Direct answer
Use binomial for successes across fixed independent equal-probability trials, Poisson for event counts over exposure under a constant-rate model, and hypergeometric for successes sampled without replacement from a finite population.
Visual explanation
Three count models, three mechanisms
What this calculation tells you
All three models produce nonnegative integer counts, but their mechanisms differ. Choosing by output type alone hides the assumptions that determine probability mass.
Ask whether there is a fixed number of trials, a continuous exposure interval, or a known finite population being sampled without replacement. Then check event probability, independence, and rate stability.
Where it is used
Quality
Separate repeated trial defects from finite-lot sampling.
Operations
Model entered event counts over documented exposure.
Reliability
Choose a count model from the event process rather than habit.
Education
Compare supports, parameters, means, variances, and assumptions.
When this guide helps
- Counting successes in fixed trials.
- Counting arrivals per time or distance.
- Sampling items from a known finite lot.
- Investigating overdispersion relative to a simple count model.
Start with the statistical question
Ask whether there is a fixed number of trials, a continuous exposure interval, or a known finite population being sampled without replacement. Then check event probability, independence, and rate stability.
Three parallel process drawings show repeated coin-like trials, events arriving on a timeline, and colored items removed from a finite urn. Assumption cards sit below each model.
Worked example
Ten independent inspections with constant defect probability p use binomial. Defects counted per kilometer may use Poisson under a constant-rate event model. Five items selected from a lot with known defect count use hypergeometric.
Assumptions that carry the result
Binomial requires fixed n and p with independent trials; Poisson requires a suitable rate process; hypergeometric requires a finite population with equal-probability sampling without replacement.
Interpret the result without overreaching
Overdispersion, varying probabilities, clustering, zero inflation, and changing exposure can invalidate simple models. Similar numerical probabilities do not make mechanisms interchangeable.
- Using Poisson only because the outcome is a count.
- Ignoring changing probabilities without replacement.
- Treating a large finite sampling fraction as independent binomial trials.
Worked case: quality sample with replacement-like trials
Inspect 20 independently produced units with constant defect probability 0.05.
X can be modeled Binomial(20, 0.05); mean defects = np = 1.
The model has support 0 through 20.
Shared batches or drifting process rates undermine independence/constant p.
Reproduce this worked caseOpen Binomial Distribution Calculator
Worked case: finite lot sample
Sample 20 units without replacement from a lot of 100 containing 5 known defectives.
Hypergeometric parameters are population 100, successes 5 and draws 20.
The model accounts for depletion and cannot exceed five defects.
A binomial approximation discards finite-population dependence.
Reproduce this worked caseOpen Binomial Distribution Calculator
count distributions: compare assumptions, not just answers
Choose from the data-generating process. Similar means do not make the distributions interchangeable.
| Case | Calculation focus | Interpretation |
|---|---|---|
| 20 process units | Independent p = 0.05 | Binomial |
| 20 from known lot | Without replacement | Hypergeometric |
| Events/hour | Rate/exposure | Poisson |
count distributions: calculation checklist
- Support checked
- Fixed n or exposure identified
- Replacement/dependence stated
- Parameter meanings preserved
- Approximation disclosed
Practical questions
Frequently asked questions
Can Poisson approximate binomial?
Sometimes when n is large and p is small, but the approximation does not erase the different process assumptions.
Why are hypergeometric draws dependent?
Removing an item changes the remaining population composition and therefore later probabilities.
What if binomial trial probabilities differ?
A Poisson-binomial or another model may be needed; ordinary binomial assumes one common p.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.