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Home › Math & Conversions › Core Math & Algebra › Monte Carlo Simulation 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 Normal (Gaussian) Uniform Triangular Custom discrete 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. Minimum (a) Maximum (b) Assumes all values between a and b are equally likely. Minimum (a) Most likely (m) Maximum (b) Must satisfy a <= m <= b. Common in project management (optimistic, most likely, pessimistic). Custom discrete values and probabilities 10:0.2 20:0.5 30:0.3 Probabilities must be non-negative and sum to 1 (within rounding). Values can be any real numbers. 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. Run simulation Clear results 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) Copy stats as CSV 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. Monte Carlo simulation – FAQ How accurate is a Monte Carlo estimate? + The accuracy improves roughly with \( 1/\sqrt{N} \), where \( N \) is the number of simulations. Doubling the number of runs reduces the standard error by about 30%. For many practical decisions, a few tens of thousands of runs are more than enough, but extremely tight confidence intervals may require more. Can I use this for financial risk analysis? + Yes, as a didactic and exploratory tool. For example, you can model annual return as a normal distribution and simulate terminal wealth. However, real financial modelling often requires correlated variables, time-series structure and more complex distributions than this simple one-variable simulator can provide. Why do results change each time I run the simulation? + The simulation uses pseudo-random numbers, so each run samples a different set of scenarios. This is expected behavior in Monte Carlo methods. If you need more stable estimates, increase the number of runs or average results from multiple runs. What if my model has several uncertain inputs? + This calculator focuses on a single aggregate random variable. In practice, you can model each input separately in a spreadsheet or script, compute the resulting outcome for each scenario, and then import the resulting values as a custom discrete distribution, or approximate them with one of the supported distributions. 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