Hypergeometric Calculator Formula

Understand the math behind the hypergeometric calculator. Each variable explained with a worked example.

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Formulas Used

P(X = k)

probability = (c_K_k * c_NK_nk) / c_N_n

Expected Value

expected = n * K / N

Variance

variance_val = n * (K / N) * ((N - K) / N) * ((N - n) / (N - 1))

Variables

VariableDescriptionDefault
NPopulation Size (N)50
KSuccess States in Population (K)10
nNumber of Draws (n)5
kObserved Successes (k)2
c_K_kDerived value= factorial(K) / (factorial(k) * factorial(K - k))calculated
c_NK_nkDerived value= factorial(N - K) / (factorial(n - k) * factorial(N - K - n + k))calculated
c_N_nDerived value= factorial(N) / (factorial(n) * factorial(N - n))calculated

How It Works

How to Calculate Hypergeometric Probability

Formula

P(X = k) = C(K,k) * C(N-K, n-k) / C(N, n)

The hypergeometric distribution models sampling without replacement from a finite population of N items containing K successes. It answers: if you draw n items, what is the probability of getting exactly k successes? Unlike the binomial, the probability changes with each draw.

Worked Example

A deck has 50 cards, 10 are red. Draw 5 cards without replacement. What is the probability of exactly 2 red cards?

N = 50K = 10n = 5k = 2
  1. 01C(10,2) = 45
  2. 02C(40,3) = 9880
  3. 03C(50,5) = 2118760
  4. 04P(X=2) = (45 * 9880) / 2118760
  5. 05= 444600 / 2118760 ≈ 0.20985

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