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Monte Carlo methods are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: [2] optimization, numerical integration, and generating draws from a probability distribution.
An illustration of Monte Carlo integration. In this example, the domain D is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (π*1.0 2) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of 4*0.8 = 3.2 ≈ π.
This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below. These Monte Carlo methods for approximating π are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained.
A Monte Carlo simulation shows a large number and variety of possible outcomes, including the least likely as well … Continue reading → The post Understanding How the Monte Carlo Method Works ...
The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics .
This can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question. [3] The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.
Machin's particular formula was used well into the computer era for calculating record numbers of digits of π, [39] but more recently other similar formulae have been used as well. For instance, Shanks and his team used the following Machin-like formula in 1961 to compute the first 100,000 digits of π: [39]
The digits of pi extend into infinity, and pi is itself an irrational number, meaning it can’t be truly represented by an integer fraction (the one we often learn in school, 22/7, is not very ...