Say I am trying to predict an election (where there are 2 candidates) by sampling N likely voters at random from some underlying large population. Suppose my estimate of the probability of a voter voting for candidate A is p and for candidate B it is (1-p). Then my confidence interval centered at the mean is :
alpha * sqrt(p * (1-p)/N)
Intuitively why should the length of the confidence interval depend on the estimate of p? If my sample size is very small and I get p = 0, the confidence interval is of length 0, why does this make sense?