Advanced Hypergeometric Probability Calculator

Solve exact, cumulative, and interval probabilities with confidence quickly. Compare expected outcomes and dispersion instantly. Generate charts, exports, and practical sampling insights for decisions.

Calculator Inputs

Population size (N)

Population successes (K)

Sample size (n)

Target successes (x)

Range lower

Range upper

Probability mode

Reset

Use whole numbers only. Hypergeometric models require a finite population and sampling without replacement.

Plotly Probability Graph

The graph shows the probability mass for every feasible number of observed successes.

Example Data Table

Scenario	N	K	n	x	Mode	Interpretation
Defective items in inspection	50	12	8	3	Exact	Probability of finding exactly 3 defectives in 8 checked items.
Survey support responses	120	44	15	6	At least	Probability of observing 6 or more supporters in a sample.
Marked cards in a draw	52	4	5	1	At most	Probability of drawing no more than 1 marked card.
Qualified applicants screened	80	25	10	2 to 4	Between	Probability of selecting between 2 and 4 qualified applicants.

Formula Used

The hypergeometric distribution models the count of successes when you draw from a finite population without replacement.

P(X = x) = [C(K, x) × C(N - K, n - x)] / C(N, n)

N = total population size
K = number of success states in the population
n = number of draws
x = observed successes in the sample
C(a, b) = combinations of choosing b items from a

Cumulative modes add the exact probabilities across the relevant feasible values of x.

How to Use This Calculator

Enter the total population size.
Enter how many population members count as successes.
Enter the sample size drawn without replacement.
Choose the target number of observed successes.
Select exact, at least, at most, or between mode.
Provide lower and upper limits when using between mode.
Press the calculate button to show results above the form.
Use the CSV or PDF buttons to export your work.

Frequently Asked Questions

1. When should I use a hypergeometric calculator?

Use it when sampling comes from a finite group without replacement. Each draw changes the remaining composition, so probabilities differ from independent binomial trials.

2. What is the difference between hypergeometric and binomial models?

The binomial model assumes independent trials with constant success probability. The hypergeometric model updates the odds after each draw because sampled items are not replaced.

3. Why must the inputs be whole numbers?

Population counts, successes, draws, and observed successes represent actual items. Because combinations use discrete counts, the model works only with nonnegative integers.

4. What does exact probability mean here?

Exact probability returns the chance of observing one specific success count, such as exactly 3 defectives in a sample of 8 inspected items.

5. What do at least and at most options calculate?

At least sums probabilities from the chosen x upward. At most sums probabilities from the minimum feasible value through the chosen x.

6. What does the feasible x range show?

It shows the lowest and highest success counts that can occur with your inputs. Values outside that range always have zero probability.

7. Why are mean and variance useful?

The mean estimates the expected number of successes. The variance and standard deviation describe how much the observed count tends to vary around that expectation.

8. Can I export my calculated results?

Yes. The page includes CSV export for numeric summaries and PDF export for a portable report of the visible results section.