# calculating sample size needed for a target confidence interval from categorical data

So I will have data sorted into one of five bins (flow cytometry sorting of cells). These bins are actually adjacent categories that quantify a single fluorscence level (0-20,20-40,40-60,60-80,80-100). I am trying to associate specific levels of fluorscence (say 15%) with a specific DNA sequence (a mutation in a gene). The sorting must be done because to measure the DNA sequence requires capture of the actual cell (I have no way of knowing the exact fluorescence value to associate with a DNA sequence other than its capture via the sorting). My question is for one specific sequence, how many replicates will I need to determine what bin it belongs to with p < 0.05 ? I don't even know where to start in terms of statistical methods.

More difficult (impossible?): is it possible to infer a more exact level based on the frequency distribution that a sequence falls into. For example, a sequence associated with 20% repair will fall into the 0-20% and 20-40% bins with equal probability, so are there methods to infer this on more complex cases (a frequency distribution of results) with an associated confidence interval on that inference? i.e, a sequence appeared to be in the 20-40% category 10 times, in 0-20% five times, and in the 40-60% category once, from that I can say there is a 95% chance the value is between 20-35%. How would I calculate this?

Hope this is clear, let me know if not and I can try to clarify.

It seems this function would work once I have the data for my second question: https://rdrr.io/cran/DescTools/man/MultinomCI.html, but right now I want to get an estimate of minimum sample size for say, a confidence interval of +/- 10%