I have been thinking recently if we assume that there is a finite population and one was taking some sort of survey. Something where it was a yes or no question which for these purposes, I am going to assign a 0 and 1. So, in this case, we have a binary system of possible responses. There are a large number of other cases where you could have this where a fractional average might mean more than what I've mentioned, but let's say we have two options: a response of 0 and a response of 1 and the population has to fall into either of these categories.
What I am wondering is: In this scenario, what can you get out of a sufficiently large data set that comes back as either all 0's or all 1's? Obviously, any sample that is not the entire population, if we assume it is 100% truthful responses, you don't know for certain the true value of probability of getting a 1 in the case that you have all 0's, but it also seems wrong to conclude that this value is 0 with an infinitely large confidence interval of 0.
So with that how does one determine what the probability that someone would respond with a 1 from a non-comprehensive set of data comprised of entirely 0's?