Monte Carlo Simulator
Estimate π and integrals by random sampling
About this calculator
The Monte Carlo Simulator estimates π or a definite integral by random sampling with a seeded, reproducible PRNG. In π mode it throws darts into the unit square and counts quarter-circle hits, so π ≈ 4 × hit fraction; in integral mode it uses the mean-value estimator (b − a)·mean f(xᵢ) for any expression. Each run reports the estimate, the 1σ Monte Carlo standard error, and the absolute and relative error against ground truth (π itself, or a high-resolution Simpson reference), making convergence visible as samples grow.
How to use the Monte Carlo Simulator calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- π with 10,000 darts (seed 1) → 7,841 hits inside, estimate 3.1364, error ≈ 0.0052
- π with 1,000,000 darts → estimate 3.143164, error shrinks to ≈ 0.0016
- ∫₀¹ x² dx with 100,000 samples → ≈ 0.33207 vs the exact 1/3
- ∫₀^π sin x dx with 50,000 samples → ≈ 2.0021 vs the exact value 2
Frequently asked questions
How does the Monte Carlo Simulator calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Monte Carlo Simulator calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.