Normal Approximation to Binomial:

$X \sim$ Binomial $(n,p)$

if both $np>5$ and $n(1-p)>5$

$X \approx X^* \sim$ Normal $(\mu = np, \sigma^2 = np(1-p))$

R Code

n=40; p=0.5 #Binomial Dist. Parameters
n*p # >5
n(1-p) # >5

dbinom(18, size=n, prob=p) #yields 0.103 / 10.3%

#find P(X>24)
#approximate:
1 - pnorm(24.5, mean=n*p, sd=sqrt(n*p*(1-p)) #yields 0.07736
#actual:
sum(dbinom(25:n, size=n, prob=p)) #yields 0.07693

Sampling Distributions