Binomial Distribution Calculator

A binomial distribution has a probability of success = 0.4

Calculate the probability of you having at least 3 successes in 10 trials:

Binomial probability formula

f(k;n,p) =	n! * p^kq^{n - k}
	k!(n - k)!

P(x >= 3) = 1 - P(x < 3) ΣP(x = k) where (0 <= k <= 3)

Calculate q:

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

q = 1 - 0.4

q = 0.6

Calculate n!:

n! = 10!

10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

10! = 3628800

Calculate P(x = 0)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 0!

0! = 1

Calculate (n - k)!:

(n - k)! = (10 - 0)!

(n - k)! = 10!

10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

10! = 3628800

Take our pieces and calculate the binomial probability:

P(x = 0) =	10! * 0.4⁰0.6^{(10 - 0)}
	0!(10 - 0)!

P(x = 0) =	3628800 * 1 * 0.6¹⁰
	1 * 3628800

P(x = 0) =	3628800 * 1 * 0.0060466176
	3628800

P(x = 0) =	21941.96594688
	3628800

P(x = 0) = 0.006

Calculate P(x = 1)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 1!

1! = 1

Calculate (n - k)!:

(n - k)! = (10 - 1)!

(n - k)! = 9!

9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

9! = 362880

Take our pieces and calculate the binomial probability:

P(x = 1) =	10! * 0.4¹0.6^{(10 - 1)}
	1!(10 - 1)!

P(x = 1) =	3628800 * 0.4 * 0.6⁹
	1 * 362880

P(x = 1) =	3628800 * 0.4 * 0.010077696
	362880

P(x = 1) =	14627.97729792
	362880

P(x = 1) = 0.0403

Calculate P(x = 2)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 2!

2! = 2 * 1

2! = 2

Calculate (n - k)!:

(n - k)! = (10 - 2)!

(n - k)! = 8!

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

8! = 40320

Take our pieces and calculate the binomial probability:

P(x = 2) =	10! * 0.4²0.6^{(10 - 2)}
	2!(10 - 2)!

P(x = 2) =	3628800 * 0.16 * 0.6⁸
	2 * 40320

P(x = 2) =	3628800 * 0.16 * 0.01679616
	80640

P(x = 2) =	9751.98486528
	80640

P(x = 2) = 0.1209

Calculate P(x = 3)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 3!

3! = 3 * 2 * 1

3! = 6

Calculate (n - k)!:

(n - k)! = (10 - 3)!

(n - k)! = 7!

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

7! = 5040

Take our pieces and calculate the binomial probability:

P(x = 3) =	10! * 0.4³0.6^{(10 - 3)}
	3!(10 - 3)!

P(x = 3) =	3628800 * 0.064 * 0.6⁷
	6 * 5040

P(x = 3) =	3628800 * 0.064 * 0.0279936
	30240

P(x = 3) =	6501.32324352
	30240

P(x = 3) = 0.215

Calculate cumulative probability

P(x > 3) = 1 - (P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3))

P(x > 3) = 1 - (0.006 + 0.0403 + 0.1209 + 0.215)

P(x > 3) = 1 - 0.3822

Excel or Google Sheets formula:

Excel or Google Sheets formula:=1-BINOMDIST(3,10,0.4,TRUE)

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

Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4)

nq = 10(0.6)

nq = 6

Calculate the mean μ (expected value)

μ = np

μ = 10 x 0.4

μ = 4

Calculate the variance σ²

σ² = np(1 - p)

σ² = 10 x 0.4 x (1 - 0.4)

σ² = 4 x 0.6

σ² = 2.4

Calculate the standard deviation σ

σ = √σ² = √np(1 - p)

σ = √2.4

σ = 1.5492

Calculate Skewness:

Skewness =	1 - 2p
	√np(1 - p)

Skewness =	1 - 2(0.4)
	√10(0.4)(1 - 0.4)

Skewness =	1 - 0.8)
	√10(0.4)(0.6)

Skewness =	0.2
	√2.4

Skewness = 0.083333333333333

Calculate Kurtosis:

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

Kurtosis =	1 - 6(0.4)(1 - 0.4)
	10(0.4)(1 - 0.4)

Kurtosis =	1 - (2.4)(0.6)
	10(0.4)(0.6)

Kurtosis =	1 - 1.44
	2.4

Kurtosis =	-0.44
	2.4

Kurtosis = -0.18333333333333

Final Answer

Kurtosis = -0.18333333333333

You have 2 free calculationss remaining

What is the Answer?

Kurtosis = -0.18333333333333

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! * p^kq^{n - 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 distribution: discrete probability distribution of the number of successes in a sequence of n independent experiments, with a success or failure outcome
continuity correction factor: the bridge between the continuous normal distribution and the discrete binomial
event: a set of outcomes of an experiment to which a probability is assigned.
factorial: The product of an integer and all the integers below it
mean: A statistical measurement also known as the average
moment: a function are quantitative measures related to the shape of the functions graph
probability: the 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 Outcomes
standard deviation: a measure of the amount of variation or dispersion of a set of values. The square root of variance
variance: How far a set of random numbers are spead out from the mean

Example calculations for the Binomial Distribution Calculator

n=10, p=0.4, x=3

Binomial Distribution Calculator Video

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