Convergence rate
Typical standard error falls with the square root of runs, so reducing noise by ten requires roughly one hundred times as many runs.
Probability
Binomial Event Simulation Calculator
Run a reproducible seeded Monte Carlo experiment for repeated binomial trials and compare simulated frequencies, mean, and threshold tail directly with exact probabilities.
SEEDED MONTE CARLO LAB
Watch finite-run frequencies converge toward an exact binomial benchmark
A deterministic seed makes the repeated experiment reproducible. Every observed count is placed beside its exact probability so simulation error stays visible.
Simulated threshold probability -
Exact threshold probability -
Signed tail error -
Simulated mean -
Exact mean -
Signed mean error -
LIVE DECISION RECORD
Simulated-versus-exact frequency table
Every run appears in exactly one success-count row; the signed difference exposes Monte Carlo noise.
| Successes k | Simulated frequency | Simulated probability | Exact probability | Signed difference |
|---|
CURRENT CALCULATION PROCESS
Formula, current substitution, intermediate values, and reconciliation
For run j: X_j=sum_i I(U_i<p); p_hat_tail=(1/M)sum_jI(X_j>=r); compare with exact P(X>=r)
| Symbol | Meaning and unit | Current value |
|---|---|---|
| trials | Trials per simulation run - Bernoulli trials inside each repeated experiment. | 24 |
| successProbabilityPct | Success probability per trial (%) - Stable event probability used by both simulation and exact benchmark. | 42 |
| simulations | Simulation runs - Repeated experiments; minimum 100. | 10000 |
| seed | Reproducibility seed - Whole unsigned 32-bit seed for the deterministic generator. | 20260804 |
| thresholdSuccesses | Threshold successes - Count defining the simulated and exact inclusive upper tail. | 13 |
Waiting for valid inputs.
WHO THIS MODEL SERVES
A scoped decision aid, not a universal forecast
Primary audience: Analysts, educators, model validators, and engineers learning or testing finite-trial Monte Carlo behavior.
Decision boundary: Use to study reproducible sampling error under an exact binomial model; it is not a substitute for exact arithmetic or a general stochastic simulator.
HOW TO RUN A REPRODUCIBLE EXPERIMENT
Five steps that preserve the simulation record
- Define the fixed trial count and stable event probability.
- Choose the threshold whose tail frequency matters.
- Set a run count large enough to reveal, but not hide, sampling variation.
- Record the seed before interpreting any apparent agreement.
- Compare simulated mean and tail with exact values, then export all count rows.
SIMULATION FUNDAMENTALS
Five controls behind Monte Carlo evidence
- Pseudo-random generator
- A deterministic algorithm that produces a repeatable sequence from a seed.
- Run
- One complete n-trial experiment yielding one success count.
- Frequency estimator
- Count of qualifying runs divided by total runs.
- Monte Carlo error
- Finite-run difference between simulated estimate and exact probability.
- Closure
- Histogram frequencies must sum to the requested run count.
ALGORITHM AND DEFAULT SUBSTITUTION
Separate the random experiment from its exact benchmark
X_j=sum I(U_i<p); simulated tail=threshold hits/M
Defaults run 10,000 experiments of 24 trials at p=0.42 with seed 20,260,804. Each run counts successes and tests X>=13. The live table compares each simulated count frequency with the exact Binomial(24,0.42) mass.
DEEPER SIMULATION ANALYSIS
Three diagnostics beyond the headline tail
Rare-event starvation
A true probability near one in a million will usually produce no hits in 200,000 ordinary runs; zero observed is not zero risk.
Reproducibility versus independence
A fixed seed supports audit. Changing seeds checks robustness, but neither proves the generator is appropriate for cryptography or high-stakes stochastic modeling.
WORKED SIMULATION CASES
Two runs that teach different lessons
Operations acceptance rehearsal
Ten thousand seeded runs give a stable demonstration of how often 13 or more successes appear. The exact tail reveals whether any apparent optimism is merely finite-run noise.
Minimum-run boundary
Fewer than 100 runs are rejected because an ultra-small demonstration would encourage false precision. Even 100 remains educational, not a rare-tail estimate.
MONTE CARLO TERMINOLOGY
Six terms used in this experiment
- Seed
- Initial state that fixes the pseudo-random sequence.
- Uniform draw
- Number on [0,1) compared with p to generate a Bernoulli result.
- Histogram
- Run counts allocated across possible success totals.
- Estimator
- Sample statistic used to approximate a model quantity.
- Bias
- Systematic expected difference, distinct from one run's random error.
- Exact benchmark
- Analytical binomial result used to audit the simulation.
EVIDENCE RETENTION
A simulation without configuration is not reproducible
Keep trial semantics, p source, run count, seed, generator/version, threshold convention, code revision, and exact benchmark. Retain failed validation states when they affect model governance.
LIMITS AND EXCLUSIONS
What the educational simulator does not claim
- Trials remain independent and equal-probability.
- The pseudo-random generator is deterministic and not cryptographic.
- Run count is capped at 200,000 for browser responsiveness.
- Ordinary Monte Carlo is inefficient for very rare tails.
- No confidence band for Monte Carlo error is provided; compare seeds and exact values.
RELIABLE SOURCES
Primary references for the benchmark distribution
SIMULATION FAQ
Questions about seeds, runs, and error
Why include a seed?
The same seed and inputs reproduce the same pseudo-random sequence, making review and bug diagnosis possible.
Is the simulated result more accurate than the exact binomial result?
No. For this model, the exact result is the benchmark; simulation demonstrates sampling error and supports workflows that later become more complex.
How many runs are enough?
It depends on the tail and acceptable Monte Carlo error. Rare events can require far more than 200,000 runs or specialized methods.
Why can two seeds disagree?
Monte Carlo estimates fluctuate because each seed selects another finite pseudo-random sample.
Does a zero error prove the simulator?
No. One rounded match can occur by chance; verify frequency closure, reproducibility, and behavior across seeds and run counts.
Can this simulate dependent trials?
Not this page. Every trial uses an independent draw with one stable p.
IMPORTANT SIMULATION NOTE
Noise is an output, not a defect
Use the exact comparison to understand finite-run variation. High-stakes simulation requires validated generators, uncertainty quantification, convergence diagnostics, and independent model review.