For the stochastic model, we will use a continuous time birth-death process. In this context, we'll often refer to $\lambda$ as the birth rate and $\mu$ as the death-rate. For convenience, we can use the language of rates, rather than probability per day, saying each cell divides at the rate of $\lambda=0.01$ per day and dies at the rate $\mu=0.005$ per day. The rate language makes even more sense when we talk about the whole population. If there are $c(t)$ cells, then the overall birth rate is $\lambda c(t)$ and the overall death rate is $\mu c(t)$.
To create a model, we will assume that a birth or death of a cell is independent of the behavior of all other cells, as well as independent of its history. (A newly born cell is just as likely to die or give birth as an older cell, and the timing of its birth or death is not influenced by the times of other cells' births or deaths.) The number $c(t)$ will jump up or down by one when a random event of a birth or death occurs. Since the events are happening in continuous time, there is a probability of zero that two events occur exactly at the same time and cause $c(t)$ to jump up or down by two.
Write down a stochastic model for the growth of $c(t)$ in continuous time. This model will allow us to march forward in time, changing $c(t)$ at every moment. We formulate the model by assuming that at time $t$, the cell count is $c(t)=n$. Then, we will write down a probability distribution for the number of cells a short time later, conditioned on the fact that $c(t)=n$. Let the length of this short interval be $\Delta t$. Hence, we specify the model by prescribing the conditional distribution
$$p(c(t+\Delta t) = m \,|\, c(t) = n).$$
In words, we assign a probability that the number of cells changes from $n$ to $m$ in a short interval $\Delta t$.
Since we are thinking of $\Delta t$ as being really small, and no two events can occur at the same time, we can ignore the possibility that two events occurred in the time interval $\Delta t$. In your model, allow $m=c(t+\Delta t)$ to be at most one different from $n=c(t)$. Your model should be based on the overall birth rate $c(t)\lambda$ (which you'll write as $\lambda n$) and the overall death rate $c(t)\mu$ (which you'll write as $\mu n$).
Hint
Since $\lambda$ is the birth rate, or the probability per day of a birth, if one multiplies by a short time interval $\Delta t$, then $\lambda \Delta t$ is the probability of a birth happening that that interval. If there are $n$ cells, each with a birth rate of $\lambda$, what is the probability that one of those cells had a birth in a short interval $\Delta t$? If at time $t$, there are $c(t)=n$ cells, how cells are there at time $t+\Delta t$ when there is a birth?
Since $\Delta t$ is small, it is most likely that there were no births or deaths in the interval. What is the probability that there were no births or deaths?
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