Sampling Without Replacement Calculator
Find exact-count and tail probabilities when drawing from a finite population without replacement. Inspect feasible success counts and exact combinatorial fractions.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Count samples with k successes by choosing k from the K success items and the remaining draws from the other items. Divide by the total number of unordered samples. Sum feasible count probabilities for a tail or range.
P(X=k)=C(K,k)C(N−K,n−k)/C(N,n)
Worked example
From10 items containing4 marked items, drawing3 without replacement gives exactly1 mark with probability C(4,1)C(6,2)/C(10,3)=60/120=1/2.
Assumptions and input details
Assume an equally likely simple random sample of distinct items; order does not affect the counted event. Population1–10000, success items0–N and draws0–min(N,100). Targets are integers within±10000; impossible events have probability0. A zero draw has one empty sample. Exact fractions and counts use integer arithmetic; percentages and plots are rounded. This differs from independent binomial trials, where the success probability remains constant.
References: OpenStax hypergeometric distribution
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Sample counts and event fractions are exact. Decimal percentages, means, standard deviations and plotted heights are rounded.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Calculate a two-draw color event with or without replacement. Follow a probability tree, multiply along each path and add the matching outcomes.
Calculate exactly k, at most k and at least k successes in independent trials. Explore the binomial distribution, expected value and standard deviation.
Count ordered or unordered selections, with or without repetition. Read the exact integer result, compare all four cases and list examples for small problems.
Reserved advertising space.