I have a problem that reduces to balls in urns (it is actually about reference and alternate alleles in populations).
Assume I have a large well-mixed urn (i.i.d draws) that can contain two colors of balls: aquamarine and robin's egg blue (a and r respectively). They are close in color, so sometimes a person classifying them makes a mistake identifying the color after drawing a ball from an urn. Let $e_r$ be the probability of an error when the ball is really r and $e_a$ when the ball is really a. Assume I know these numbers (I think they're less than 0.01 but still need to check) and I have chosen a significance.
In an experiment, my companion draws $n$ balls from the urn and identifies $r$ balls as color r and $a$ as a ($n=r+a$). He then tells me $r$ and $a$. I want to test $H_0$ that all balls are r versus $H_a$ the urn contains at least one a ball given the numbers of balls drawn.
My goal is to perform the test at 2 different levels to give a "star" rating to the strength of the reported results. Could not reject at 0.05 = 2 stars, rejected at 0.05 = 3 stars and rejected at 0.01 = 4 stars.
What test can I use for this problem? (Though I've put this in conventional terms, I'd be happy with getting a Bayes factor and setting up thresholds based on that. I'm also happy with tests that require a certain number of measurements for validity - I can just classify samples that are too small as "could not reject")
Note this is different than testing a proportion because those tests don't have error in measurement (and don't work for proportion = 0 or 1). I thought of trying to set a non-zero $H_0$ proportion using some kind of fudge factor based on the error rate and the sample size(e.g. testing $H_0=P \le e_r$ where $P$ is the true proportion, but I couldn't come up with a well-justified number). I also started trying to derive my own test, but it was taking quite a while and this seems like the kind of problem someone would have investigated before.
Edit Rewrote the question slightly to clarify that I don't know the sequence of draws/classifications