What is the hypergeometric distribution used for?

It models the number of successes in a sample drawn without replacement from a finite population (N total items with K successes).

What is the difference between PMF and CDF?

PMF is P(X=k). CDF is P(X≤k), the cumulative probability up to k.

How do I compute P(X≥k)?

Use the upper tail: P(X≥k)=1−P(X≤k−1). This calculator shows tails directly in the table.

What is the valid range (support) for k?

k_min=max(0, n−(N−K)) and k_max=min(n, K). Outside that range the probability is 0.

When can I approximate hypergeometric with binomial?

A common rule-of-thumb is when the sample is small compared with the population (often n ≤ 0.05N), using Binomial(n, p=K/N) as an approximation.

Hypergeometric Distribution Calculator: PMF, CDF & Tails

← Back to RNG & Probability Tools

📌 Overview — Hypergeometric Distribution Calculator (PMF, CDF, Tails, Range & “Outside”)

The hypergeometric distribution models a classic “finite population” situation: you sample without replacement from a population and count how many “successes” you get. This is the right model for many real-world questions that students (and analysts) search for: hypergeometric distribution calculator, hypergeometric CDF, right-tail probability P(X ≥ k), left-tail probability P(X ≤ k), and range probability P(k1 ≤ X ≤ k2).

In stats notation, the setup is: N = population size, K = number of success items in that population, n = sample size (draws), and the random variable X counts successes observed in the sample. This tool lets you compute multiple query types in one place and also shows the full distribution table plus PMF/CDF charts for intuition and homework checking.

Exact PMF: P(X = k) for any valid k.
CDF: P(X ≤ k) for “at most” questions.
Right tail: P(X ≥ k) for “at least” questions.
Range: P(k1 ≤ X ≤ k2) for “between” questions.
Outside: P(X ≤ k1 OR X ≥ k2) (common in two-sided discrete “rejection regions”).

Prefer a story/visual framing (balls, colors, urns)? Use the Ball & Urn Probability Calculator. For card-first problems (hands, suits, ranks), try the Deck of Cards Probability Calculator. If your problem is about independent trials (like coin flips), compare with the Coin & Dice Probability Lab.

How to read your results (quick intuition):

Probability is shown as a decimal (e.g., 0.2995).
Percent is the same value × 100 (e.g., 29.9474%).
“1 in N” odds is a friendly inverse: 1 / Probability (useful for gut-checking).
Mean E[X] is the expected number of successes in your sample.
SD (standard deviation) tells you typical variability around the mean.

✅ Educational note: This tool is built for learning, analytics, and coursework. It does not predict outcomes and is not a betting system.

📘 Formula & Methodology — Support, PMF, CDF, Tails, Mean & Variance

1) Valid support (the only k values that can happen)

Many errors come from requesting a probability for an impossible k. The hypergeometric support is:

k_min = max(0, n - (N - K))
k_max = min(n, K)

This tool displays the support in the results and (helpfully) shows a Valid support hint under k / k1 / k2. If you enter k outside the support, the probability is 0 (and the tool explains why).

2) Hypergeometric PMF (probability mass function)

The PMF is the probability of seeing exactly k successes in n draws without replacement:

P(X = k) = [ C(K, k) * C(N - K, n - k) ] / C(N, n)

Where C(a, b) is “a choose b” (combinations). Internally, this calculator uses a numerically stable log-combination approach and then normalizes the distribution so it stays accurate even when values get large.

3) CDF, right tail, ranges, and “outside”

Once the PMF is known, the tool computes:

P(X ≤ k) = Σ P(X = i) for i = k_min..k
P(X ≥ k) = Σ P(X = i) for i = k..k_max
P(k1 ≤ X ≤ k2) = Σ P(X = i) for i = k1..k2
P(X ≤ k1 OR X ≥ k2) = P(X ≤ k1) + P(X ≥ k2)

Tip: “At least one” questions are often easiest by complement: P(X ≥ 1) = 1 - P(X = 0). You can compute it either way and cross-check your work.

4) Mean and standard deviation (finite population correction)

The expected number of successes is:

E[X] = n * (K / N)

The variance includes a finite-population correction (because draws are without replacement):

Var[X] = n * (K/N) * (1 - K/N) * ((N - n) / (N - 1))   (for N > 1)
SD = sqrt(Var[X])

5) When should I use hypergeometric vs binomial?

Model	Replacement?	Probability changes each draw?	Common wording	Typical examples
Hypergeometric	No (without replacement)	Yes	“Draw n items from N…”	Cards, audits, batch sampling
Binomial	Independent trials / with replacement	No (constant p)	“Repeat n trials with probability p…”	Coin tosses, repeated independent checks

This page includes a Binomial approximation option (p = K/N) and clearly labels it as an approximation. A common rule-of-thumb is n ≤ 0.05N (small sample relative to population).

