Binomial Distribution Calculator

A binomial distribution has a probability of success = 0.2

Calculate the probability of you having at least 1 successes in 8 trials:

Binomial probability formula

f(k;n,p)  =	n! * pkqn - k
	k!(n - k)!

P(x >= 1) = 1 - P(x < 1) ΣP(x = k) where (0 <= k <= 1)

Calculate q:

q = 1 - p (q represents the probability of failure)

q = 1 - 0.2

q = 0.8

Calculate n!:

n! = 8!

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

8! = 40320

Calculate P(x = 0)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 0!

0! = 1

Calculate (n - k)!:

(n - k)! = (8 - 0)!

(n - k)! = 8!

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

8! = 40320

Take our pieces and calculate the binomial probability:

P(x = 0)  =	8! * 0.200.8(8 - 0)
	0!(8 - 0)!

P(x = 0)  =	40320 * 1 * 0.88
	1 * 40320

P(x = 0)  =	40320 * 1 * 0.16777216
	40320

P(x = 0)  =	6764.5734912
	40320

P(x = 0) = 0.1678

Calculate P(x = 1)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 1!

1! = 1

Calculate (n - k)!:

(n - k)! = (8 - 1)!

(n - k)! = 7!

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

7! = 5040

Take our pieces and calculate the binomial probability:

P(x = 1)  =	8! * 0.210.8(8 - 1)
	1!(8 - 1)!

P(x = 1)  =	40320 * 0.2 * 0.87
	1 * 5040

P(x = 1)  =	40320 * 0.2 * 0.2097152
	5040

P(x = 1)  =	1691.1433728
	5040

P(x = 1) = 0.3355

Calculate cumulative probability

P(x > 1) = 1 - (P(x = 0) + P(x = 1))

P(x > 1) = 1 - (0.1678 + 0.3355)

P(x > 1) = 1 - 0.5033

Excel or Google Sheets formula:

Excel or Google Sheets formula:=1-BINOMDIST(1,8,0.2,TRUE)

Calculate nq to see if we can use the Normal Approximation:

Since q = 1 - p, we have n(1 - p) = 8(1 - 0.2)

nq = 8(0.8)

nq = 6.4

Calculate the mean μ (expected value)

μ  =  np

μ  =  8 x 0.2

μ = 1.6

Calculate the variance σ2

σ2  =  np(1 - p)

σ2  =  8 x 0.2 x (1 - 0.2)

σ2  =  1.6 x 0.8

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(0.2)
	√8(0.2)(1 - 0.2)

Skewness  =	1 - 0.4)
	√8(0.2)(0.8)

Skewness  =	0.6
	√1.28

Skewness = 0.46875

Calculate Kurtosis:

Kurtosis  =	1 - 6p(1 - p)
	np(1 - p)

Kurtosis  =	1 - 6(0.2)(1 - 0.2)
	8(0.2)(1 - 0.2)

Kurtosis  =	1 - (1.2)(0.8)
	8(0.2)(0.8)

Kurtosis  =	1 - 0.96
	1.28

Kurtosis  =	0.04
	1.28

Kurtosis = 0.03125

Final Answer

Kurtosis = 0.03125

What is the Answer?

Kurtosis = 0.03125

How does the Binomial Distribution Calculator work?

Free Binomial Distribution Calculator - Calculates the probability of 3 separate events that follow a binomial distribution. It calculates the probability of exactly k successes, no more than k successes, and greater than k successes as well as the mean, variance, standard deviation, skewness and kurtosis.
Also calculates the normal approximation to the binomial distribution with and without the continuity correction factor
Calculates moment number t using the moment generating function
This calculator has 4 inputs.

What 3 formulas are used for the Binomial Distribution Calculator?

q = 1 - p
f(k;n,p) = n! * pkqn - k/k!(n - k)!
Z = X - np/√np(1 - p)

For more math formulas, check out our Formula Dossier

What 10 concepts are covered in the Binomial Distribution Calculator?

binomial distributiondiscrete probability distribution of the number of successes in a sequence of n independent experiments, with a success or failure outcomecontinuity correction factorthe bridge between the continuous normal distribution and the discrete binomialeventa set of outcomes of an experiment to which a probability is assigned.factorialThe product of an integer and all the integers below itmeanA statistical measurement also known as the averagemomenta function are quantitative measures related to the shape of the functions graphprobabilitythe likelihood of an event happening. This value is always between 0 and 1.
P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomesstandard deviationa measure of the amount of variation or dispersion of a set of values. The square root of variancevarianceHow far a set of random numbers are spead out from the mean

Example calculations for the Binomial Distribution Calculator

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