Quick guide
How to use this calculator
- Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
- Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
- Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the drawing without replacement calculator works
Without replacement use C(K,k)C(N−K,n−k)/C(N,n); with replacement use C(n,k)p^k(1−p)^(n−k).
Without replacement changes later draw probabilities; with replacement preserves the same target probability on every independent draw.
Worked example
Drawing Without Replacement example
Drawing 5 items without replacement from 50 containing 10 targets uses the hypergeometric rather than binomial model.
Without replacement use C(K,k)C(N−K,n−k)/C(N,n); with replacement use C(n,k)p^k(1−p)^(n−k).
Supported inputs
Precision and limits
Model and design
Items are sampled uniformly, target membership is fixed, and sample size cannot exceed the population without replacement.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
Without replacement changes later draw probabilities; with replacement preserves the same target probability on every independent draw.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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