The Hypergeometric Distribution
The hypergeometric probability distribution has the form
f(x) = h(x;n, n 1 , n 2 ) =
(
n 1
x
)(
n 2
n−x
)
(
n 1 +n 2
n
) , x = 0, 1 , 2 , 3 ,... , n (11 .71)
where x is an integer satisfying 0 ≤x ≤n, n 1 represents the number of successes
and n 2 represents the number of failures, where nitems are selected from (n 1 +n 2 )
items without replacement. This is a probability distribution with three parameters,
n, n 1 and n 2. The hypergeometric probability distribution is used in quality control,
estimates of animal population size from capture-recapture data, the spread of an
infectious disease when a fixed number of individuals are exposed to an illness.
Note that the binomial distribution is used in sampling with replacement while
the hypergeometric distribution is applicable for problems where there is sampling
without replacement. The hypergeometric distribution has mean
μ=
nn 1
n 1 +n 2
and variance given by
σ^2 = nn^1 n^2 (n^1 +n^2 −n)
(n 1 +n 2 )^2 (n 1 +n 2 −1)
The equation (11.71) represents the probability of x successes and n−x failures
selected from n 1 +n 2 items where the sampling is without replacement. For example,
to find the probability of selecting two aces from a standard deck of 52 playing cards
in 6 draws with no replacement of cards selected one would select the following
parameters for the hypergeometric distribution. Here there are 6 draws so that
n= 6. There are 4 aces in the deck so n 1 = 4 is the number of successes in the deck
and n 2 = 48 is the number of failures in the deck, with n 1 +n 2 = 52 the total number
of cards in the deck. The hypergeometric distribution gives the probability of x= 2
successes in n= 6 draws as
h(2; 6, 4 ,48) =
(
4
2
)(
48
4
)
(
52
6
) =^621
10829
= 0. 0573
