Given an error bound, I can calculate the sample size to estimate proportion. E.g. for 10% the sample size is 100. But this relies on maximum variance being $\sqrt(0.25/n)$ which is achieved when p=0.5. Now say the result on the sample is 1%. So the actual p can be in the range (0%, 11%). But for this range, the variance is lower than the above and is $0.11*0.89/n=0.0979$. This yields that the error bound is in fact lower, leading to a lower maximum variance etc.
What is the formula to determine the actual error bound given a sample size and estimated proportion?