The Monte Carlo method is actually a broad class of research and analysis methods, with the unifying feature being a reliance on random numbers to investigate a problem. The fundamental premise is that while certain things might be entirely random and not useful over small samples, over large samples they become predictable and can be used to solve various problems.
A simple example of the Monte Carlo method can be seen in a classic experiment, using random dart throws to determine an approximate value of pi. Let’s take a circle and cut it into quarters. Then we’ll take one of those quarters and place it within a square. If we were to randomly throw darts at that square, and discount any that fell out of the square, some would land within the circle, and some would land outside. The proportion of darts that landed in the circle to darts that landed outside would be roughly analogous to one-fourth of pi.
Of course, if we only threw two or three darts, the randomness of the throws would make the ratio we arrived at also fairly random. This is one of the key points of the Monte Carlo method: the sample size must be large enough for the results to reflect the actual odds, and not have outliers affect it drastically. In the case of randomly throwing darts, we find that somewhere in the low-thousands of throws the Monte Carlo method starts to yield something very close to pi. As we get into the high thousands the value becomes more and more precise.
Of course, actually throwing thousands of darts at a square would be somewhat difficult. And making sure to do them entirely randomly would be more or less impossible, making this more of a thought experiment. But with a computer we can make a truly random “throw,” and we can quickly do thousands, or tens of thousands, or even millions of throws. It is with computers that the Monte Carlo method becomes a truly viable method of calculation.
One of the earliest thought experiments like this is known as the Buffon’s Needle Problem, which was first presented in the late-18th century. This presents two parallel strips of wood, with the same width, laying on the floor. It then assumes we drop a needle on to the floor, and asks what the probability is that the needle will land at such an angle that it crosses a line between two of the strips. This can be used to calculate pi to an impressive degree. Indeed, an Italian mathematician, Mario Lazzarini, actually did this experiment, tossing the needle 3408 times, and arrived at 3.1415929 (355/113), an answer remarkably close to the actual value of pi.
The Monte Carlo method has uses far beyond the simple calculation of pi, of course. It is useful in many situations where exact results can’t be computed, as a sort of shorthand answer. It was most famously used in Los Alamos during the early nuclear projects of the 1940s, and it was these scientists who coined the term Monte Carlo method, to describe the randomness of it, as it was similar to the many games of chance played in Monte Carlo. Various forms of the Monte Carlo method can be found in computer design, physical chemistry, nuclear and particle physics, holographic sciences, economics, and many other disciplines. Any area where the power needed to calculate precise results, such as the movement of millions of atoms, can potentially be greatly assisted by utilizing the Monte Carlo method.