Negative Hypergeometric Distribution

The distribution of failures observed before a fixed number of successes when sampling without replacement from a finite population.

Version

v2 · 2026-09-06 · History

Domain-specific #

2359

Origin domain

probability theory

Subdomain

finite population sampling

Aliases

Inverse hypergeometric distribution

Core Idea

Consider a finite population of \(N\) items, exactly \(K\) labeled successes and \(N-K\) failures. Draw uniformly without replacement until the \(r\)-th success appears, where \(1\le r\le K\). If \(X\) is the number of failures observed before that stopping draw, then \(X\) has a negative hypergeometric distribution:

\[ \Pr(X=x)= \frac{\binom{r+x-1}{x}\binom{N-r-x}{K-r}} {\binom{N}{K}}, \qquad x=0,1,\ldots,N-K. \]

This convention makes the stopping target a success. Some sources exchange “success” and “failure” or count total draws \(T=X+r\); parameters must therefore be locked before formulas are compared. The distribution is the without-replacement analogue of the negative binomial distribution.^[1][2]

Structural Signature

• Finite population: \(N\) total items.
• Binary composition: \(K\) target successes and \(N-K\) failures.
• Uniform random order: sampling without replacement is equivalent to a random permutation.
• Fixed target count: stop on the \(r\)-th success.
• Random waiting count: \(X\) counts failures before stopping.
• Finite support: \(0\le X\le N-K\).
• Combinatorial mass: arrangements before the last success and remaining population determine probabilities.
• Depletion dependence: draw probabilities change after every observation.
• Dual convention: success/failure labels and failure-target formulations transform parameters.
• Asymptotic relation: with a large population and stable success proportion, negative binomial behavior emerges.

Recognition test. Verify three roles together: sampling without replacement, a finite binary population, and stopping after a fixed number of one category. With replacement gives a negative binomial law; a fixed sample size gives a hypergeometric law.

What It Is Not

It is not the ordinary hypergeometric distribution, which counts successes in a predetermined number of draws. Here the success target is fixed and sample size is random.

It is not the negative binomial distribution, which models independent Bernoulli trials or sampling with replacement until a target count. Finite-population depletion reduces uncertainty as items are removed.

It is not a negative-valued distribution; “negative” denotes inversion of what is fixed and what is random. It is also not automatically the beta-binomial distribution, although parameter transformations can produce related identities.

Scope of Application

The law models destructive inspection: inspect a finite lot until a required number of defective or conforming items is found, and count the opposite type encountered. It models card draws without replacement, ecological capture from a closed enumerated population, and randomized search through a finite marked set.

In sequential acceptance sampling, the distribution can calculate expected inspection burden under a stop-after-\(r\)-targets rule. The stopping rule here is fixed by count, not optimized; decision thresholds and early rejection may create different distributions.

Finite-population survey designs use the same mathematics when sampling continues until a quota of a known category is reached. However, if category membership is learned only after measurement and the frame's total \(K\) is unknown, the ordinary negative-hypergeometric likelihood is conditional on a quantity the analyst does not possess. Conditioning, estimation, and prediction should be distinguished.

Order-statistic interpretation is often useful. Place the \(K\) successes at random positions among \(N\). Then \(T=X+r\) is the position of the \(r\)-th success. This ties the distribution to finite-population order statistics and gives moments directly.^[2]

Clarity

The probability mass can be derived by fixing \(T=r+x\). Among the first \(r+x-1\) positions, choose the \(x\) failures, equivalently the \(r-1\) successes; position \(r+x\) must be a success; the remaining successes occupy later positions. Counting full success-position sets gives the displayed formula.

The expectation and variance under this convention are

\[ \mathbb E[X]=r\frac{N-K}{K+1}, \]

\[ \operatorname{Var}(X)= \frac{r(N+1)(N-K)(K-r+1)} {(K+1)^2(K+2)}. \]

The mean is not \(r(1-p)/p\) exactly because finite sampling stops from a depleted population; the negative-binomial expression appears asymptotically.

Manages Complexity

One named law replaces enumeration of every possible stopping sequence. From \(N,K,r\), one obtains the full waiting-count distribution, mean inspection effort, tail risks, and quantiles.

The finite-support property is operationally valuable: unlike an infinite Bernoulli sequence, all failures have been exhausted after \(N-K\), and the target is guaranteed because \(r\le K\).

Tail probabilities answer concrete capacity questions: how many inspections must be budgeted so that the target is reached with a stated probability? The hypergeometric cumulative relation computes this without summing every waiting path. Quantiles remain discrete and may not attain an exact nominal probability, so reporting the smallest qualifying integer is clearer than interpolating.

The compression assumes a uniformly random order and correct binary labels. Biased inspection order, misclassification, replacement, an unknown population composition, or adaptive selection require other models.

Abstract Reasoning

The random-permutation view proves normalization because each set of \(K\) success positions among \(N\) is equally likely, and the events indexed by the \(r\)-th success position partition those sets.^[3]

The cumulative event \(X\le x\) means that at least \(r\) successes occur among the first \(r+x\) draws. Thus negative-hypergeometric tails can be computed through a hypergeometric distribution:

\[ \Pr(X\le x)= \Pr\!\left(H_{r+x}\ge r\right), \]

where \(H_n\) counts successes in \(n\) draws without replacement.

