This online hypergeometric distribution calculator computes the probability of the exact outcome of an hypergeometric experiment (hypergeometric probability P), given the population size N, the number of successes in the population K, the sample size n and the number of successes in the sample k. It can also possible to compute cumulative hypergeometric probabilities P for no more than k successes or for no less than k successes. The parameters N, K, n and k, used in our hypergeometric calculator, must be non-negative integers and satisfy the conditions N > 0, n > 0, k ≤ n ≤ N and 0 ≤ k ≤ K ≤ N ≤ 1000.
Hypergeometric distribution formula
A hypergeometric experiment is a statistical experiment when a sample of size n is randomly selected without replacement from a population of N items. Assume that in the above mentioned population, K items can be classified as successes, and N − K items can be classified as failures. A hypergeometric variable k is the number of successes in the sample that result from an hypergeometric experiment.
The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes. The respective hypergeometric distribution probability mass function can be expressed by the following hypergeometric distribution formula:
$$P(k)=\frac { C(K,k)\cdot C(N-K,n-k) }{ C(N,n) }, $$
where \(C(n,k)\) is the binomial coefficient:
$$C(n,k) = \frac { n! }{ k!(n-k)! }.$$
A cumulative hypergeometric probability refers to the probability that the hypergeometric experiment outcomes fall within a specified range. Thus, the lower cumulative hypergeometric probability is the probability of no more than k successes:
$${ P }_{ l }(k)=\sum _{ x=0 }^{ k }{ P(x) } ,$$
and the upper cumulative hypergeometric probability is the probability of no less than k successes:
$${ P }_{ u }(k)=\sum _{ x=k }^{ n }{ P(x) } .$$
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