Why this relationship works
Hypergeometric probability C(K,k)C(N−K,n−k)/C(N,n) counts successes in a sample drawn without replacement.
For this page, hypergeometric distribution is interpreted under the stated convention and input order.
Math calculator
Hypergeometric Distribution Calculator
Calculate successes in a sample drawn without replacement. Beside sample probability, the output shows the operation used to obtain it.
Hypergeometric Distribution inputs
Build the expression
Follow one set of numbers
From N=50 with K=12, drawing 8 and observing 3 successes has probability about .2057.
Manual method
Choose requested successes and failures separately, then divide by every sample of size n. Hypergeometric Distribution also leads to replacement comparison.
Where the shortcut stops
All counts must be mutually possible. Unlike binomial trials, draws are dependent because objects are not returned.
Compare the result with the worked example's scale before relying on the reported sample probability.
Where this calculation appears
It fits card hands, lot inspection, roster samples, and finite inventories. A related application of Hypergeometric Distribution is sample counts.
A closer look at result behavior
The cleanest input audit is to restate the problem using population size n, population successes k, draw count n and sample successes k. If that sentence sounds wrong, correct the assignment before asking for sample probability.
A larger sampling fraction makes the without-replacement dependence increasingly important. The pattern also provides a quick estimate of whether a revised result is plausible.
The formula and result serve different readers: the formula shows what was done, and the rounded sample probability communicates scale. Keep both when the work needs review.
Binomial uses approximately constant p; hypergeometric tracks the finite population exactly. The wording of the problem should decide which operation is appropriate.
A screenshot is unnecessary when the shown formula and the supplied inputs (population size n, population successes k, draw count n and sample successes k) are saved in plain text. Include the rounding rule if the reported sample probability is approximate.
Precision and scale in Hypergeometric Distribution
Exact arrangement counts from Hypergeometric Distribution should not be shortened to a decimal approximation. Probability calculations based on Population size N, Population successes K, Draw count n, Sample successes k may be rounded for presentation, but their intermediate products and cumulative sums need extra digits so the final tail or complement is not distorted.
For Hypergeometric Distribution, keep decimal and percentage forms visibly separate. A result of 0.08 means 8%, not 0.08%, and an input described as 8% normally enters the formula as 0.08. This scale check catches an error much larger than ordinary rounding.
Questions about Hypergeometric Distribution
Are draws independent?
No.
When is binomial close?
When the sampling fraction is small.
Can k exceed K?
No.
Why divide by C(N,n)?
It counts all equal-size samples.