What is a Monte Carlo simulation? A card-deck explanation
· 6 min read
The idea in one sentence
A Monte Carlo simulation answers “how likely is this?” by doing the random experiment many times on a computer and counting how often it happens.
A worked example
Question: what is the chance a five-card hand contains at least one ace? Simulation recipe: shuffle, deal five cards, check for an ace, record yes or no, and repeat. After 100,000 repetitions about 34,100 contain an ace, so the estimate is about 34.1%.
Exact counting gives 1 − C(48, 5) ÷ C(52, 5) = 34.116%. Because a deck can be counted exactly, it is an ideal place to see how good a simulation is.
estimate = (hands with an ace) ÷ (hands dealt)How accurate is it?
The typical error of an estimated probability p after n trials is √(p(1 − p) ÷ n). To make the estimate ten times more precise you need one hundred times more trials. That slow improvement is the price of the method.
Why use it at all?
For card questions we can often count exactly. Monte Carlo matters when exact counting is too hard: weather models, engineering reliability, finance and epidemics all rely on it. Practising on cards builds the right instincts.
Reproducibility
Computers use pseudo-random numbers that start from a seed. Using the same seed repeats the same simulation exactly, which makes experiments checkable. Every simulator on this site shows its seed.