Number of succes S is a Binomal variable $n=50$ and $p=0.75$

Question

Approximate(with normal curve and correction for continuity) the probability that S is bigger then 39? Approximate Pr(S=40) and compare it with the exact calculation? R can be used

Answer 1

First, I have no idea what "correction for continuity" means here. Please explain.

Here is the discrete binomial, the continuous binomial (red), and approximating normal (green):

obtained with Mathematica

Show[
 DiscretePlot[
  PDF[BinomialDistribution[50, .75], k], 
  {k, 25, 50}],
 Plot[
Binomial[50, x] .75^x .25^{50 - x},
  {x, 25, 50},
  PlotStyle -> Red],
 Plot[
PDF[NormalDistribution[.75 50, Sqrt[50 (.75) (1 - .75)]], x],
  {x, 25, 50},
  PlotStyle -> Green]
 ]

The approximate probability $s>39$ comes from an error function for a given $s$:

$$0.5 - 0.11547 \left( \frac{(s - 37.5) \text{erf}\left( 0.23094 (37.5 -s)\right)}{8.66025 - 0.23094 s}\right)$$

and is:

(* 0.256815 *)

The exact probability that $x=40$ is:

PDF[BinomialDistribution[50, .75], 40]

(* 0.0985184 *)

The normal approximation:

PDF[NormalDistribution[.75 50, Sqrt[50 (.75) (1 - .75)]], 40]

(* 0.0933597 *)