Background: We begin with a population of one individual. Each individual has a probability $p$ to survive until it replicates into two independent and identical individuals.
Then the probability $\pi$ that the population ultimately goes extinct can be found by first step analysis over the first event: $$ \pi=p\pi^2+1-p \quad\Rightarrow\quad \pi=\begin{cases}1, & p<\frac12 \\ p^{-1}-1, & p\geq\frac12.\end{cases} $$
Likewise, if $p<\frac12$ such that extinction is assured, then the average number of replications $R$ that occur prior to extinction can be found by the same approach: $$ R=p(2R+1) \quad\Rightarrow\quad R=\begin{cases}\frac{1}{p^{-1}-2}, & p<\frac12 \\ \infty, & p\geq\frac12,\end{cases} $$
where the average number of replications $R$ is infinite if the extinction probability $\pi$ is nonzero. Instead, I would like to condition on extinction so that $R$ is always finite except at precisely $p=\frac12$.
Question: For $p>\frac12$, conditional on a population that ultimately goes extinct, what is the average number of replications $R$ that occur prior to extinction?
Numerics: In case it might be useful, I have simulated this birth-death process numerically and obtained the following table of $p$ and $R$ values:
