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Suppose we start with a total of $N$ offspring of a certain animal. So we start with $N$ and no new offspring is introduced. Each year each individual of the offspring dies with probability $p$ independently.

Let $S(n$) be the first year all of the offspring is dead. Give the distribution function of $S(n)$.

My attempt: I tried to compute the probability that all the offspring is dead in the year $k$. Since for any given animal the probability that it dies in year $K$ is equal to $(1-p^k)$. Then the probability that all of them are dead is equal to $(1-p^k)^n$.

Now i don't know how to make sure $k$ was really the first year that they were all dead.

(in the my attempt part i used p as the probability an individual survives the year, i made a mistake typing the question)

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    $\begingroup$ The probability that it dies in year $k$ is $(1-p)^{k-1}p$, the probability that it lives for $k-1$ years, and then dies in year $k$. $\endgroup$ – saulspatz Apr 16 '20 at 22:51
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For a given animal, we don't care what year it dies, merely that it is not alive in year $k$. This is the complement of the probability that it survives for $k$ years:$$ 1-(1-p)^k$$

The probability that all $n$ offspring die by year $k$ is $$P_k:=(1-(1-p)^k)^n\tag1$$

What you say at the end about the first year that all offspring are dead doesn't seem to have anything to do with the question posed, but if you wanted to compute it it is is $P_k-P_{k-1}$, where $P_k$ is defined in $(1)$.

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  • $\begingroup$ Could you explain why it is $P_k-P_{k-1}$? $\endgroup$ – Jed berry Apr 16 '20 at 23:27
  • $\begingroup$ @Jedberry It's the probability that all were dead by year $k$, but not but year $k-1$, so the last one died in year $k$. $\endgroup$ – saulspatz Apr 17 '20 at 1:49

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