# Binomial Distribution Calculator

 Enter Number of Occurrences (n) Enter probability of success (p) Enter Number of successes (k) Moment Number (t) (Optional)

A binomial distribution has a probability of success = 0.8. Calculate the probability of exactly 3 successes in 8 trials:

The binomial probability formula is as follows:
 f(k;n,p) = n! * pkqn - k k!(n - k)!

Calculate q:
q = 1 - p (q represents the probability of failure)
q = 1 - 0.8
q = 0.2

Calculate n!:
n! = 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8! = 40320

Calculate k!:
k! = 3!
3! = 3 * 2 * 1
3! = 6

Calculate (n - k)!:
(n - k)! = (8 - 3)!
(n - k)! = 5!
5! = 5 * 4 * 3 * 2 * 1
5! = 120

Take our pieces and calculate the binomial probability:
 P(X = 3) = 8! * 0.830.2(8 - 3) 3!(8 - 3)!

 P(X = 3) = 40320 * 0.512 * 0.25 6 * 120

 P(X = 3) = 40320 * 0.512 * 0.00032 720

 P(X = 3) = 6.60603 720

P(X = 3) = 0.0092

In Microsoft Excel or , you write this function as =BINOMDIST(3,8,0.8,FALSE)

Calculate the mean μ (expected value):
μ = np
μ = 8 x 0.8
μ = 6.4

Calculate nq to see if we can use the Normal Approximation:
Since q = 1 - p, we have n(1 - p) = 8(1 - 0.8)
nq = 8(0.2)
nq = 1.6

Calculate the variance σ2
σ2 = np(1 - p)
σ2 = 8 x 0.8 x (1 - 0.8)
σ2 = 6.4 x 0.2
σ2 = 1.28

Calculate the standard deviation σ:
σ = √σ2 = √np(1 - p)
σ = √1.28
σ = 1.1314

Calculate Skewness:
 Skewness = 1 - 2p √np(1 - p)

 Skewness = 1 - 2(0.8) √8(0.8)(1 - 0.8)

 Skewness = 1 - 1.6) √8(0.8)(0.2)

 Skewness = -0.6 √1.28

Skewness = -0.46875

Calculate Kurtosis:
 Kurtosis = 1 - 6p(1 - p) np(1 - p)

 Kurtosis = 1 - 6(0.8)(1 - 0.8) 8(0.8)(1 - 0.8)

 Kurtosis = 1 - (4.8)(0.2) 8(0.8)(0.2)

 Kurtosis = 1 - 0.96 1.28

 Kurtosis = 0.04 1.28

Kurtosis = 0.03125