Drag N, K and n under Parameters: population size, success states, number of draws. K and n never exceed N, and N never drops below K or n. Bars, statistics and every calculator result update at once. Learn more about the population parameters
Statistics shows Expected Value, Variance, Std Deviation and Mode for the current parameters; Show Formulas prints the rule under each card. Learn more about the statistics panel
In the Probability Calculator, tap P(X = k), P(X < k), P(X ≤ k), P(X > k) or P(X ≥ k) and type k; the box below shows the probability to six decimals. Learn more about exact probabilities
Adjust N (population size) to define the total number of items in your finite population. The slider accommodates populations from 5 to 100, suitable for quality control batches and sampling scenarios.
Set K (success states) to specify how many items in the population have the desired characteristic. This must be ≤ N and defines the proportion K/N of successes available.
Configure n (number of draws) for your sample size. This must be ≤ N and determines how many items you're selecting without replacement from the population.
Understanding Parameter Dependencies
The three parameters create constraints that the sliders automatically enforce. As you increase N, both K and n can potentially increase. Decreasing N may force K or n to decrease to maintain K ≤ N and n ≤ N.
The proportion p = K/N plays a crucial role similar to the binomial's p parameter. When the population is homogeneous (p near 0 or 1), the distribution concentrates at extreme values. Balanced populations (p near 0.5) create more spread-out distributions.
Watch how the support (possible values) changes with parameters. The minimum possible successes is max(0, n-(N-K)) and maximum is min(n, K), creating a restricted range compared to binomial.
Reading the Statistics Panel
The Statistics panel sits in the left column, under Parameters. Its four cards show Expected Value, Variance and Std Deviation to four decimals, and Mode as a whole number of successes. Every card recomputes on each slider move, so the panel always describes the bars in the middle column; the red E[X] line on the chart sits at the Expected Value number.
Show Formulas prints the rule under each card: E[X]=n(K/N), Var(X)=n(K/N)(1−K/N)(N−n)/(N−1), σ=Var(X) and Mode=⌊(n+1)(K+1)/(N+2)⌋. The button then reads Hide Formulas. The numbers are the same either way; the formulas only show where they come from.
At the opening values, N=50, K=20, n=10, the cards read 4.0000, 1.9592, 1.3997 and 4. Variance 1.9592 is smaller than the binomial np(1−p)=2.4 for the same proportion, because of the factor (N−n)/(N−1)=40/49: each draw leaves the population. At K=10, n=25 the cards read 5.0000, 2.0408, 1.4286 and 5.
The Parameters cards above the panel repeat each value with its symbol and a one-line meaning: Population Size (N), Success States (K), Number of Draws (n) and Proportion of Successes (K/N). For where the formulas come from, see Mean, Variance, and Finite Population Correction.
Interpreting the PMF Visualization
DemoPMF, CDF and Table tabs
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The PMF bars show P(X = k) only for feasible values in the support. You might notice the display doesn't start at 0 if max(0, n-(N-K)) > 0, reflecting the fact that some success counts are impossible given the parameters.
The distribution's shape depends on n and the ratio K/N. Larger samples (n close to N) create distributions more concentrated around the expected value nK/N. Smaller samples allow more variability.
Compare the hypergeometric shape to binomial with p = K/N. They're similar when N is large relative to n, but hypergeometric shows less spread due to the finite population correction.
The Explorer at Its Opening Parameters
Drawing n=10 from a population of N=50 containing K=20 successes gives eleven bars, k=0 to 10, summing to exactly 1. The peak is at k=4 with P=0.2801, which is also where the red mean line sits: E[X]=nNK=4.
The mean is the same as a binomial with n=10 and p=K/N=0.4. The spread is not.
Hypergeometric, N = 50, K = 20, n = 10: PMF above, CDF below
Peak at k = 4, which is the mean n*K/N. Variance 1.959 against a binomial's 2.4 at the same mean - the difference is the finite population correction (N-n)/(N-1) = 40/49.
The variance here is nNK(1−NK)N−1N−n=2.4×4940=1.959, against the binomial's 2.4. That trailing factor is the finite population correction, and it is always below 1 once n>1.
The reason is that sampling without replacement is self-correcting: each success drawn leaves fewer successes for the next draw, so outcomes cluster more tightly around the mean than independent trials would. Draw the entire population and the variance falls to zero — you would know the answer exactly.
As N grows with n fixed the correction tends to 1 and the hypergeometric converges to the binomial. That is why polling a city is treated as binomial while sampling ten items from a batch of fifty is not.
Using the CDF Display
The CDF steps only at feasible values in the support. Unlike binomial which has support from 0 to n, hypergeometric CDF may start above 0 and end before n depending on your parameters.
