Calculator Inputs
Example Data Table
Example uses N = 20, K = 7, and n = 5. The table lists exact probabilities for each feasible success count.
| Successes in Sample (x) | Exact Probability P(X = x) |
|---|---|
| 0 | 0.0830108359 |
| 1 | 0.3228199174 |
| 2 | 0.3873839009 |
| 3 | 0.1760835913 |
| 4 | 0.0293472652 |
| 5 | 0.0013544892 |
Formula Used
Probability mass function:
P(X = x) = [C(K, x) × C(N - K, n - x)] / C(N, n)
Here, N is population size, K is successes in the population, n is sample size, and x is observed successes in the sample.
Mean: E(X) = n × (K / N)
Variance: Var(X) = n × (K/N) × (1 - K/N) × ((N - n) / (N - 1))
Standard deviation: σ = √Var(X)
Support limits:
min x = max(0, n - (N - K))
max x = min(n, K)
Lower-tail and upper-tail probabilities are sums of exact probabilities across the support.
How to Use This Calculator
- Enter the total population size, the number of successes in that population, and the sample size drawn without replacement.
- Provide the target success count x for exact, lower-tail, and upper-tail probability calculations.
- Enter interval bounds a and b if you also want the probability of falling inside a success range.
- Select the primary result you want highlighted after submission.
- Press the calculate button to show the result section above the form, including the graph and distribution table.
- Use the CSV and PDF buttons to export the summary and distribution values.
FAQs
1) What does the hypergeometric distribution model?
It models the number of successes in a fixed sample drawn without replacement from a finite population containing a known number of successes.
2) When should I use this instead of the binomial distribution?
Use the hypergeometric model when draws are made without replacement. Use the binomial model when draws are independent and the success probability stays constant.
3) Why do support limits matter?
Not every success count is possible. The support limits identify the feasible values of X, preventing invalid probability requests and improving interpretation.
4) What is the exact probability output?
It is the probability of getting exactly x successes in the sample, not fewer and not more.
5) What do lower-tail and upper-tail probabilities show?
Lower-tail probability gives the chance of getting at most x successes. Upper-tail probability gives the chance of getting at least x successes.
6) Why is there a finite population correction in the variance?
Because sampling without replacement reduces variability. Each draw changes the remaining population, so the variance is smaller than the comparable binomial case.
7) Can this calculator handle interval probabilities?
Yes. Enter a start and end value, and the calculator sums all exact probabilities between those feasible bounds, inclusive.
8) What is the mode in this distribution?
The mode is the most likely success count. In rare cases, the distribution may have two adjacent modes instead of one.