🧪 Worked Examples — Accurate Numbers You Can Verify in the Tool

Example 1: Cards — exactly 1 ace in 5 cards

A standard deck has N = 52 cards and K = 4 aces. You draw n = 5 cards (without replacement). Compute P(X = 1).

P(X = 1) = [ C(4,1) * C(48,4) ] / C(52,5)
= 0.2994736356  (29.9474%)

In the calculator: set N=52, K=4, n=5 → choose P(X = k) → set k=1. With the default 4-decimal display, you should see 0.2995 (≈ 29.9474%) and “1 in N” odds ≈ 1 in 3.34. The support is k ∈ [0, 4].

Example 2: “At least one ace” (right tail)

Same setup (N=52, K=4, n=5), but now compute P(X ≥ 1).

P(X ≥ 1) = 1 - P(X = 0)
= 0.3411580017  (34.1158%)

In the calculator: choose P(X ≥ k) and set k=1. You should see 0.3412 (≈ 34.1158%). Use this example to practice “tail thinking”: “at least” = right tail.

Example 3: Quality control — “at most 1 defective” in a sample

A batch has N = 200 items with K = 12 defectives. You sample n = 10 items without replacement. Find P(X ≤ 1) (at most 1 defective).

P(X ≤ 1) = P(X=0) + P(X=1)
= 0.5383004057 + 0.3481370146
= 0.8864374203  (88.6437%)

In the calculator: set N=200, K=12, n=10 → choose P(X ≤ k) → set k=1. You should see 0.8864 (≈ 88.6437%). The distribution table is especially useful here because it shows the two terms P(X=0) and P(X=1) and how they accumulate.

✅ Accuracy tip: Always confirm your k is inside the support. If the tool says the support is [k_min, k_max], any k outside that range is impossible.

📊 Infographic & Visual Guide — PMF vs CDF, Support Bounds, and “Tail Thinking”

The infographic below is a quick “cheat sheet” for reading hypergeometric results correctly. It summarizes the parameters (N, K, n, X), the PMF formula, the valid support range, and the most common query types students search for: P(X = k), P(X ≤ k), P(X ≥ k), range, and outside.

Hypergeometric Distribution Cheat Sheet: parameters (N, K, n, X), PMF formula, valid support, and how to interpret exact / tail / range / outside queries.

Fast interpretation tips (avoid the most common mistakes):

Check support first: if your chosen k is outside [k_min, k_max], then P = 0 (the outcome is impossible).
PMF vs CDF: PMF answers “exactly k”. CDF answers “at most k” by accumulating probabilities.
Right tail: “at least k” means P(X ≥ k). A quick cross-check is P(X ≥ k) = 1 - P(X ≤ k - 1).
Range queries: “between k1 and k2” is inclusive: P(k1 ≤ X ≤ k2).
Outside queries: “outside the middle” means two tails: P(X ≤ k1 OR X ≥ k2).

This tool intentionally shows the distribution in multiple ways, because hypergeometric questions are usually about regions: “at least k”, “at most k”, “between k1 and k2”, or “outside a middle band”. The calculator highlights the relevant k-values in the table and charts so you don’t have to re-sum by hand.

Query you choose	What it means in words	What to look at
P(X = k)	Exactly k successes	One PMF bar / one table row
P(X ≤ k)	At most k successes	CDF value at k (left tail)
P(X ≥ k)	At least k successes	Right tail (tail column)
P(k1 ≤ X ≤ k2)	Between two bounds (inclusive)	Highlighted block of rows (range)
P(X ≤ k1 OR X ≥ k2)	Outside a middle region (two tails)	Two highlighted tails

Bonus: If your page includes the optional Monte Carlo simulation, use it to see how empirical frequencies converge toward the exact PMF as the number of trials increases.

🧩 Practice Lab — Try Hypergeometric “Without Replacement” Experiments at Home (and Check Your Results)

If you’re learning hypergeometric probability, the fastest way to build intuition is to run the sampling experiment yourself (cards, slips of paper, colored beads) and then use this calculator to evaluate how your observed results compare to the exact PMF/CDF and the Monte Carlo simulation. This section is designed for students, teachers, and self-study practice.

Key idea: “Without replacement” means you do not put the item back during one trial. But when you repeat the experiment many times, you typically reset the population between trials (shuffle again / refill the bag), so each trial is comparable.

