You sample n people, and a proportion p of them are found to be carrying a red flag (like some political party, prefer a brand of soap etc.). Textbooks say that the estimate of the proportion carrying the red flag in the total population is p with a variance of n.p.(1-p). This would indicate that p close to 0 or 100 pct can be estimated with smaller samples than p around 50 pct with the same confidence.
But suppose we have carried out these samplings repeatedly and past results show that the proportion carrying the red flag always comes in between 0 and say 15 pct. We can even estimate a histogram distribution of p from past samples. If we now make a new sampling of n items - and we wish to rely on the past sampling results, how would the mean and variance estimates change?