Hypergeometric Calculator Formula
Understand the math behind the hypergeometric calculator. Each variable explained with a worked example.
Formulas Used
P(X = k)
probability = (c_K_k * c_NK_nk) / c_N_nExpected Value
expected = n * K / NVariance
variance_val = n * (K / N) * ((N - K) / N) * ((N - n) / (N - 1))Variables
| Variable | Description | Default |
|---|---|---|
N | Population Size (N) | 50 |
K | Success States in Population (K) | 10 |
n | Number of Draws (n) | 5 |
k | Observed Successes (k) | 2 |
c_K_k | Derived value= factorial(K) / (factorial(k) * factorial(K - k)) | calculated |
c_NK_nk | Derived value= factorial(N - K) / (factorial(n - k) * factorial(N - K - n + k)) | calculated |
c_N_n | Derived 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?
- 01C(10,2) = 45
- 02C(40,3) = 9880
- 03C(50,5) = 2118760
- 04P(X=2) = (45 * 9880) / 2118760
- 05= 444600 / 2118760 ≈ 0.20985
Frequently Asked Questions
How does hypergeometric differ from binomial?
The binomial assumes independent draws (with replacement), so success probability stays constant. The hypergeometric models draws without replacement, where each draw changes the composition of the remaining pool.
When can I approximate hypergeometric with binomial?
When the population N is much larger than the sample n (rule of thumb: n < 0.05*N), the hypergeometric is well approximated by the binomial with p = K/N because removing a few items barely changes the population composition.
What are real-world applications?
Quality control (defective items in a batch), card games (drawing specific cards), ecological capture-recapture studies, and audit sampling are all modeled by the hypergeometric distribution.
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Open Hypergeometric Calculator