Question

A single cell can either die, with probability .1, or split into two cells, with probability .9, producing a new generation of cells. Each cell in the new generation dies or splits into two cells independently with the same probabilities as the initial cell. Find the probability distribution for the number of cells in the next generation.

Answer 1

its kind of like the binomial distribution, except each generation must have an even number of cells

\[\large P(X=k) = \left(\begin{matrix}n \\ k/2\end{matrix}\right)(0.9)^{k/2} (0.1)^{n-k/2}\]
where 0<k<2n and n is number of cells in previous generation
k must be even