Given the number of draws $x$ from a binomial distribution with known probability parameter $\pi$, is there a function which gives distribution of likely $n$ from which these $x$ were sampled? For instance, let's say we have $x=315$ items randomly selected with known probability $\pi=0.34$ from a population of $n$ items. Here most likely value is $\hat{n}=926$ but what is probability distribution for $n$. Is there a distribution which gives $p(n)$?
I know that $p(\pi | x,n)$ is given by the beta distribution and that $p(x |\pi, n)$ is the binomial distribution. I'm looking for that third creature, $p(n |\pi, x)$, properly normalized of course such that $\sum_{n=x}^{\infty} p(n)=1$
first "attempt" at this, given the normal approximation to binomial distribution is $p(x|\pi, n)=\mathcal{N}(x/\pi,x\pi(1-\pi))$, is that $p(n|\pi,x)\approx\mathcal{N}(x/\pi,x\pi(1-\pi))$?