Let's say I own a bakery that, among other desserts, sells one really tasty chocolate cake. It's so good that I've estimated that I've estimated 80% of my daily customers will buy a piece of cake, that is $p=0.8$.
However, due to special ingredient requirements and intensive labor, I can only make a single cake per day before open. In order to distribute equally, I count the number of customers waiting at the door and slice my single cake into $N$ discrete pieces, where $N \sim Uniform \{1,1000\}$. This means that on any given day, I will sell between 0% and 100% of my chocolate cake, with a mean of 80%=p. But of course if N=1 on any day, it becomes binary and I would never sell anything other than all or nothing.
I want to get an approximate range for my expected revenue on any given day/month/year. How might I go about estimating some sort of interval for the proportion of my cake I will sell on any given day?
As an additional complication, how would that prediction interval change if $p$ is now a prediction instead of known/estimated? For example, what if via some good-but-not-perfect predictive model, I predict that 90% of my Saturday customers will want cake instead of 80%?
There are some other questions asked on here that are somewhat in the same topic but not quite asking the same questions, for example here, and here.
I have been able to get estimates by simulating from assigned/estimated distributions for $p$ and $N$ many times and looking at the results as if a bootstrap, but wondering if there are any methods that are more rigorous, especially if the distributions for $p$ or $N$ are predicted with error from a model. This seems like it could maybe be done with some sort of beta distribution for the proportion if some simplifying assumptions could be made, but I'm not sure how that would look.
