Results
PMF formula
P(X=k)=C(K,k)·C(N−K,n−k)/C(N,n)
Tip: “at least one” is P(X≥1)=1−P(X=0).
Distribution (PMF table & bar chart)
| k | P(X=k) | CDF |
|---|
Simulation (Monte Carlo)
Use trials and seed to reproduce runs. For large ranges, the tool bins the histogram to stay fast.
Worked examples & interpretation
What the inputs mean
- N: population size (total items).
- K: number of “successes” in the population (items you care about).
- n: number of draws (sample size).
- k: successes in your sample (the random variable X).
When to use this (vs. binomial)
- Hypergeometric: sampling without replacement (the success probability changes after each draw).
- Binomial: independent trials with a fixed success probability (often “with replacement” or a very large population).
- Rule of thumb: if your sampling fraction is small (n/N is small), hypergeometric and binomial with
p = K/Ncan be close.
Worked examples (try the presets)
Cards: draw 5 from 52, how likely to get exactly 2 aces?
Set N=52, K=4, n=5, query “Exactly” with k=2.
Inspection: 10 defectives in 100, sample 8 — probability of at least 1 defective?
Set N=100, K=10, n=8, query “At least” with k=1.
“At least one” quickly (complement trick)
Compute P(X ≥ 1) = 1 − P(X = 0). You can also use the helper button “P(X≥1)”.
Common pitfalls
- Invalid k: the support is
k_min=max(0,n−(N−K))andk_max=min(n,K). Outside it,P(X=k)=0. - Define “success” clearly: “success” is just the label for the item type you are counting (e.g., “defective”, “red”, “ace”).
- Large ranges: the PMF table can be omitted for performance; use the probability result and/or simulation.
References
How to use this calculator effectively
This guide helps you use Hypergeometric distribution calculator (without replacement) in a repeatable way: define a baseline, change one variable at a time, and explain each output using explicit assumptions before sharing results.
How it works
The calculator applies deterministic formulas to your input values and only rounds at the final display layer. This makes it useful for comparative analysis: keep one scenario as a baseline, then vary assumptions and measure the delta in both absolute terms and percentage terms. If a change appears too large or too small, verify units, period conventions, and sign direction before interpreting the result.
When to use
Use this page when you need a fast planning estimate, a classroom check, or a reproducible scenario that teammates can review. It is most effective at the decision-prep stage, where you need to compare options quickly and decide which assumptions deserve deeper modeling or external validation.
Common mistakes to avoid
- Mixing units such as percent vs decimal, or monthly vs yearly settings.
- Changing multiple fields at once, which hides the real cause of result movement.
- Comparing outputs across tools without aligning constants and default conventions.
- Treating rounded display values as exact inputs for downstream calculations.
Interpretation and worked example
Start with a baseline case and save that output. Next, edit one assumption to reflect your realistic alternative, then compare both the direction and size of change. If the direction matches domain intuition and magnitude is plausible, your setup is likely coherent. If not, check hidden defaults, unit conversions, boundary conditions, and date logic before drawing conclusions.
See also
FAQ
What is the hypergeometric model used for?
It models draws from a finite population where each item is not returned, so probabilities change after each draw.
How do I set inputs?
Set population size N, number of success items K, draw size n, and target success count k (or a range) before checking tails and cumulative values.
Why does this differ from binomial?
Binomial assumes independent draws with replacement-like replacement assumptions, while hypergeometric handles finite population depletion and is accurate for sampling without replacement.
How should I interpret the tail result?
Tail results summarize 'at least / at most' scenarios and are useful for risk checks, thresholds, and decision boundaries.
Can I use results for decisions?
Use them as a quantitative signal for scenario comparison, then validate against context assumptions and operational limits before final decisions.
How it’s calculated
C(K,k)·C(N−K,n−k)/C(N,n)computed in log-space for stability.k_min=max(0,n−(N−K)),k_max=min(n,K).