Calculator Inputs
Formula Used
The calculator uses the Poisson probability model. It is appropriate when defects or events occur randomly and independently within a fixed inspection opportunity.
Poisson probability:
P(r) = (e-λ × λr) / r!
For zero occurrences:
P(r=0) = e-λ
When using defect rate and sample size:
λ = u × n
How to Use This Calculator
- Select the calculation method.
- Enter either defect rate and sample size, or direct λ.
- Choose decimal precision for the displayed answer.
- Set a chart maximum to compare nearby scenarios.
- Press Calculate P(r=0).
- Review λ, P(r=0), P(r≥1), and the interpretation box.
- Use the chart to see how probability changes as λ grows.
- Export the summary with the CSV or PDF buttons.
Example Data Table
| Average defects per unit (u) | Sample size (n) | λ = u × n | P(r=0) | P(r≥1) |
|---|---|---|---|---|
| 0.05 | 5 | 0.25 | 0.7788 | 0.2212 |
| 0.08 | 10 | 0.80 | 0.4493 | 0.5507 |
| 0.12 | 5 | 0.60 | 0.5488 | 0.4512 |
| 0.20 | 4 | 0.80 | 0.4493 | 0.5507 |
| 0.30 | 3 | 0.90 | 0.4066 | 0.5934 |
Frequently Asked Questions
1. What does P(r=0) represent?
It represents the probability of observing zero defects, zero failures, or zero counted events during the inspected opportunity. In practice, it estimates how often a sample appears completely clean.
2. Why is the Poisson model used here?
The Poisson model is common for defect counts when events are rare, independent, and randomly distributed over a fixed area, time period, lot, or sample.
3. What is λ in this calculator?
λ is the expected number of occurrences in the inspected opportunity. It may come from a direct estimate, or from multiplying average defects per unit by the number of units inspected.
4. Can I use defect rate and sample size together?
Yes. Choose the rate-and-sample option. The calculator multiplies the average defects per unit by the inspected units to produce λ automatically before finding P(r=0).
5. What if defects are clustered or dependent?
Then the Poisson assumption may be weak. Results can mislead when events are not random or independent. In that case, review other count models or use real process data.
6. Does a larger sample always lower P(r=0)?
If the defect rate stays constant, yes. A larger inspected sample increases λ, and higher λ reduces the probability of observing zero occurrences.
7. How should I interpret a very small P(r=0)?
A very small value means a defect-free sample is unlikely under the stated average rate. It suggests your process usually produces at least one observed occurrence.
8. Can I export the calculated results?
Yes. After calculation, use the CSV button for spreadsheet work or the PDF button for reports, reviews, and documentation.