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

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

• What have you tried so far? It looks from the problem posed that you are asking this site to do a homework problem for you, which is OK as long as you have tried and just need help to get over some sticking point. – Mark Fischler Dec 5 '19 at 20:47
• What have you tried? – Ahmad Bazzi Dec 5 '19 at 21:32

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 *)