The Monte Carlo method uses random numbers to investigate problems and is useful in situations where exact results cannot be calculated. It was famously used in the early nuclear projects of the 1940s and has applications in various disciplines, including computer design, physics, economics, and more. A simple example is using random throws of darts to determine an approximate value for pi. The sample size must be large enough for the results to reflect actual probabilities. The method was named after the many games of chance played in Monte Carlo.
The Monte Carlo method is actually a broad class of research and analysis methods, with the unifying feature of relying on random numbers to investigate a problem. The basic premise is that while some things might be completely random and not useful in small samples, in 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 throws of darts to determine an approximate value for pi. We take a circle and cut it into quarters. So we’re going to take one of those quarters and place it inside a square. If we were to randomly throw darts at that square and discard any that fell out of the square, some would land inside 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 a quarter of pi.
Of course, if we only threw two or three darts, the randomness of the throws would also make the ratio we arrived at quite 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 probabilities and not have outliers that drastically affect it. In the case of randomly thrown darts, we find that somewhere out of thousands of throws the Monte Carlo method starts producing something very close to pi. As we get to the thousands, the value gets more and more precise.
Sure, throwing thousands of darts into a square would be a bit difficult. And making sure to run them completely randomly would be more or less impossible, making it more of a thought experiment. But with a computer we can make a truly random “throw” and we can quickly make thousands, or tens of thousands, or even millions of throws. It is with computers that the Monte Carlo method becomes a really good calculation method.
One of the first thought experiments like this is known as Buffon’s needle problem, first presented in the late 18th century. This has two parallel wooden slats, of the same width, lying on the floor. He then assumes that we drop a needle on the floor and asks what is the probability that the needle lands at such an angle that it crosses a line between two of the strips. This can be used to calculate pi impressively. In fact, an Italian mathematician, Mario Lazzarini, actually did this experiment, flipping the needle 18 times, and arrived at 3408 (3.1415929/355), an answer remarkably close to the actual value of pi.
The Monte Carlo method has uses far beyond just calculating pi, of course. It is useful in many situations where exact results cannot be calculated, as a sort of shorthand answer. It was most famously used at Los Alamos during the early nuclear projects of the 1940s, and it was these scientists who coined the term Monte Carlo method, to describe its randomness, 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 required to calculate precise results, such as the motion of millions of atoms, can potentially be greatly assisted using the Monte Carlo method.
Protect your devices with Threat Protection by NordVPN
