diff(pbinom(c(44,55),100,.5))
[1] 0.728747

diff(pbinom(c(449,550),1000,.5))
[1] 0.9986083

par(mfrow=c(2,1))

x = 0:100;  PDF = dbinom(x, 100, .5)
plot(x, PDF, type="h", lwd=2, main="BINOM(100,.5) with Approx. Normal Density")
  abline(h = 0, col="green2")
  abline(v = c(44.5,55.5), col="red", lwd=2, lty="dotted")
  curve(dnorm(x, 50, 5), add=T, lwd=2, col="blue")

x = 400:600;  PDF = dbinom(x, 1000, .5)
plot(x, PDF, type="h", main="BINOM(1000,.5) with Approx. Normal Density")
  abline(h = 0, col="green2")
  abline(v = c(449.5,550.5), col="red", lwd=2, lty="dotted")
  curve(dnorm(x, 500, sqrt(250)), add=T, lwd=2, col="blue")

par(mfrow=c(1,1))