Negative Hypergeometric Distribution¶
The distribution of failures observed before a fixed number of successes when sampling without replacement from a finite population.
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:
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.
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.
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.
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 \(r\le K\).
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.
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:
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.
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)
- Negative Hypergeometric Distribution → Probability Distribution → Probability → Measure → Set and Membership
- Negative Hypergeometric Distribution → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Negative Hypergeometric Distribution → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Negative Hypergeometric Distribution → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Negative Hypergeometric Distribution sits in a sparse region of the domain-specific corpus (89th 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