Monte Carlo Simulation

Turn Complex Uncertainty into Visible Results

Monte Carlo simulation is not "random guessing", but uses extensive random sampling to approximate true probabilities, means, and risk distributions. This page breaks it into two intuitive experiments: one shows how estimation converges, and the other shows how distributions support decision-making.

Random Sampling Repeated Trials Probability Distribution Risk Decision
Enter Interactive Experiment

The Core Idea is Simple

When a problem is hard to solve analytically, rewrite it as a repeatable random experiment.

1. Define ranges or distributions for uncertain factors 2. Randomly sample and run the model once 3. Repeat many times and store results 4. Read mean, quantiles, loss probability, and optimal decisions
Mean Look at long-term average performance, not just one lucky outcome.
Quantiles See what happens in the worst 10% and best 10% cases.
Convergence More samples usually mean more stable estimates.
Decision Don't just give answers, give risk profiles.
Understanding the Method

The essence of Monte Carlo is turning "hard problems" into "repeatable experiments"

As long as you can describe input uncertainties and write the logic for one trial, the rest is just extensive sampling. The final result is no longer a single number, but an entire distribution that explains risk.

1

Define Uncertain Factors First

Such as demand, errors, price fluctuations, random positions. Identify what "can change".

2

Generate Scenarios via Random Sampling

Each sample creates a "possible world" - it's just one possibility, but realistic enough.

3

Repeat the Same Rules

Execute formulas, simulation logic, or business rules thousands of times, collecting output each time.

4

Read Conclusions from Distribution

Mean tells you the general level, quantiles show tail risks, optimal decisions come from comparing alternatives.

General Structure

Random Input → Run Model Repeatedly → Get Output Distribution → Read Expectation / Confidence Band / Loss Probability / Optimal Solution

Interactive Lab

First see "convergence" with π, then see "decision" with profit

Both scenarios allow real-time parameter adjustment. The first helps you understand the method itself, the second shows why it's useful in real problems.

Random Points & Area Relationship
Green points are inside the quarter-circle, red points are outside. When the proportion stabilizes, the π estimate stabilizes too.
Estimate Convergence Curve
Blue line is cumulative estimate, orange line is true π. More samples usually mean smaller fluctuations.
Recent Batch Local Fluctuations
Even as overall results stabilize, individual batches still have random fluctuations - this is the source of Monte Carlo error.
Profit Distribution Histogram
Each bar shows how often a profit range occurred. Wider distribution means higher uncertainty.
Break Even
P10
P50
P90
P10：— P50：— P90：—
Inventory Strategy Comparison
Compare expected profits across different stock levels. Green dot is optimal, orange is your current choice.
How to Read Results
The real value of Monte Carlo isn't a single answer, but giving you the "risk profile".

Waiting for simulation results...

When to Use

When You Care About More Than "Average Answer", But "What Could Happen"

Monte Carlo is often ideal when problems involve significant uncertainty, complex interactions, nonlinear outcomes, or tail risks.

①

Inputs Fluctuate

When demand, costs, errors, arrival times, prices, or failure rates aren't constant, single-point calculations easily distort reality.

②

You Want the Full Distribution

Monte Carlo is natural when you need worst-case, best-case, quantiles, and loss probabilities—not just averages.

③

You Need to Compare Decisions

Different inventories, parameters, and strategies can be compared on the same random samples for more robust choices.