binomial distribution confidence interval versus successes

Question

Can you explain why the confidence interval of binomial distribution shrinks with the number of successes terms?

n <- 100
res <- vector(length = n)
for ( i in 1:n) {
  p <- binom.test(i, 1000, conf.level = 0.95, alternative = "two.sided")
  res[i] <- diff(p$conf.int)
}
plot(res, type = "l", ylim=c(0, 0.09))
grid()

For example single success has 5x less wide confidence interval than 50 successes. I would have thought large number of successes should have a narrower confidence interval than single success. I am aware of the variance formula of $p(1-p)/N$ but it sounds a bit counterintuitive when less number of success are having a narrower confidence band

Answer 1

Since you already know the mathematical explanation I will try to give an intuitive one. 50 successes out of 100 could be generated by a wide range of success parameters, but 1 success out of 100 is close to the boundary of the success parameter at 0. Large success parameters are unlikely with only one success, whereas smaller ones are just mathematically impossible. These restrictions lead to a narrow confidence interval.

Note also that the confidence interval is only narrower on the linear scale. On this scale success parameters of 0.01, 0.001 and 0.0001 are close together, although they represent vastly different success probabilities. If the true success parameter is 0.5, 51 successes out of 100 trials is a very likely outcome. If the true success parameter is 0.01, then 2 successes out of 100 is much less likely, since this means twice as many successes as expected. Still the absolute difference in success probability is the same. Mostly in statistics one compares ratios of probabilities rather than differences for this reason.