Hypergeometric Distribution Calculator

Publish date: 2024-06-04

Using a hypergeometric distribution, calculate the Probability that if you choose 10 items out of 60 with a success count of 24,
you will get 4 items of the success count.

The hypergeometric probability is listed below:

P(x;n,N,k) =	(kCx) * (N - kCn - x)
	NCn

Calculate Numerator 1

kCx =	k!
	x!(k - x)!

24C4 =	24!
	4!(24 - 4)!

Calculate k!:

24! = 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
24! = 6.2044840173324E+23

Calculate x!:

4! = 4 x 3 x 2 x 1
4! = 24

Calculate (x - k)!:

k - x = 24 - 4 = 20
20! = 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
20! = 2432902008176640000

24C4 =	6.2044840173324E+23
	24(2432902008176640000)

24C4 =	6.2044840173324E+23
	5.8389648196239E+19

24C4 = 10626

Calculate Numerator 2

N - kCn - x =	(N - k)!
	(N - k - n + x)!(n - x)!

60 - 24C10 - 4 =	(36)!
	(60 - 24 - 10 + 4)!(10 - 4)!

36C6 =	(36)!
	(30)!(6)!

Calculate 6!:

6! = 6 x 5 x 4 x 3 x 2 x 1
6! = 720

Calculate 36!:

36! = 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
36! = 3.719933267899E+41

Calculate 30!:

30! = 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
30! = 2.6525285981219E+32

36C6 =	3.719933267899E+41
	720(2.6525285981219E+32)

36C6 =	3.719933267899E+41
	1.9098205906478E+35

36C6 = 1947792

Calculate Denominator

NCn =	N!
	m!(N - m)!

60C10 =	60!
	10!(60 - 10)!

Calculate N!:

60! = 60 x 59 x 58 x 57 x 56 x 55 x 54 x 53 x 52 x 51 x 50 x 49 x 48 x 47 x 46 x 45 x 44 x 43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
60! = 8.3209871127414E+81

Calculate n!:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
10! = 3628800

Calculate (N - n)!:

50! = 50 x 49 x 48 x 47 x 46 x 45 x 44 x 43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
50! = 3.0414093201713E+64

60C10 =	8.3209871127414E+81
	3628800(3.0414093201713E+64)

60C10 =	8.3209871127414E+81
	1.1036666141038E+71

60C10 = 75394027566

Calculate Probability

P(4;10,60,24) =	10626 x 1947792
	75394027566

P(4;10,60,24) =	20697237792
	75394027566

P(4;10,60,24) = 0.2745

You have 1 free calculations remaining

Calculate the mean μ:

μ =	nk
	N

μ =	10 x 24
	60

μ =	240
	60

μ = 4

Calculate the variance σ2

σ2 =	nk(N - k)(N - n)
	N2(N - 1)

σ2 =	(10)(24)(60 - 24)(60 - 10)
	602(60 - 1)

σ2 =	(240)(36)(50)
	3600(59)

σ2 =	432000
	212400

σ2 = 2.0339

Calculate the standard deviation σ:
σ = √σ2
σ = √2.0339
σ = 1.4261

What is the Answer?

P(4;10,60,24) = 0.2745

How does the Hypergeometric Distribution Calculator work?

Free Hypergeometric Distribution Calculator - Calculates the probability of drawing x objects out of a subgroup of k with n possibilities in a total group of N using the hypergeometric distribution.
This calculator has 4 inputs.

What 3 formulas are used for the Hypergeometric Distribution Calculator?

P(x;n,N,k) = (kCx) * (N - kCn - x)/NCn
μ = nk/N
σ2 = nk(N - k)(N - n)/N2(N - 1)

For more math formulas, check out our Formula Dossier

What 10 concepts are covered in the Hypergeometric Distribution Calculator?

combinationa mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter
nPr = n!/r!(n - r)!distributionvalue range for a variableeventa set of outcomes of an experiment to which a probability is assigned.factorialThe product of an integer and all the integers below ithypergeometric distributiondiscrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in ndraws, without replacementmeanA statistical measurement also known as the averagepermutationa way in which a set or number of things can be ordered or arranged.
nPr = n!/(n - r)!probabilitythe 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