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1 Introduction to Monte Carlo Methods

Numerical methods that are known as Monte Carlo methods can be loosely described as statistical simulation methods, where statistical simulation is defined in quite general terms to be any method that utilizes sequences of random numbers to perform the simulation. Monte Carlo methods have been used for centuries, but only in the past several decades has the technique gained the status of a full-fledged numerical method capable of addressing the most complex applications. The name ``Monte Carlo'' was coined by Metropolis (inspired by Ulam's interest in poker) during the Manhattan Project of World War II, because of the similarity of statistical simulation to games of chance, and because the capital of Monaco was a center for gambling and similar pursuits. Monte Carlo is now used routinely in many diverse fields, from the simulation of complex physical phenomena such as radiation transport in the earth's atmosphere and the simulation of the esoteric subnuclear processes in high energy physics experiments, to the mundane, such as the simulation of a Bingo game or the outcome of Monty Hall's vexing offer to the contestant in ``Let's Make a Deal.'' The analogy of Monte Carlo methods to games of chance is a good one, but the ``game'' is a physical system, and the outcome of the game is not a pot of money or stack of chips (unless simulated) but rather a solution to some problem. The ``winner'' is the scientist, who judges the value of his results on their intrinsic worth, rather than the extrinsic worth of his holdings.