Monte-Carlo Simulations

Monte Carlo simulation uses randomness and the Law of Large Numbers to estimate quantities that are hard to compute analytically. A simple example is estimating π by throwing random (x,y) darts at a unit square and counting hits inside the inscribed quarter circle; the hit ratio times four approximates π, but error falls only like 1/√N so high precision requires huge sample sizes (the post notes ~10 billion samples for four-decimal reliability). Monte Carlo excels at numerical integration and high-dimensional problems. It also handles intractable combinatorics: with the 81-card game SET, there are C(81,12)=70,724,320,184,700 twelve-card hands and Knuth found 2,284,535,476,080 have no set, giving a 3.23% chance; a 10 million-run Monte Carlo reproduced 0.0323, and the same approach estimates the 15-card no-set probability at about 0.00037.

Monte Carlo is indispensable for financial projections because averages hide sequence risk and volatility. A naive deterministic model with 4% nominal return and 2% inflation gives a ~1.96% real return and would deplete $1M after ~55 years when withdrawing $30k/year, but stochastic simulations tell a different story. Running 1,000,000 trials that draw returns and inflation from realistic distributions shows wide variation: 25% of scenarios exhaust funds by ~22 years and only about 60% last 30 years, revealing substantially higher risk than the simple average implies. All code for the examples is available on GitHub.