Monte Carlo method

The Monte Carlo method emerged from a game of solitaire during illness. In 1946, Stanislaw Ulam lay convalescing at Los Alamos and wondered about the probability of a Canfield solitaire coming out successfully. Combinatorial calculations proved intractable, but he realized laying out the game a hundred times and counting successes would give a practical estimate. Immediately, he thought: "How to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations."

The adjacent possible was computational. ENIAC, completed in 1945, could perform 5,000 additions per second—10,000 times faster than human calculators. It weighed 30 tons, consumed 150 kilowatts, and required days to reprogram via plugboards, but it could execute the millions of random trials that Monte Carlo required. The problem demanding solution was neutron diffusion through fissionable material—calculating radiation shielding for hydrogen bomb development. Scientists had all the necessary data about neutron travel distances and collision probabilities but could not solve the problem using conventional differential equations.

Ulam described his insight to John von Neumann during an especially long conversation in a government car driving from Los Alamos to Lamy. Von Neumann immediately recognized the potential. On March 11, 1947, he sent a handwritten letter to Robert Richtmyer containing the first formulation of Monte Carlo computation for an electronic computer. Nicholas Metropolis suggested the codename "Monte Carlo" because Ulam's uncle had frequently borrowed money from relatives to gamble at the casino in Monaco. By April-May 1948, the first Monte Carlo calculations ran on ENIAC, with Klára von Neumann coding for 32 straight days in Maryland.

The cascade extended far beyond nuclear physics. In 1964, David Hertz introduced Monte Carlo to corporate finance in Harvard Business Review. By 1977, Phelim Boyle had pioneered derivatives valuation using random sampling. Today, Monte Carlo methods price trillions in financial derivatives, calculate Value at Risk for global banks under Basel requirements, train deep neural networks when gradients fail, simulate climate uncertainty for IPCC reports, design drug molecules through binding affinity prediction, and model radiation dose distributions for cancer therapy. The method invented to calculate hydrogen bomb physics now underpins everything from Wall Street risk models to protein folding algorithms—all because a mathematician played solitaire while sick.