Run-count precision
Standard error falls approximately with the square root of runs. Four times as many runs roughly halves Monte Carlo error, subject to the draw cap.
Probability
Defect Rate Simulation Calculator
Run a reproducible seeded Monte Carlo model of independent defect counts and compare simulated rejection risk with the exact binomial benchmark.
SEEDED DEFECT-COUNT EXPERIMENT
Simulate repeated lots and audit the result against exact probability
The page generates a reproducible sequence of independent unit outcomes, builds a defect-count histogram, and separates simulation uncertainty from the assumed process rate.
Simulated rejection risk-
Exact rejection risk-
Simulated mean-
Exact mean-
Tail standard error-
Total random draws-
LIVE DECISION RECORD
Simulated and exact defect-count histogram
Observed run count, simulated share, exact binomial mass, and simulation-minus-exact difference for every material count.
| Defects | Simulated lots | Simulated mass | Exact mass | Difference |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
X_j = sum Bernoulli(p); qHat = count(X_j >= r)/R; SE(qHat) = sqrt(qHat(1-qHat)/R)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| n | Units per simulated lot | 50 |
| p | Assumed defect rate, percent | 4 |
| R | Simulation runs | 10000 |
| r | Rejection defect count | 5 |
| seed | Pseudo-random initial state | 20260804 |
Waiting for valid inputs.
FIVE-STEP SIMULATION
Run an experiment another analyst can repeat
- Define a lot size and a stable per-unit defect probability for the scenario.
- Set the rejection count using the actual screening rule, including whether equality rejects.
- Choose enough runs for the desired Monte Carlo precision without exceeding five million draws.
- Record the seed before interpreting the histogram.
- Compare simulated mean and tail with exact binomial values, then export both.
FIVE SIMULATION FUNDAMENTALS
What randomness on this page represents
- Bernoulli draw
- One independent unit is defective with probability p.
- Virtual lot
- The sum of n Bernoulli draws in one run.
- Seed
- An integer that fixes the pseudo-random sequence.
- Monte Carlo estimate
- The fraction of simulated lots meeting the event definition.
- Benchmark
- The exact binomial result used to audit simulation behavior.
DEFAULT SUBSTITUTION
Simulate 10,000 fifty-unit lots at 4%
Exact E[X] = np = 50 x 0.04 = 2 defects; reject when X >= 5
The seeded engine makes 500,000 unit draws. It reports `qHat` from the simulated rejection count, compares it with the exact upper tail, and gives `sqrt(qHat(1-qHat)/10000)` as the simulation-only standard error.
THREE DEEPER MODULES
Separate model risk from run noise
Rare-tail visibility
A zero simulated count does not prove zero exact risk. When the event is rare relative to R, the exact comparator is essential.
Seed sensitivity
Changing seeds should move results within ordinary simulation error. Large systematic differences suggest too few runs or a problem in scenario framing.
TWO SIMULATION CASES
Ordinary variability and a deterministic edge
Rejection planning
A production lead uses the histogram to explain why identical-rate lots can produce different observed counts, then uses the exact tail for the policy estimate.
Zero assumed rate
At p=0, every simulated lot has zero defects. A positive rejection threshold has zero simulated and exact risk; a threshold of zero rejects every lot by definition.
SIMULATION GLOSSARY
Six terms in the reproducibility record
- Pseudo-random
- Deterministic sequence designed to emulate random draws.
- Histogram
- Counts of virtual lots at each observed defect total.
- Upper tail
- Probability of the threshold count or more defects.
- Standard error
- Run-to-run uncertainty of the simulated event fraction.
- Exact mean
- `np`, the binomial expected count.
- Reproducibility
- Ability to regenerate the same result from inputs and seed.
LIMITS AND EVIDENCE
Simulation precision is not process validity
- Units and lots are independent with one constant defect probability.
- The Park-Miller generator supports reproducibility, not cryptographic randomness.
- Clustering, drift, inspection error, rework, and finite-lot depletion are excluded.
- The five-million-draw domain protects responsive live recalculation.
- Use governed acceptance rules for release decisions.
Retain: inputs, seed, code/version date, run count, exact comparator, histogram, sampling basis for p, and threshold authority.
RELIABLE SOURCES
Primary distribution and sampling references
DEFECT SIMULATION FAQ
Questions about seeds, runs, and exact checks
Why require a seed?
The seed makes the pseudo-random sequence reproducible so another reviewer can rerun the same scenario and obtain the same histogram.
What is Monte Carlo error?
A finite number of runs introduces random estimation noise. The reported tail standard error quantifies that noise for the simulated rejection fraction.
Why also show an exact result?
This scenario has a known binomial benchmark. Comparing simulation with exact probability distinguishes sampling noise from a coding or interpretation problem.
Can simulation model process drift?
Not on this page. Every unit uses one stable probability. Drift, clustering, and batch-to-batch random effects require a different simulation design.
Why is there a five-million-draw limit?
The live page executes one Bernoulli draw per unit per run. The cap keeps keyboard interaction and mobile recalculation responsive.
Does more simulation fix a wrong defect rate?
No. More runs reduce Monte Carlo noise around the assumed model; they cannot make an unsupported input probability representative.
IMPORTANT SIMULATION NOTE
More runs refine the assumed world
A precise simulation of an unrepresentative defect rate is still unrepresentative. Validate the model inputs before increasing run count.