I am trying to understand the stochastic model of growth cell.
In this model the cell cycle time $c$ is assumed to be constant. However. each cell independently of every other cell gives birth to a random number $0$,$1$ or $2$ offspring with probabilities $p_0$,$p_1$,$p_2$ respectively.
From the fig. it is evident that the number of offspring in the first generation is not determined with certainty, but only that $Z_1 = 0,1,2$ with probabilities $p_0$,$p_1$ ,$p_2$ respectively.
Now, here the situation is more complicated for generation $2$.The book said that the probabilities of $Z_2$ (where the number of descendants $Z_k$ of a single cell or ancestor in the $k$th generation is not a constant but a random variable)
Here, I really don't understand how they get these probabilities: $$P(Z_2=0) = p_0+p_1p_0+p_2p_0^2,$$ $$P(Z_2=1) = p_1^2+2p_2p_1p_0,$$ $$P(Z_2=2) = p_1p_2+2p_2^2p_0+p_2p_1^2,$$ $$P(Z_2=3) = 2p_1p_2^2,$$ $$P(Z_2=4) = p_2^3,$$
I am biologist, and this is a little bit difficult, Could someone can help me to know how the book get these probabilities?
Thank you in advance for your time and help.
As an illustrative example of the first case using conditional probability
$$P(Z_2=0) \\ = P(Z_1=0,Z_2=0) \qquad + \qquad P(Z_1=1,Z_2=0) \qquad+ \qquad P(Z_1=2,Z_2=0) \\ = P(Z_1\!=\!0)P(Z_2\!=\!0 \!\mid\! Z_1\!=\!0) \!+\! P(Z_1\!=\!1)P(Z_2\!=\!0 \!\mid\! Z_1\!=\!1) \!+\!P(Z_1\!=\!2)P(Z_2\!=\!0 \!\mid\! Z_1\!=\!2) \\= p_0 \qquad\times\qquad 1 \qquad+\qquad p_1 \qquad\times\qquad p_0 \qquad+\qquad p_2 \qquad\times\qquad p_0^2 \\= p_0+p_1p_0+p_2p_0^2$$