The CDF represents P(X ≤ k), crucial for quality control questions: "What's the probability of finding k or fewer defects in my sample?" The curve's steepness around the mean indicates sampling precision.
Notice how the finite population correction affects CDF shape. When sampling a large fraction of the population (n/N is significant), the CDF rises more steeply, showing less variability than the equivalent binomial.
Using the Table Tab
Table is the third tab above the chart, after PMF and CDF. It replaces the chart with three columns, k, P(X = k) and P(X ≤ k), one row per value on the chart's axis, every value to six decimals. The rows follow the sliders: move one and the whole table recomputes.
Use it where the chart is too small to read. At the opening values, N=50, K=20, n=10, the table runs from k=0 to k=10, the support max(0,n−(N−K)) to min(n,K). Row 4 is the largest, 0.280059; row 10 is 0.000018, and P(X≤k) ends at 1.000000.
The P(X ≤ k) column is the CDF as numbers, and each P(X = k) entry is one bar of the PMF chart. Subtract two neighbouring P(X ≤ k) rows to recover a P(X = k) entry. For a value the table does not list, or a range of values, use the Probability Calculator below the chart.
Calculating Exact Probabilities
DemoPoint and cumulative probabilities
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Enter k in the P(X = k) row of the Probability Calculator to compute P(X = k) using: [C(K,k) × C(N-K, n-k)] / C(N,n). The three binomial coefficients account for all ways to select k successes and n-k failures.
The denominator C(N,n) counts all possible samples of size n from population N. The numerator counts samples with exactly k successes: choose k from K successes and n-k from N-K failures.
Try N = 52, K = 13, n = 5 (drawing 5 cards from a deck, counting spades). P(X = 2) ≈ 0.274 gives the probability of exactly 2 spades, accounting for sampling without replacement.
Computing Cumulative and Range Probabilities
DemoRange probabilities
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Use P(X ≤ k) to find the probability of k or fewer successes in your sample. This sums point probabilities over the range [min support, k], providing cumulative mass up to k.
P(X ≥ k) gives the tail probability of k or more successes, useful for acceptance sampling: "If k or more items are defective, reject the lot."
Range calculations P(a ≤ X ≤ b) answer questions about intervals: "What's the probability of finding between 3 and 7 items with the characteristic?" Four boundary options handle inclusion/exclusion of endpoints.
Key Properties and Real-World Applications
The right column holds two lists. Key Properties has 5 rows, each ending in ▼: Support: X ∈ {0, 1, ..., 10}, Sampling Without Replacement, Finite Population Correction, Approximates Binomial Distribution, Symmetric Under Complement. Row titles here are those at the opening values. Tap a row to open its explanation; tap another and the first closes, so one explanation is open at a time; tap the open row to close it.
The support row shows the current smallest and largest k, and Finite Population Correction quotes (N−n)/(N−1) (0.816 at the opening values). Open a row, move a slider, and the numbers in the open explanation follow.
Real-World Applications, below, lists 5 situations the hypergeometric distribution models, starting with Drawing cards from a deck without replacement and Quality control: selecting items from a batch without replacement. This list is fixed text and does not change with the sliders.
The hypergeometric distribution models sampling without replacement from a finite population with two types of items (successes and failures). Each draw changes the population composition, creating dependence between draws.
Unlike the binomial distribution where draws are independent, hypergeometric accounts for depletion - each success drawn reduces remaining successes, altering probabilities for subsequent draws.
Applications include quality control sampling, card game probabilities, ecological sampling (capture-recapture), lottery odds, and poll accuracy analysis. For theoretical foundations and derivations, see hypergeometric distribution theory page.
Hypergeometric vs Binomial
Use hypergeometric for sampling without replacement from small populations. Use binomial for sampling with replacement or when the population is large enough that depletion is negligible.
When N ≥ 10n (population is at least 10 times the sample size), hypergeometric approximates binomial with p = K/N. The approximation improves as N/n increases.
The key difference: binomial assumes constant probability p across trials, while hypergeometric adjusts probabilities after each draw. This distinction matters most when sampling a substantial fraction of the population (n/N > 0.1).
Mean, Variance, and Finite Population Correction
The mean equals n(K/N) = np, matching the binomial mean. With N = 50, K = 20, n = 10, expect 4 successes on average.
When sampling without replacement, you're dividing the population into your sample and the remainder. This "information" about the population reduces uncertainty, decreasing variance compared to independent sampling.
Related Distributions and Calculators
The binomial distribution is the limit as N → ∞ with K/N = p fixed. When population is large relative to sample, binomial provides a good approximation with simpler calculations.
In quality control, hypergeometric models acceptance sampling where you inspect n items from a lot of N, deciding acceptance based on defect count. Standards like MIL-STD-105 use hypergeometric principles.