A population starts with a single amoeba. For this one and for the generations thereafter, there is a probability of $p$ that an individual amoeba will split to create two amoebas, and a $(1-p)$ probability that it will die out without producing offspring. Now, can we find a probability function at $n_{th}$ generation, which will let me know how many members will we find at that generation? For example, that at $2nd$ generation no member will survive has cumulative probability of $(1-p)+p(1-p)^2$. Also, what distribution will this follow?
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What exactly do you want? The expected number of amoebas in generation $n$? The probability that there are no amoebas in generation $n$? The probability distribution of generation $n$? The first question is easy; I'm not sure about the others. – saulspatz Jan 02 '21 at 21:45
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If I know the first one, I can know the second one. But I want to know if the first question follows any specific probability distribution. There is an answer below based on a recursive code, but I wanted to know if there was a generalised formula. – Elin Das Jan 03 '21 at 08:55
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There is only two choices, the offspring number is either $0$ or $2$. So we model the problems as such: for non negative integers $n,k$ with $k\le 2^n$ and where $2^n-k$ is even, let $f(n,k)$ be the probability that the $n$-th generation yields an amoeba population of size $k$.
Then $f$ can be computed recursively as in the Maple code shown below.
Applying the recursion, here are the results for $0\le n\le 3$: \begin{array}{|c|c|c|} \hline n&k&f(n,k)\\ \hline 0&1&1\\ \hline 1&0&1-p\\ &2&p\\ \hline 2&0&1-2p^2+p^3\\ &2&2p^2-2p^3\\ &4&p^3\\ \hline 3&0&1-4p^3+2p^4+4p^5-4p^6+p^7\\ &2&4p^3-4p^4-8p^5+12p^6-4p^7\\ &4&2p^4+4p^5-12p^6+6p^7\\ &6&4p^6-4p^7\\ &8&p^7\\ \hline \end{array}