✅ Step-by-step: run a real hypergeometric experiment (3 easy setups)

Pick a physical setup (choose one):
- Cards: Use a standard deck. “Success” could be “ace”, “heart”, “face card”, etc.
- Bag / cup: Use 20–100 small objects (beans, Lego, beads). Mark K of them as “success”.
- Paper slips: Write “S” on K slips and “F” on N−K slips, mix in a box.
Map your setup to the hypergeometric parameters:
- N = total items in the population (deck size, total slips, total beads).
- K = number of “success” items in the population (aces, red beads, “S” slips).
- n = how many you draw per trial (sample size).
- X = number of successes in your sample (count them each trial).
Run one trial: draw n items without replacement, count successes = X, record X.
Reset the population (important!):
- Cards: put the cards back and reshuffle.
- Bag/slips: return all items and mix thoroughly.
Repeat T times (start with T=30, then 100+). Your results will get closer to the exact distribution as T increases.

📌 Use the calculator to “score” your practice results (compare observed vs exact)

After you record your trial outcomes (your list of X values), use this tool to check whether your hands-on experiment is behaving as expected:

Check the most common outcome (“mode”): Your most frequent X should usually be near the tool’s mean E[X] = n(K/N) (not always identical, but close).
Compare exact probabilities: For each k you observed often, look up P(X = k) in the distribution table. Your observed frequency count(k)/T should be close for larger T.
Practice tails: If your homework asks “at least k”, compute P(X ≥ k) and compare it with (# trials with X ≥ k) / T.
Use the built-in simulation (optional): Turn on Monte Carlo simulation with the same N, K, n. It gives a second “empirical” comparison that should track your hands-on results.

🧪 Mini “home worksheet” template (copy into Notes / Excel)

You can copy this table into a spreadsheet and tally counts automatically.

Trial #	Setup (N, K, n)	Observed X	Notes (mistakes? reshuffle?)
1	e.g., N=52, K=4, n=5	0–4	Returned cards + shuffled?
2	same	0–4	—
…	…	…	…

🎯 Practice challenges (great for students & classroom demos)

Challenge A (Cards): N=52, K=4 (aces), n=5. Run T=50 trials. Compare your observed P(X ≥ 1) with the tool (Example 2).
Challenge B (Bag): Build N=30 items with K=6 successes. Draw n=5 each trial. Ask: “What is P(X = 0)?” and verify by counting how often you got zero.
Challenge C (Quality control intuition): Pretend a batch has 10% defectives (e.g., N=100, K=10), sample n=8. Practice both: P(X ≤ 1) and P(X ≥ 2).
Challenge D (Range thinking): Choose a setup and compute P(1 ≤ X ≤ 3). Then estimate it empirically with your trials and compare.

⚠️ Common practice mistakes (and how to fix them)

Forgetting to reset between trials: You must return items and remix/shuffle before the next trial, or your trials won’t be comparable.
Changing the success definition mid-way: Decide what counts as success (e.g., “ace”, “red”, “defective”) and keep it constant.
Using replacement by accident: If you put items back during the same trial, that’s no longer hypergeometric (it’s closer to binomial).
Too few trials: Small T is noisy. If your observed frequency looks “off,” increase trials (T=100+ usually looks much more stable).
Comparing to the wrong query type: “At least k” means P(X ≥ k), “at most k” means P(X ≤ k), and “between” means inclusive range P(k1 ≤ X ≤ k2).

Want a more visual story version while practicing? Try the Ball & Urn Probability Calculator. For card-first setups, the Deck of Cards tool matches common homework word problems closely.

Note: Hands-on experiments are inherently random. Use this practice lab to learn probability and validate logic — not to “predict” outcomes.

🧠 Use Cases — Where Hypergeometric Probabilities Show Up

Statistics homework & exams: PMF/CDF/tail/range queries with correct support bounds.
Quality control & acceptance sampling: defect counts in an inspection sample from a finite lot.
Audit sampling: probability of finding ≥k flagged items in a random subset of records.
Cards & games (educational): hand probabilities like “exactly 1 ace in 5 cards”.
Batch testing / lab sampling: drawing a subset from a finite batch (interpret results with domain rules).
Teaching binomial vs hypergeometric: show how “without replacement” changes variability and tails.

If you want the same math with a more visual “urn” story, the Ball & Urn tool is a great companion. For card-style problems, use the Deck of Cards tool. For independent-trial intuition, compare with the Coin & Dice Probability Lab.

❓ FAQ — Hypergeometric Distribution Calculator (Real Questions People Ask)

What do N, K, n, and X (or k) mean in the hypergeometric distribution?

N is the population size, K is how many “success” items are in the population, n is the number of draws (sample size), and X is the random variable counting successes in the sample. In the calculator, k is the specific value of X you’re asking about.

