Every hour, a cell will attempt to clone itself. The cell will succeed with probability $p$ and fail and die with probability $1-p$. On a certain petri dish, there are $n$ cells. I am trying to find the expected number of cells $C_i$ after $i$ hours have elapsed using iterated expectation.
I already know that, say if there is $1$ cell at first, the expected number of cells after $1$ hour is $2(p) + 0(1-p) = 2p$. I also know that if there are $m$ cells at first, the expected number of cells after $1$ hour is $\sum_{i=1}^{m}E[C_1] = 2mp$ since whether each cell duplicates or dies is independent of other cells. Does this necessarily mean that the expected number after $2$ hours would be $E[C_2] = E[C_1]2p = 4mp^2$?
From the formula, the law of iterated expectation states that $E[C_i] = E[E[C_i|Y]]$. I'm a bit lost as to what exactly I should be conditioning $C_i$ on.