# Listing B.20 Exact binomial sample size calculation
errorProb <- function(targetRate,difference,size) {
  pbinom(
    ceiling(( targetRate - difference) * size),
    size = size, prob = targetRate)
}

print(errorProb(0.045 , 0.004, est))    ## [1] 0.04153646

binSearchNonPositive <- function(fEventuallyNegative) {
  low <- 1
  high <- low + 1
  while(fEventuallyNegative(high) > 0) {
    high <- 2 * high
  }
  while(high > low + 1) {
    m <- low + (high-low) %/% 2
    if(fEventuallyNegative(m) > 0) {
      low <- m
    } else {
      high <-m
    }
  }
  high
}

actualSize <- function(targetRate, difference, errorProb) {
  binSearchNonPositive(function(n) {
    errorProb(targetRate, difference, n) - errorProb
  })
}

size <- actualSize(0.045, 0.004, 0.05)

print(size) ## [1] 7623 print(errorProb(0.045,0.004,size)) ## [1] 0.04983659