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Monte Carlo simulation
Definition
Monte Carlo simulation uses repeated random sampling to explore the behavior of a model or estimate a quantity that is difficult to calculate directly. Each run uses sampled inputs and produces an outcome. Many outcomes are then summarized to estimate averages, probabilities, ranges, or other features. The results reflect the model and its assumptions, rather than independently revealing what the future must contain.
What it means
Instead of calculating one outcome from one fixed set of inputs, the analyst represents selected inputs with probability distributions and repeatedly samples from them. For example, uncertain task durations can produce a distribution of project completion times. Randomness is a computational tool here: it allows many plausible combinations to be explored according to an explicit sampling design. The quality of the exercise depends on whether those combinations and their probabilities represent the question sensibly.
How it works
First define the outcome and the model that maps inputs to it. Specify distributions for uncertain inputs and any dependencies between them. Draw a sample, run the model, record the output, and repeat. Summaries may include the mean, percentiles, or the share of runs exceeding a threshold. Assess numerical sampling error and check whether conclusions stabilize as the number of runs increases. Keep this error distinct from uncertainty about distributions, parameters, and the model itself. More runs can reduce sampling noise without correcting a mistaken assumption.
Three practical examples
- Scheduling: an operations team simulates uncertain task durations, respecting dependencies, to examine the chance of completing work by a target date.
- Measurement: researchers propagate specified input uncertainties through a calculation to examine the resulting output distribution.
- Capacity planning: a team samples demand and service times to investigate congestion under a proposed staffing arrangement, then checks the simulation against observed operations.
Think of it this way
Think of rehearsing a plan under many deliberately varied conditions. The collection of rehearsals shows how the plan behaves across the conditions represented. The analogy stops at realism: repeating the wrong rehearsal thousands of times cannot reveal a situation that the model never allowed.
Worked example
Consider a toy project with two sequential tasks. Task A takes one day with probability 0.5 or three days with probability 0.5. Task B takes two days with probability 0.75 or four days with probability 0.25. Assume the durations are independent. A simulation samples both durations and adds them. The possible totals are three days with probability 0.375, five days with probability 0.5, and seven days with probability 0.125. Thus the exact probability of exceeding five days is 12.5%, and the exact mean is 4.5 days.
Suppose an illustrative 1,000-run simulation contains 128 totals above five days. Its estimate is 12.8%, rather than exactly 12.5%, because random sampling introduces variation. The approximate standard error of this independent-run proportion is the square root of 0.128 times 0.872 divided by 1,000, about 1.06 percentage points. This toy example permits exact enumeration, which provides a useful check on the simulation. Realistic projects may have many more combinations and dependent durations, making direct enumeration difficult.
Illustrative case study
In a fictional service center, managers simulate call arrivals and handling times to examine waiting times. Their initial model samples each hour independently, but actual busy periods tend to persist. They revise the model to represent that dependence and compare simulated patterns with held-out observations. The revised exercise informs discussion about staffing alternatives without promising an exact future queue. This case shows that realistic relationships between inputs can matter more than simply increasing the number of runs.
Common misconceptions and mistakes
A simulation percentile is not automatically a confidence interval for an estimated mean. A random seed supports reproducibility but does not validate the assumptions. Independent sampling is inappropriate when important inputs are dependent. Rare outcomes may be poorly represented by a modest number of ordinary runs. Monte Carlo simulation is also broader than Markov chain Monte Carlo, which uses dependent samples for particular estimation tasks. A smooth output chart can hide uncertain inputs and structural model errors.
Business use
Use simulation when uncertainty materially affects a decision and a defensible model can connect inputs to outcomes. Explain the origin of distributions, dependence assumptions, and excluded factors. Compare alternatives using appropriate metrics and investigate sensitivity to important assumptions. Report the number of runs and numerical precision without presenting extra decimal places as extra knowledge. Validate simplified cases where exact answers are available. Review tail behavior when unusually severe outcomes matter, because average performance may not capture the operational concern.
People also ask
Does Monte Carlo simulation require AI?
No. It is a general computational method and can use a straightforward mathematical or operational model.
Will more runs make a wrong model correct?
No. They can improve numerical estimation within that model.
Can two runs of the same simulation differ?
Yes, when random samples differ; controlled random seeds can make a specified implementation reproducible.
Check your understanding
Why can a simulated exceedance estimate of 12.8% be reasonable when the toy model's exact value is 12.5%?
Answer: A finite random sample varies. The difference can reflect Monte Carlo sampling error rather than a change in the underlying model probability.
Related terms
- Probability distribution
- Sensitivity analysis
- Standard error
- Markov chain Monte Carlo
- Random sampling
- simulation model
- uncertainty propagation
- random seed
Learn next
Enumerate a small discrete model exactly, then simulate it and compare the estimates across sample sizes. Separate sampling error from model uncertainty. Next, introduce dependent inputs and inspect how the outcome distribution changes.
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Sources
NIST, Monte Carlo uncertainty tool overview: https://www.nist.gov/news-events/news/2020/01/new-tool-account-uncertainty NIST, consistency in Monte Carlo uncertainty analyses: https://www.nist.gov/publications/consistency-monte-carlo-uncertainty-analyses NIST, sampling-quality and uncertainty guidance: https://www.nist.gov/publications/best-practices-quantification-uncertainty-and-sampling-quality-molecular-simulations
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