Hypergeometric Distribution Calculator

Publish date: 2024-06-27

Using a hypergeometric distribution, calculate the Probability that if you choose 10 items out of 80 with a success count of 32,
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)!

32C4 =	32!
	4!(32 - 4)!

Calculate k!:

32! = 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
32! = 2.6313083693369E+35

Calculate x!:

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

Calculate (x - k)!:

k - x = 32 - 4 = 28
28! = 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
28! = 3.0488834461171E+29

32C4 =	2.6313083693369E+35
	24(3.0488834461171E+29)

32C4 =	2.6313083693369E+35
	7.3173202706811E+30

32C4 = 35960

Calculate Numerator 2

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

80 - 32C10 - 4 =	(48)!
	(80 - 32 - 10 + 4)!(10 - 4)!

48C6 =	(48)!
	(42)!(6)!

Calculate 6!:

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

Calculate 48!:

48! = 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
48! = 1.2413915592536E+61

Calculate 42!:

42! = 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
42! = 1.4050061177529E+51

48C6 =	1.2413915592536E+61
	720(1.4050061177529E+51)

48C6 =	1.2413915592536E+61
	1.0116044047821E+54

48C6 = 12271512

Calculate Denominator

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

80C10 =	80!
	10!(80 - 10)!

Calculate N!:

80! = 80 x 79 x 78 x 77 x 76 x 75 x 74 x 73 x 72 x 71 x 70 x 69 x 68 x 67 x 66 x 65 x 64 x 63 x 62 x 61 x 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
80! = 7.1569457046264E+118

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)!:

70! = 70 x 69 x 68 x 67 x 66 x 65 x 64 x 63 x 62 x 61 x 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
70! = 1.197857166997E+100

80C10 =	7.1569457046264E+118
	3628800(1.197857166997E+100)

80C10 =	7.1569457046264E+118
	4.3467840875987E+106

80C10 = 1646492110120

Calculate Probability

P(4;10,80,32) =	35960 x 12271512
	1646492110120

P(4;10,80,32) =	441283571520
	1646492110120

P(4;10,80,32) = 0.268

You have 1 free calculations remaining

Calculate the mean μ:

μ =	nk
	N

μ =	10 x 32
	80

μ =	320
	80

μ = 4

Calculate the variance σ2

σ2 =	nk(N - k)(N - n)
	N2(N - 1)

σ2 =	(10)(32)(80 - 32)(80 - 10)
	802(80 - 1)

σ2 =	(320)(48)(70)
	6400(79)

σ2 =	1075200
	505600

σ2 = 2.1266

Calculate the standard deviation σ:
σ = √σ2
σ = √2.1266
σ = 1.4583

What is the Answer?

P(4;10,80,32) = 0.268

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