If \(N,K\to\infty\) with \(K/N\to p\) and \(0<p<1\) while \(r\) remains fixed, depletion over the relevant early draws becomes negligible and \(X\) converges to failures-before-\(r\)-successes negative binomial.

The limiting statement does not erase the finite-population distinction. For fixed \(N\), every draw changes the remaining success fraction, the support is bounded, and tail probabilities differ from independent Bernoulli sampling. The approximation becomes credible only when the stopping time typically uses a small portion of the population relative to \(N\). A diagnostic comparison should therefore report sampling fraction and verify that the without-replacement law, not a convenient infinite-population substitute, matches the design.

Knowledge Transfer

The same law transfers among cards, inspections, marked objects, and any finite random permutation with two labels. Only the interpretations of success, failure, and stopping change.

Transfer fails for unequal draw probabilities, hidden changing labels, replacement, or population replenishment. Those violations alter exchangeability and invalidate the combinatorial denominator.

Examples

1. Cards. Shuffle a deck containing four aces and stop at the second ace; \(X\) counts non-aces before it with \(N=52,K=4,r=2\).
2. Quality inspection. A lot has \(K\) defective units; inspect random units until \(r\) defects are found.
3. First success. For \(r=1\), \(X\) is the number of failures before the first success in a finite population.
4. All successes. For \(r=K\), stopping occurs at the last success; failures after it are not counted.
5. Total draws. \(T=X+r\) shifts the distribution's support to \(r,\ldots,r+N-K\).
6. Non-example. Replacing each inspected unit returns constant success probability and produces a negative binomial distribution instead.

Structural Tensions

• Convention symmetry vs. formula mismatch: sources swap target labels or counted quantities. Diagnostic: define \(N,K,r,X\) before quoting moments.
• Finite depletion vs. independent-trial intuition: probabilities change after draws. Diagnostic: do not substitute a negative-binomial likelihood.
• Fixed composition vs. uncertain lot: the law conditions on known \(K\). Diagnostic: add a prior or hierarchical model when composition is unknown.
• Random order vs. selection bias: targeted search breaks exchangeability. Diagnostic: audit the sampling mechanism.
• Stopping description vs. decision rule: fixed count is not optimal stopping. Diagnostic: separate stochastic law from policy optimization.
• Autonomy vs. Hypergeometric: both use sampling without replacement, but one fixes sample size and the other fixes successes. Diagnostic: identify which quantity is random.

Structural–Framed Character

The abstraction is structural in the random-permutation stopping position and its combinatorial law, but framed by finite-population probability. Binary labels and without-replacement sampling are literal.

Its autonomy comes from inversion of the hypergeometric sampling plan, not from a surface naming convention.

Structural Core vs. Domain Accent

The core is waiting to a target count under depletion. The domain accent is uniform finite-population sampling and the resulting combinatorial mass. Removing depletion yields negative binomial; removing stopping yields hypergeometric.

All legitimate uses occupy one probability lineage, so it is domain-specific.

Instantiates / Related Primes

Probability Distribution is the proposed minimal parent: this is a strict discrete finite-population distribution. Sampling and Stopping Rule describe generative roles but are broader. Optimal Stopping is not a parent because the stop time is prescribed rather than optimized.

Relationships to Other Abstractions

Current abstraction Negative Hypergeometric Distribution Domain-specific

Parents (1) — more general patterns this builds on

• Negative Hypergeometric Distribution is a kind of Probability Distribution Domain-specific

  Probability Distribution is the proposed minimal parent: this is a strict discrete finite-population distribution.

Hierarchy paths (5) — routes to 3 parentless roots

• Negative Hypergeometric Distribution → Probability Distribution → Random Variable → Function (Mapping)

Show alternative paths (4)

Neighborhood in Abstraction Space

Negative Hypergeometric Distribution sits in a sparse region of the domain-specific corpus (89^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Poisson binomial distribution — 0.80
• Empirical Measure — 0.79
• Particle Filter — 0.79
• Independent and Identically Distributed Random Variables — 0.78
• Murphy's Law — 0.78

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Hypergeometric distribution: successes in a fixed sample size.
• Negative binomial distribution: waiting count with replacement or independent trials.
• Beta-binomial distribution: random-probability overdispersed binomial model.
• Inverse sampling: broader class of sampling plans.
• Optimal stopping rule: chooses a stopping time to maximize reward.
• Total-draw convention: shifted variable \(T=X+r\).

References

[1] “Negative Hypergeometric Distribution,” Encyclopedia of Mathematics, citing Johnson and Kotz, current online edition, https://encyclopediaofmath.org/wiki/Negative_hypergeometric_distribution. registry ↩

[2] Norman L. Johnson, Adrienne W. Kemp, and Samuel Kotz, Univariate Discrete Distributions, 3^rd ed., Wiley, 2005, https://doi.org/10.1002/0471715816. registry ↩a ↩b

[3] William Feller, An Introduction to Probability Theory and Its Applications, vol. 1, 3^rd ed., Wiley, 1968, sections on sampling without replacement. registry ↩