What does “without replacement” mean, and why does it matter?

“Without replacement” means once an item is drawn, it is not returned. That makes draws dependent: the success probability changes slightly after each draw. This is exactly why the hypergeometric distribution differs from the binomial distribution.

What is the support (k_min..k_max), and why does the tool show it?

The support is the set of k values that can actually occur. Hypergeometric outcomes are constrained by both the sample size and the number of successes available. If you enter k outside the support, the true probability is 0. The calculator shows the support to prevent common homework mistakes.

How do I calculate “at least k successes”?

Choose P(X ≥ k). The table also includes a tail column that already represents P(X ≥ k) for each row k. This is one of the most common hypergeometric queries in coursework.

How do I calculate “at most k successes”?

Choose P(X ≤ k). This is the left tail (CDF). You can also read it directly from the CDF chart at the chosen k.

How do I calculate a range like P(2 ≤ X ≤ 4)?

Choose P(k1 ≤ X ≤ k2) and enter k1 and k2. The calculator sums the PMF over that inclusive range and highlights the included rows in the table.

What does the “outside” option mean: P(X ≤ k1 OR X ≥ k2)?

“Outside” combines two tails: it adds P(X ≤ k1) and P(X ≥ k2). This is commonly used in two-sided discrete setups where the “extreme” outcomes are considered evidence against a baseline.

Why does the tool show both PMF and CDF charts?

PMF shows the probability of each exact k (the “shape”). CDF answers “≤ k” questions at a glance by accumulating probability as k increases. For many students, seeing both is what makes hypergeometric problems finally “click”.

What does “1 in N odds” mean?

It’s an intuitive inversion: 1 / P. For example, if P = 0.25, the odds are “1 in 4”. This is helpful for sanity checks, but probability/percent are the primary outputs.

Why is my hypergeometric distribution skewed?

Skew depends on the success rate K/N and sample size n. If successes are rare, most probability mass sits near small k. If successes are common, it shifts toward larger k. The PMF chart makes skew obvious.

Is hypergeometric the same as binomial when N is large?

They become similar when the population is large and the sample is small relative to N. That’s why many courses allow a binomial approximation when n ≤ 0.05N. This tool can show that approximation and flags whether the rule-of-thumb is met.

What does the “Monte Carlo simulation” option do?

It simulates drawing n items without replacement many times and estimates the PMF/CDF empirically. Exact probabilities are still computed analytically; simulation is for learning, intuition, and validation.

Why does the simulation differ slightly from the exact result?

Random simulation has sampling noise. Increasing the number of trials reduces the difference (typically shrinking like 1/√trials). That’s exactly what makes simulation a good teaching tool.

What is the mean E[X] telling me?

E[X] = n(K/N) is the average number of successes you’d expect over many repeated samples under the same setup. It’s a baseline: results far from the mean are less likely (though discrete distributions can still have meaningful tail mass).

What does SD (standard deviation) tell me here?

SD describes typical spread around the mean. Hypergeometric SD includes a finite-population correction (because draws are without replacement), which often makes SD slightly smaller than a comparable binomial model.

What is the “z-style position” number in the results?

It’s a quick standardized location: roughly (k − mean) / SD. It is not claiming the distribution is normal— it’s simply a compact way to see whether your queried k is below/near/above the expected value.

Why is the tool sometimes disabling charts or the full table?

If the support range becomes extremely large, rendering thousands of rows or large charts can freeze mobile devices. The tool uses a performance safety limit and recommends exporting CSV instead for very large distributions.

How can I export results for homework or a report?

Use Copy Results for a clean text summary, Export to TXT for a file, and Save as CSV for the full distribution table. You can also Print or Save as PDF for a formatted snapshot with charts.

Does the shared link preserve my inputs?

Yes. “Share Calculation Link” generates a URL with your inputs encoded so the page can load the same setup and re-calculate automatically.

Is this tool appropriate for real-money gambling or “predicting” outcomes?

No. This calculator is designed for education, analysis, and understanding probability. It does not predict real-world outcomes and is not gambling advice.

Important Disclaimer:
This hypergeometric distribution calculator is provided for educational and informational purposes only. It does not provide gambling advice, does not predict outcomes, and should not be used for security-critical or high-stakes decision-making without appropriate domain review and professional validation.

Reviewed: December 2025 — formulas, examples, and on-page interpretation guidance checked for clarity and accuracy.

Hypergeometric Distribution Calculator: PMF, CDF & Tails

Enter parameters

Probability query

Inputs

Results

Distribution

PMF (P(X = k))

CDF (P(X ≤ k))

PMF / CDF / Tail Table