Hypergeometric Calculator
Calculate the probability of drawing exactly k successes in n draws from a finite population without replacement.
P(X = k)
0.20983972
P(X = k) vs Population Size (N)
公式
## 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.
计算示例
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
常见问题
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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