Enter plain numbers without measurement units. Datasets accept commas, spaces, semicolons, or line breaks and are limited to 10,000 values. Results stay in this browser.
Statistics result
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What the drawing without replacement calculator is calculating
Counting outcomes across repeated trials. Repeated-event probability combines the count of qualifying outcome sequences with the probability of each sequence. Replacement and dependence determine whether trial probabilities stay fixed.
1Specify trials and possible outcomes2Count qualifying sequences3Weight sequences with the correct trial probabilities
Probability rule
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).
Every probability must remain between 0 and 1, and overlapping regions must be jointly feasible. The calculator rejects combinations that violate the stated model.
How to interpret it
Without replacement changes later draw probabilities; with replacement preserves the same target probability on every independent draw.
Model boundary: Items are sampled uniformly, target membership is fixed, and sample size cannot exceed the population without replacement.
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.
