Probabilities apply only if the sample is random, items are not replaced, and the success classification is fixed. “Two or more” includes every feasible count through min(K,n); “two or less” includes zero, one and two. Overlapping tail percentages should not be added together.
Practical use: People usually open this page when they need to find probabilities for successes in a sample drawn without replacement without building a spreadsheet for a single question. The hypergeometric model fits finite populations where each item is success or non-success and sampling changes what remains. It differs from the binomial model, which assumes independent trials or replacement. For a real decision, calculate a central case and then change the most uncertain input to a reasonable lower and upper value. Seeing that range is often more informative than treating one rounded answer as permanent.
A useful result should be repeatable: keep the original inputs, unit choices and relevant date beside the exact and tail probabilities. If a second source uses another convention, reconcile that difference before comparing the numbers.
When sharing the answer, include a short label such as “exact and tail probabilities from the stated inputs.” That label keeps the number attached to its meaning. It also prevents a reader from mistaking a converted value, model output or planning estimate for a direct observation.
Scope: It assumes random sampling without replacement and two fixed item classes.
