Overview
Monte Carlo Simulation samples random inputs or events and summarizes the resulting distribution of outcomes.
FoundationSource checked
Monte Carlo Simulation
Estimates outcomes by running many random trials.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Simulation
- Difficulty
- Beginner
- Family
- Stochastic Simulation
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 7
- Code example
- Not ready
- Visualization
- Ready
- Aliases
- Monte Carlo Method
Core Concepts
random trialsample averageuncertainty estimaterepeatable seedsimulated staterandom samplingsample selectionrandom draw
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(t)
Average
O(t)
Worst
O(t)
Space
O(1)
When to Use
- Use it when a quantity can be estimated by repeated random trials.
- Use it when exact analysis is difficult but sampling the process is straightforward.
- Use it with confidence intervals or error estimates when reporting results.
When Not to Use
- Avoid it when deterministic exact computation is cheap.
- Avoid it when sampling from the required distribution is not possible or is biased.
- Avoid reporting estimates without uncertainty or sample-count context.
What It Does
- Runs repeated random experiments
- Estimates averages, probabilities, or risk
Use Cases
- Risk analysis
- Physics and finance models
- Estimating hard integrals
Inputs
- Random input model
- Trial count
- Outcome function
Outputs
- Estimated statistic
- Outcome distribution
Key Formulas (1)
Monte Carlo estimate
estimate = ∑(sampleresults) / samplecount
estimate = sum(sample_results) / sample_countApproximates a quantity by averaging many random trials.
Pseudocode
- 1.Define a random experiment
- 2.Run the experiment many times
- 3.Record the result of each trial
- 4.Aggregate the results
- 5.Return the estimated statistic
Canonical Example
- Input
- 10000 random points in the unit square
- Output
- estimate of pi from the fraction inside the quarter circle
- Explanation
- The area fraction inside x^2 + y^2 <= 1 estimates pi / 4.
Implementation Notes
- Define the random experiment and statistic before running trials.
- Use a seedable random source for tests and experiments.
- Track the number of samples, mean, and uncertainty estimate together.
Common Pitfalls
- Sampling from the wrong distribution and getting a precise but biased estimate.
- Using too few trials for the expected variance.
- Failing to control random seeds for reproducible tests.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow many sampled outcomes accumulating into an estimate.
Step 1 of 5: Define a random experiment
random-trialsNotes
Monte Carlo error usually decreases proportionally to one over the square root of the sample count.
Sources (1)
What each reference supports for this entry.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
AlternativesEuler-Maruyama Method
Numerical Methods · Extended
Both model randomness but at different levels of simulation detail.
Compare with thisAlternativesQ-Learning
Machine Learning · Specialized
Q-learning learns value estimates from experience rather than pure simulation averages.
Compare with thisUsesMersenne Twister
Randomness & Sampling · Core
Monte Carlo simulation requires pseudorandom samples from a generator.
Compare with thisRelationships
View full relationship pageUsesMersenne Twister
Monte Carlo simulation requires pseudorandom samples from a generator.
GeneralizesMonte Carlo Integration
Monte Carlo integration is a numerical integration use of Monte Carlo simulation.
Related Algorithms
Tags
SimulationStochasticSamplingRandomness