red cell. At the end of one minute, the red cell dies and is replaced by 2 red cells with probability $\frac{1}{4}$, with 1 red and 1 white cell with probability $\frac{2}{3}$, and with 2 white cells with probability $\frac{1}{12}$. Each red cell lives for one minute and gives birth to offspring in the same way as the parent cell. Each white cell lives for one minute and dies without reproducing.
I need to find a probability generating function for this. I don't have any problems with finding the pgf; however, I am unsure what the offspring distribution is to be used in the pgf summation. What I am thinking is this:
Let $X$ be the number of off-springs. Then $ \mathbb{P}(X=0) = \frac{1}{12} \qquad \mathbb{P}(X=1) = \frac{2}{3} \qquad \mathbb{P}(X=2) = \frac{1}{4} $ I am unsure if this is correct because if we have 2 red cells, then won't the total number of offspring be 4? So don't we need an $X=4$ case?
If I am correct with my distribution, then the pgf will be $\frac{1}{12} + \frac{2}{3} s + \frac{1}{4} s^2$.
Thanks for the help.
EDIT:
I am trying to figure out the probability that the entire culture dies out (using the pgf). I am not exactly sure what random variable I need for that...