# Hypergeometric distribution probabilities

Calculates probabilities under a Hypergeometric distribution of parameters M, n and N.

n <= M
N <= M
a <= N
optional (b > a)

## Hypergeometric distribution Formulas

We draw without replacement N objects from a set of M objects.
Among the N drawn objects, n objects have a specific feature. We call a "success" a draw of an object with the specific feature.
We have drawn,
• n objects corresponding to a success and,
• N-n objects corresponding to a failure

Let X be the random variable that counts the total number of successes.
Then X follows an hypergeometric distribution of parameters M, n and N and is written X ~ N(M, n, N).

### Probability Mass function (PMF)

P(X = k) = (([n], [k])*([M-n], [N-k]))/(([M],[N]))
([n], [k]) is the binomial coefficient also called 'n choose k',
([n], [k]) = (n!)/(k! * (n-k)!)

For P(X = k) to b a real number between 0 and 1, k must satisfy,
max(0,N-M+n) <= k <= min(n,N)   (1)
(k must be less or equal to n in ([n], [k]) for the three binomial coefficients in above formula).

### Cumulative distribution function (CDF)

P(X <= k) = sum_(i=max(0,N-M+n))^k (([n], [k])*([M-n], [N-k]))/(([M],[N]))  k satifying equation (1)

### Survival function

P(X >= k) = sum_(i=k)^(min(n,N)) (([n], [k])*([M-n], [N-k]))/(([M],[N]))  k satisfying equation (1)