Sampling fraction
When the sample is tiny relative to the lot, a binomial approximation may be close. As the fraction grows, without-replacement dependence materially reduces variance.
Probability
Defect Rate Distribution Calculator
Calculate the exact hypergeometric distribution for defective units in a random sample drawn without replacement from a finite lot.
FINITE-LOT SAMPLE DISTRIBUTION
Enumerate every feasible defect count without replacement
Use a fixed lot composition and a simple random sample. The calculator normalizes stable log-combination weights, reports exact point and tail probabilities, and reconciles the full outcome table to the analytic mean.
P(X = k)-
P(X <= k)-
P(X >= k)-
Expected defects-
Standard deviation-
Feasible support-
LIVE DECISION RECORD
Exact finite-lot outcome distribution
Every feasible sample defect count with exact probability, cumulative probability, upper tail, and distance from the expected count.
| Sample defects x | P(X = x) | P(X <= x) | P(X >= x) | Position |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
P(X=x) = C(D,x) C(N-D,n-x) / C(N,n); E[X] = nD/N
| Symbol | Meaning and unit | Current value |
|---|---|---|
| N | Finite lot size, units | 100 |
| D | Known lot defective units | 8 |
| n | Sample size without replacement | 10 |
| k | Selected sample defect count | 2 |
Waiting for valid inputs.
FIVE-STEP USE
Frame a finite-lot probability question
- Confirm the physical lot is finite and sampling removes units from the remaining population.
- Enter the lot size and the fixed defect count used by the scenario.
- Enter a simple random sample size no larger than the lot or 5,000-unit reporting limit.
- Choose an exact defect count and decide whether the point, lower-tail, or upper-tail question matches the decision.
- Export the complete outcome table and retain the randomization procedure with the lot record.
FIVE FINITE-LOT FUNDAMENTALS
Why the distribution changes after every draw
- Without replacement
- An inspected unit is not returned before the next selection.
- Fixed composition
- The scenario conditions on exactly D defective units in N total units.
- Feasible support
- Counts run only from `max(0,n-(N-D))` through `min(n,D)`.
- Point probability
- The mass assigned to exactly one sample defect count.
- Finite correction
- Variance shrinks as the sample consumes a larger share of the lot.
FORMULA AND DEFAULT SUBSTITUTION
Trace a 10-unit sample from a 100-unit lot
P(X=2) = C(8,2) C(92,8) / C(100,10) = 0.15056142397
For `N=100`, `D=8`, and `n=10`, the expected count is `10 x 8/100 = 0.8`. Summing masses through 2 gives 96.8612%; summing from 2 upward gives 18.1950%. The overlap at exactly 2 belongs to both inclusive tails.
THREE DEEPER MODULES
Read the exact table in context
Inclusive-tail overlap
`P(X<=k) + P(X>=k)` equals `1 + P(X=k)`, not one. This is a useful reconciliation and prevents treating overlapping tails as complements.
Lot-composition scenario
If D is not known, rerun plausible compositions. The distribution then shows conditional detection power rather than estimating the unknown lot defect count.
TWO SAMPLING CASES
Partial and complete inspection
Supplier containment sample
From a 100-unit lot believed to contain 8 defectives, a 10-unit draw has a nontrivial chance of seeing none or one. The table makes the miss risk visible before a containment rule is approved.
Inspect the whole lot
With `n=N`, the sample defect count must equal D. Variance becomes zero and the outcome table collapses to one row with probability 100%.
FINITE-LOT GLOSSARY
Six terms for the sampling record
- Lot
- The bounded set from which units are selected.
- Simple random sample
- Selection giving eligible samples equal probability.
- Hypergeometric variable
- Defect count in a fixed-size sample without replacement.
- Combination
- The number of unordered ways to select a subset.
- Cumulative probability
- Total mass at or below a selected count.
- Upper tail
- Total mass at or above a selected count.
LIMITS AND EVIDENCE
Random selection is part of the model
- The lot composition is fixed and supplied; it is not inferred from the sample.
- Sampling must be without replacement and reasonably random across the lot.
- Inspection misclassification, clustering, stratification, and process drift are excluded.
- Samples above 5,000 are blocked because the page reports the complete exact support.
- The result is a conditional probability calculation, not a regulatory acceptance plan.
Retain: lot identity, N and D basis, randomization method, sample identifiers, inspection disposition, selected tail rule, and exported exact table.
RELIABLE SOURCES
Primary finite-sampling references
DEFECT DISTRIBUTION FAQ
Questions about support, tails, and replacement
Why is this not a binomial distribution?
The finite lot is sampled without replacement, so each draw changes what remains. The hypergeometric model includes that dependence and finite-population correction.
What does known lot defects mean?
It is the fixed defective count assumed to exist in the entire lot for this scenario. If it is unknown, the page is a what-if model, not an estimate.
Why can some sample counts be impossible?
A sample cannot contain more defects than the lot holds or more conforming units than the lot holds. The feasible support enforces both constraints.
Which tail should I use?
`P(X>=k)` answers at least k defects; `P(X<=k)` answers at most k. The point probability answers exactly k and should not be confused with either tail.
Why limit the sample to 5,000 units?
The page prints every feasible exact outcome for review. The limit protects browser responsiveness and PDF readability while allowing large finite lots.
Does a small tail probability prove the lot composition is different?
Not by itself. The result is conditional on simple random sampling and the assumed fixed lot composition; inspection errors and selection bias can dominate.
IMPORTANT SAMPLING NOTE
An exact formula cannot repair a biased draw
The probabilities are exact for the stated finite-lot model. A convenience sample from accessible cartons can still make the operational conclusion unreliable.