# Poisson or binomial confidence level for large samples and low observation rate, and low true value of p

Consider a more extreme case for a follow on to this question here about the Prussian soldier-face kicking example for Poisson random variables

Relating the binomial probability distribution to the Poisson Distribution in an example

Given an experiment to identify a probability p, for n trials, without prior knowledge of p... in cases where any where between 1, $$10^2$$, or $$10^3$$ over N samples may represent the success rate, and n observed events may range anywhere between $$10^3$$ - $$10^8$$. Events remain independent of other events throughout.

p may have true orders of magnitude of 0.005, 0.00001, but generally no less than $$10^-8$$, but there is a clearly an unknown order of magnitude to which p belongs.

Lets say I want to come up with a confidence level for p... and I do not know if I have sampled sufficiently in my experiment with N samples and observing n events.

1. Does a binomial confidence interval using incomplete beta distribution apply (Clopper-Pearson CI), or Inv-Chi Square distribution better represent the uncertainty / dispersion of the true mean? (As both represented in Univariate Discrete Distributions by Johnson, Kemp, and Kotz)
2. How does one know to assume a Poisson process, and that the sample number $$n\rightarrow\infty$$ applies for a general case, when it is unknown what the true number of N should be to approach the true probability $$p= n/N$$?

I recognize an estimator for p, is not the same as the true probability p as above I referred to them interchangeably.