What can I do with this Monte Carlo simulation calculator?

This calculator lets you choose a probability distribution (normal, uniform, triangular, or custom discrete), run thousands of simulations in your browser, and obtain summary statistics such as the mean, standard deviation, percentiles, and the probability of exceeding a chosen threshold.

How many simulations should I run?

It depends on the precision you need and the complexity of your model. For quick intuition, a few thousand runs are often enough. For more stable estimates, 10,000 to 50,000 simulations are common. This calculator caps simulations at 200,000 per run to keep performance reasonable in a browser.

What distributions does this tool support?

The tool supports normal (Gaussian) distribution, uniform distribution, triangular distribution, and a custom discrete distribution where you specify values and their probabilities. You can use these to approximate many real-world situations in risk and uncertainty analysis.

Is the Monte Carlo simulation result exact?

No, Monte Carlo results are approximate and subject to sampling variability. Running more simulations typically reduces this variability, but the results will always be estimates of the true underlying probabilities and statistics.

Monte Carlo Simulation Calculator

Run Monte Carlo simulations directly in your browser. Choose a probability distribution, define the number of runs, and obtain an empirical distribution of outcomes with summary statistics, percentiles and the probability of exceeding a threshold.

This tool is suitable for statistics students, engineers, analysts and decision-makers who need a quick way to explore uncertainty and risk without installing any software.

1. Choose distribution and parameters

Distribution type

Start with a simple distribution. For more complex models you can approximate the outcome as a single random variable and simulate it here.

Mean (μ)

Standard deviation (σ)

σ must be positive. Many finance and engineering models are approximated with a normal distribution.

2. Simulation settings

Number of simulations

Up to 200,000 runs per execution.

Threshold (optional)

If set, the tool will estimate P(X ≥ threshold).

Decimals in output

Controls rounding in the statistics table.

Pure client-side – no data uploaded

Monte Carlo results

Summary statistics

Simulations (n)
Mean
Standard deviation
Minimum
Maximum
Median (50th pct)
5th percentile
95th percentile
Threshold
P(X ≥ threshold)

Simulation settings used

Results are empirical: repeat the simulation (or increase n) to see how estimates stabilize.

Approximate histogram (10 bins)

Bin range	Count	Relative freq.	Cumulative freq.

Histogram bins are computed from the simulated minimum and maximum using equal-width bins. For publication-quality plots, export the CSV statistics and use a dedicated plotting tool.

What is a Monte Carlo simulation?

A Monte Carlo simulation is a numerical method that uses repeated random sampling to estimate the behavior of a system with uncertainty. Instead of solving a problem analytically, you model the uncertain inputs as random variables and simulate many possible scenarios.

After many runs, you get an empirical approximation of the distribution of outcomes: averages, percentiles, and probabilities. This is extremely useful when the model is too complex for closed-form formulas, or when you want to visualize risk in a way that stakeholders can understand.

Basic Monte Carlo procedure

Define a model \( Y = g(X_1, X_2, \dots, X_k) \) where \( X_i \) are random inputs.
Choose probability distributions for each \( X_i \) (e.g. normal, uniform, triangular).
For each simulation run \( j = 1, \dots, N \):
- Sample \( x_1^{(j)}, x_2^{(j)}, \dots, x_k^{(j)} \) from their distributions.
- Compute the outcome \( y^{(j)} = g(x_1^{(j)}, \dots, x_k^{(j)}) \).
Use the simulated outcomes \( y^{(1)}, \dots, y^{(N)} \) to estimate:
- Mean and standard deviation.
- Percentiles (e.g. 5th, 50th, 95th).
- Probabilities of events, e.g. \( \mathbb{P}(Y \geq y_0) \).

Supported distributions in this calculator

Normal (Gaussian) distribution

Defined by a mean \( \mu \) and standard deviation \( \sigma \), with density

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right). \]

This is widely used for modelling measurement errors, aggregated effects and approximate returns.

Uniform distribution

All values in \([a,b]\) are equally likely. This is useful when you only know a range, not a shape.

Triangular distribution

Defined by minimum \( a \), most-likely value \( m \) and maximum \( b \). It approximates expert estimates of “optimistic–most likely–pessimistic” scenarios and is common in project management and cost estimation.

Custom discrete distribution

You specify a finite set of values \( x_i \) and probabilities \( p_i \) such that \( p_i \ge 0 \) and \( \sum_i p_i = 1 \). This covers scenarios like demand levels, failure modes, or aggregated outcomes from more complex models.