Threshold cliff
Moving r by one count can materially change risk because the distribution is discrete. Any threshold change needs policy authority, not post-result tuning.
Probability
Defect Rate Risk Calculator
Calculate exact binomial rejection probability, expected rejected lots, and consequence-weighted exposure from an assumed defect rate and lot threshold.
FORWARD LOT-REJECTION RISK
Turn an assumed unit defect rate into a lot-level tail event
The calculator sums the exact probability of every defect count at or above the rejection boundary, then scales that per-lot risk across the planned program and consequence assumption.
Lot rejection probability-
Expected rejected lots-
Expected exposure-
Expected defects per lot-
Defect-count SD-
Risk band-
LIVE DECISION RECORD
Lot risk and program exposure ledger
Per-lot probability, program frequency, consequence, and complementary probability reconcile on the current threshold.
| Risk component | Probability or count | Scale | Weighted result | Decision role |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
X~Binomial(n,p); q=P(X>=r); rejected lots=Lq; exposure=LqC
| Symbol | Meaning and unit | Current value |
|---|---|---|
| n | Units per lot | 200 |
| p | Assumed defect probability, percent | 1.5 |
| r | Inclusive rejection count | 7 |
| C | Consequence per rejected lot | 4000 |
| L | Planned lots | 40 |
Waiting for valid inputs.
FIVE-STEP RISK WORKFLOW
Connect the unit model to a governed lot boundary
- Set the lot size and document the data or scenario supporting p.
- Enter the exact inclusive defect count that rejects a lot.
- Define one consequence amount on a consistent accounting basis.
- Enter comparable planned lots in the same risk horizon.
- Review per-lot tail, expected rejected lots, and exposure together before exporting.
FIVE LOT-RISK FUNDAMENTALS
From unit probability to program consequence
- Binomial lot
- n independent units sharing one defect probability.
- Inclusive threshold
- The rejection event includes r and every larger count.
- Upper tail
- Exact sum of all rejection-state probability masses.
- Expected rejected lots
- Per-lot tail multiplied by comparable planned lots.
- Weighted exposure
- Expected rejected lots multiplied by one consequence.
DEFAULT SUBSTITUTION
Screen forty 200-unit lots at 1.5%
E[X]=200 x 0.015=3; q=P(X>=7)=0.03237109209
Expected rejected lots are `40 x q = 1.2948437`. At $4,000 each, expected exposure is about $5,179.37. The 3.2371% tail is exact for the stated binomial model and includes the mass at seven.
THREE DEEPER MODULES
Find the leverage point behind the tail
Rate sensitivity
Small changes in p can multiply a far-tail probability. Evaluate plausible rate bands when the assumed input is uncertain.
Program dependence
Expected rejected lots uses comparable independent lots in expectation. Shared machines, raw material, or shifts can cluster actual rejections.
TWO RISK CASES
A normal tail and a policy boundary
Supplier program reserve
The default tail implies roughly 1.29 expected rejected lots across forty. Operations can use exposure to reserve review capacity while keeping the exact probability visible.
Zero-count rejection
If r=0, every lot rejects because every lot has at least zero defects. A 100% result reveals an invalid or intentionally absolute policy, not a numerical failure.
LOT-RISK GLOSSARY
Six terms in the exposure record
- Assumed rate
- Forward defect probability supplied by the user.
- Rejection count
- Smallest defect count in the adverse set.
- Tail probability
- Total mass at or above the threshold.
- Complement
- Probability of remaining below the rejection count.
- Planning horizon
- The set of comparable lots included in L.
- Exposure
- Probability-weighted consequence across planned lots.
LIMITS AND EVIDENCE
A stable p is the central hypothesis
- Units and lots use one constant independent defect probability.
- The rejection event is count-based and inclusive at r.
- Consequence is fixed per rejected lot with no severity or timing distribution.
- Finite-lot sampling, inspection error, clustering, and process drift are excluded.
- This scenario does not override a governed acceptance or safety plan.
Retain: p source, lot definition, threshold authority, consequence basis, planning horizon, process-stability evidence, and exported tail reconciliation.
RELIABLE SOURCES
Primary binomial and quality references
DEFECT RISK FAQ
Questions about rates, tails, and exposure
How is this different from the confidence calculator?
This page starts with an assumed defect probability and predicts a future lot tail. The confidence page starts with observed counts and infers an interval for the unknown probability.
Does rejection probability equal expected defect rate?
No. The defect rate applies to units; rejection probability applies to the event that a whole lot reaches the count threshold.
Why is the threshold inclusive?
A rejection count of 7 means seven or more defects reject. The exact upper tail therefore includes the probability mass at seven.
What does expected rejected lots mean?
It is planned lots multiplied by per-lot rejection probability. It may be fractional because it is a planning average over repeated comparable programs.
Is consequence-weighted exposure an invoice forecast?
Only under the entered constant consequence per rejected lot. Variable severity, shared-cause events, delays, and capacity effects need a richer model.
What happens when rejection count is zero?
Every lot has at least zero defects, so rejection probability is 100% regardless of the assumed defect rate.
IMPORTANT RISK NOTE
Forward risk is conditional on the supplied p
If the process rate is inferred from limited or stale data, preserve that uncertainty through multiple scenarios instead of treating one input as known.