Monte Carlo Simulation

Monte Carlo methods replace a difficult calculation with many simulated trials. First define the quantity and an input model, then draw values according to that model, compute the result for each trial and summarize the results. MIT's introductory lecture describes the method as estimating an unknown quantity using sampling and inferential statistics. Draws describe the specified model, whose assumptions may be wrong.

A geometric example estimates π. Draw points uniformly from a square with x and y each between −1 and 1. The unit circle inside has area π while the square has area four, so the fraction of points with x² + y² ≤ 1 approaches π/4 as independent draws accumulate. Multiply that fraction by four. If an invented run puts 785 of 1,000 points inside, its estimate is 4 × 785/1,000 = 3.14. That is one noisy result, not a new exact value of π. A different random run will generally differ.

The same pattern can model uncertainty in a project or a measured quantity: draw uncertain inputs, propagate them through a calculation and inspect the output distribution. Choose distributions that reflect evidence and preserve important dependencies. If two costs rise together, sampling them independently can badly distort the risk estimate. Rare outcomes also need enough trials to be represented; the absence of a rare event in a small run is not proof it cannot occur.

Repeating more independent trials reduces ordinary sampling noise, often at a rate proportional to one over the square root of the number of draws. Roughly four times as many draws can halve the standard error for a simple sample mean, not divide it by four. Record the random seed for reproducibility and compare several runs or uncertainty summaries. When an exact calculation is available and cheap, use it as a check. A simulation is a tool for exploring assumptions, not a guarantee of accuracy or future performance.