Basic Sampling - Provide confidence of estimate I tried searching the site but nothing came up.
I have a simple situation for a business problem - We have a population of 50 million files. We want to review a sample and see whether a file contains or does not contain "x". So it's binary. We will calculate a proportion of yes:no.
We want to be able to quantify how accurate our estimate is of the true population. "We are 95.. 99% confident that the true proportion is "x/y" +/- x%. From my research, confidence intervals cannot be interpreted in such a way. Rather, confidence intervals are supposed to be interpreted as "in the long-run, if we conduct many experiments, we are 95% confident that the CIs calculated for those samples will contain the true population parameter". Do we just take an arbitrary "large enough" random sample, calculate the proportion and assume that is the true proportion? I came across the binomial credible interval that can be interpreted as the former, but I know nothing of Bayesian inference unfortunately.

Confidence intervals can absolutely provide the inference you are interested in.  A confidence interval is a set of plausible hypotheses in the parameter space given the observed data.  The confidence level is based on the frequency probability of the experiment.  This allows you to make the claim, "I am 95% confident that the CI computed from my sample covers the unknown fixed true population parameter."  It's analogous to knowing the bias of a coin is 0.95 in favor of heads (95% of the time the coin lands heads) and the confidence this knowledge of the long-run proportion imbues regarding the outcome of a single flip.  If asked how confident you are that the coin will land heads (or has already landed heads), you would say you are 95% confident based on its long-run performance.
Often times papers are published with examples that do not use all of the available information in the likelihood when constructing a confidence interval and this is presented as evidence against using a confidence procedure.  This is analogous to using only a fraction of the information in the likelihood when constructing a posterior credible interval.  Both the credible interval and the confidence interval reside in the parameter space.  The choice between constructing a Bayesian credible interval or a frequentist confidence interval is a matter of what you want to measure using probability, the experimenter or the experiment.
The most precise solution to your problem is to construct a confidence interval for the population proportion by inverting the CDF of a binomial distribution.  Here are some related threads that discuss frequentist confidence intervals for a Bernoulli proportion compared to Bayesian credible intervals when the unknown fixed true proportion is near 0 or 1 [1], [2].
