Download Digital Dice: Computational Solutions to Practical by Paul J. Nahin PDF

By Paul J. Nahin

A few likelihood difficulties are so tricky that they stump the neatest mathematicians. yet even the toughest of those difficulties can frequently be solved with a working laptop or computer and a Monte Carlo simulation, within which a random-number generator simulates a actual strategy, reminiscent of one million rolls of a couple of cube. this is often what Digital Dice is all approximately: tips to get numerical solutions to tricky likelihood difficulties with no need to unravel advanced mathematical equations.

Popular-math author Paul Nahin demanding situations readers to resolve twenty-one tricky yet enjoyable difficulties, from picking the percentages of coin-flipping video games to realizing the habit of elevators. difficulties construct from rather effortless (deciding even if a dishwasher who breaks many of the dishes at a restaurant in the course of a given week is clumsy or simply the sufferer of randomness) to the very tricky (tackling branching strategies of the type that needed to be solved through big apple venture mathematician Stanislaw Ulam). In his attribute type, Nahin brings the issues to existence with attention-grabbing and abnormal ancient anecdotes. Readers study, for instance, not only the right way to confirm the optimum preventing aspect in any choice technique yet that astronomer Johannes Kepler chosen his moment spouse via interviewing 11 women.

The e-book indicates readers easy methods to write effortless computing device codes utilizing any universal programming language, and gives options and line-by-line walk-throughs of a MATLAB code for every problem.

Digital Dice will attract somebody who enjoys well known math or desktop technological know-how. In a brand new preface, Nahin wittily addresses the various responses he bought to the 1st variation.

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Extra info for Digital Dice: Computational Solutions to Practical Probability Problems

Example text

M has done well! m) to an obvious question: What if we had used not a million but fewer simulations? Theoretical analyses of the underlying mathematics of the Monte Carlo technique show that the error of the method decreases as N−1/2 , where N is the number of simulations. So, going from N = 100 = 102 to N = 10,000 = 104 (an increase√in N by a factor of 102 ) should reduce the error by a factor of about 102 = 10. To illustrate more exactly just what this means, take a look at Figure 1, where you’ll see a unit square in the first quadrant, with an inscribed quarter-circle.

Let’s try another geometric probability problem, this one of historical importance. In the April 1864 issue of Educational Times, the English mathematician J. J. Sylvester (1814–1897) submitted a question in geometric probability. As the English mathematician M. W. Crofton (1826–1915), Sylvester’s one-time colleague and protégé at the Royal Military Academy in Woolwich, wrote in his 1885 Encyclopaedia Britannica article on probability,8 “Historically, it would seem that the first question on [geometric] probability, since Buffon, was the remarkable four-point problem of Prof.

3 generator’s cycle length. To gain some historical appreciation of modern random number generators, consider the following complaint made by Lord Kelvin (Scottish engineer, physicist, and mathematician William Thomson [1824–1907]) in a lecture given in April 1900 at the Royal Institution of Great Britain (you can find his talk in the July 1901 issue of The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science under the title, “Nineteenth Century Clouds over the Dynamical Theory of Heat and Light’’).

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