The plot below shows the sample size needed to detect a proportion with a precision 0.01 for various true proportions:
This assumes an infinite population size, and the confidence intervals are fixed at 0.95.
If the true proportion is very small (or equivalently, very large) then we need a sample size of under 1000 to detect the proportion. However, if the true proportion is 0.5 then we need a sample size of almost 10,000.
I understand mathematically why this is from using the formula to calculate population size. But I would like an intuitive understanding of why this is the case.
It seems more intuitive to me that small proportions would need larger samples to detect them. Is this because I have kept the precision fixed (1% ± 1% is a much larger percentage error than 50% ± 1%)?