The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer. The method applies to problems with no probabilistic content as well as to those with inherent probabilistic structure. Among all numerical methods that rely on N-point evaluations in M-dimensional space to produce an approximate solution, the Monte Carlo method has absolute error of estimate that decreases as N superscript -1/2 whereas, in the absence of exploitable special structure all others have errors that decrease as N superscript -1/M at best.

History of Monte Carlo method

The method is called after the city in the Monaco principality, because of a roulette, a simple random number generator. The name and the systematic development of Monte Carlo methods dates from about 1944.
There are however a number of isolated and undeveloped instances on much earlier occasions.
For example, in the second half of the nineteenth century a number of people performed experiments, in which they threw a needle in a haphazard manner onto a board ruled with parallel straight lines and inferred the value of PI = 3.14… from the observation of the number of intersections between needle and lines. An account of this playful diversion (indulged in by certain Captain Fox, amongst others, whilst recovering from wounds incurred in the American Civil War) occurs in a paper Hall (A. HALL 1873. " On an experimental determination of PI"). The author of this WEB page has developed the software for simulating of this experiment. Try JAVA implementation of Buffon's needle experiment for the determination of PI.
In 1899, Lord Rayleigh showed that a one-dimensional random walk without absorbing barriers could provide an approximate solution to a parabolic differential equation.
In 1931, Kolmogorov showed the relationship between Markov stochastic processes and certain integro-differential equations.

In early part of the twentieth century, British statistical schools indulged in a fair amount of unsophisticated Monte Carlo work. Most of this seems to have been of didactic character and rarely used for research or discovery. Only on a few rare occasions was the emphasis on original discovery rather than comforting verification. In 1908, student (W. S. Gosset) used experimental sampling to help him towards his discovery of the distribution of the correlation coefficient. In the same year Student also used sampling to bolster his faith in his so-called t-distribution, which he had derived by a somewhat shaky and incomplete theoretical analysis.
The real use of Monte Carlo methods as a research tool stems from work on the atomic bomb during the second world war. This work involved a direct simulation of the probabilistic problems concerned with random neutron diffusion in fissile material; but even at an early stage of these investigations, von Neumann and Ulam refined this particular "Russian roulette" and "splitting" methods. However, the systematic development of these ideas had to await the work of Harris and Herman Kahn in 1948. About 1948, Fermi, Metropolis and Ulam obtained Monte Carlo estimates for the eigenvalues of Schrodinger equation.
In about 1970, the newly developing theory of computational complexity began to provide a more precise and persuasive rationale for employing the Monte Carlo method. The theory identified a class of problems for which the time to evaluate the exact solution to a problem within the class grows, at least, exponentially with M. The question to be resolved was whether or not the Monte Carlo method could estimate the solution to a problem in this intractable class to within a specified statistical accuracy in time bounded above by a polynomial in M. Numerous examples now support this contention. Karp (1985) shows this property for estimating reliability in a planar multiterminal network with randomly failing edges. Dyer (1989) establish it for estimating the volume of a convex body in M-dimensional Euclidean space. Broder (1986) and Jerrum and Sinclair (1988) established the property for estimating the permanent of a matrix or, equivalently, the number of perfect matching in a bipartite graph.

How to use this Applet
We have to look at the unit circle (radius = 1) within a square with sides equal to 2 (see figure 1). Now if we pick a random point (x, y), where both x and y are between -1..1, the probability of that this random point lies inside the unit circle is given as the proportion between the area of the unit circle and the square :


If we pick a random point N times and M of those times the point lies inside the unit circle, the probability of that a random point lies inside the unit circle is given as :


where the dot indicates that this is a discreet distribution (because M and N are integers). But if N becomes very large (theoretically infinite), the two probabilities will become equal and we can write :

---