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Digital Dice: Computational Solutions to Practical Probability Problems

4.4 out of 5 stars 7 customer reviews
ISBN-13: 978-0691126982
ISBN-10: 0691126984
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Editorial Reviews

Review

"The problems are accessible but still realistic enough to be engaging, and the solutions in the back of the book will get you through any sticky spots. Writing your own versions of a few of these programs will acquaint you with a useful approach to problem solving and a novel style of thinking."--Brian Hayes, American Scientist

"[T]he book is targeted at teachers and students of probability theory or computer science, as well as aficionados of recreational mathematics, but anyone who is familiar with the basics of probability and is capable of writing simple computer programs will have no problem working their way through this interesting and rewarding book."--Physics World

"[An] enjoyable read, as [Nahin] writes clearly, with humour and is not afraid to include equations where necessary. Nahin spices the book throughout with factual and anecdotal snippets. Digital Dice will appeal to all who like recreational mathematics."--Alan Stevens, Mathematics Today

"Digital Dice will appeal to recreational mathematicians who have even a limited knowledge of computer programming, and even nonprogrammers will find most of the problems entertaining to ponder."--Games Magazine

"After the appearance of the author's earlier book on probability problems, [Duelling Idiots And Other Probability Puzzlers], one has high expectations for this book, and one is not disappointed. . . . The book will certainly have great appeal to all three of the targeted audiences."--G A. Hewer, Mathematical Reviews

"This well-written entertaining collection of twenty-one probability problems presents their origin and history as well as their computer solutions. . . . These problems could be used in a computer programming course or a probability course that includes Monte Carlo simulations."--Thomas Sonnabend, Mathematics Teacher

"All of the books by Nahin and Havil are worth having, including others not listed here. I particularly recommend Digital Dice for the task of teaching undergraduates in mathematics the fundamentals of computation and simulation."--James M. Cargal, The UMAP Journal

From the Back Cover


"Paul Nahin's Digital Dice is a marvelous book, one that is even better than his Duelling Idiots. Nahin presents twenty-one great probability problems, from George Gamow's famous elevator paradox (as corrected by Donald Knuth) to a bewildering puzzle involving two rolls of toilet paper, and he solves them all with the aid of Monte Carlo simulations and brilliant, impeccable reasoning."--Martin Gardner


"Nahin's new book is a rich source of tantalizing, real-life probability puzzles that require considerable ingenuity, and in most cases computer simulation, to solve. Though written to be delved into rather than read cover-to-cover, Digital Dice has an engaging and often witty style that makes each chapter a pleasurable read."--Keith Devlin, author of The Math Gene and The Math Instinct


"Open this delightful, matchless book to be sucked into a treasure trove of wonderful conundrums of everyday life. Then, persuaded by straightforward Monte Carlo simulation exercises, emerge refreshed, invigorated, and fully satisfied by the unique experience of learning from Nahin's marvelous Digital Dice."--Joseph Mazur, author of The Motion Paradox


"One of the strengths of Digital Dice is its wealth of historical information. Nahin carefully notes the origin of each problem and traces its history. He also tells a number of amusing anecdotes. I found all the problems interesting, especially Parrondo's Paradox. Anyone who has not met this paradox will be amazed by it! Digital Dice is a very enjoyable read."--Nick Hobson, creator of the award-winning Web site Nick's Mathematical Puzzles


"By presenting problems for which complete theoretical analysis is difficult or currently impossible, Digital Dice is a reminder that mathematics is often advanced by investigation, long before theoretical tools are brought to bear. The book's choice of problems is eclectic and interesting, and the explanations are clear and easy to read. A welcome addition to popular mathematical literature."--Julian Havil, author of Nonplussed!: Mathematical Proof of Implausible Ideas


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Product Details

  • Hardcover: 280 pages
  • Publisher: Princeton University Press (March 23, 2008)
  • Language: English
  • ISBN-10: 0691126984
  • ISBN-13: 978-0691126982
  • Product Dimensions: 9.4 x 6.4 x 0.9 inches
  • Shipping Weight: 1.2 pounds
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (7 customer reviews)
  • Amazon Best Sellers Rank: #976,433 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

Format: Hardcover
I have liked Nahin's other books so I bought this one since I wanted to learn more about simulating probability problems. All seemed well, the examples were written in Nahin's literate style and seemed mathematically sound. Then I got interested in a particular problem, No. 12, "How many runners in a marathon" where you try to estimate the total size of a population from samples. I ran into major problems in many areas of the discussion.

First, in the theory, Nahin does not show the derivation of the key formula that E{Xmax} = n*(N+1)/(n+1) even though he obviously had a simplified derivation available in a reference. I found the reference "Estimating the Size of a Population" by R. W. Johnson on google scholar but to get online access to the article cost more than the price of the book! Not all of us have access to the online subscriptions through a university library like he does. This formula is intrinsic to his implementation but we don't know why it works since he made no attempt to explain it.

The implementation is quite poor both in style and in accuracy. In style, Nahin uses names of builtin Matlab function names, like size() and error(), as array names. Matlab's compiler is smart enough to do the right thing but this causes major confusion for anyone trying to read the code.

There are several other major problems. First, in trying to generate samples without replacement, Nahin first uses matlab's randperm function. That's good; this is a great function. But then he inexplicably generates random samples from the randperm output using a clumsy implementation of an obscure algorithm by Bebbington, which he does not explain either. But since randperm randomizes the sample, a random sample without replacement of length n of x = randperm(N) is simply x(1:n).
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Format: Hardcover
Rather than write yet another book on probability in which the math is explained in yet another way, the author has chosen to tackle the problem of understanding probability via writing simulations and poking around a bit until you find the pattern and thus the solution. The author presents 21 problems in probability in the first half of the book, and shows his solutions in the second half with programs written in MATLAB. The idea is that you should try writing your solutions first before reading the second half of the book and seeing how the author solves the problem. Sometimes the author goes into detail in his reasoning, somethimes he just goes through a detailed explanation of his MATLAB code without really telling you how he arrived at his solution. Sometimes a little theory goes a long way, and this lack of theory at some points is the only real shortcoming of the book.

Of course, the problem here is, if you can't trust your intuition to solve a probability problem, how do you know that the computer program you wrote using that same intuition is also trustworthy? I found this to double as a book on helping you reason out the simulation of probability problems and also as just a good algorithm book on the solution of probability problems. You can take the approach to these specific problems and extend them to many other computational probability problems you are likely to encounter. Overall, highly recommended.

The reader should already understand fundamental probability theory and also have some experience in both Excel and MATLAB.
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Format: Hardcover
This is a delightful book showing how probability can be made to come alive by using Monte Carlo simulation. Wonderful examples are given to demonstrate this. A little experience in Excel or Matlab suffices to solve by simulation interesting probability problems that are otherwise not easily amenable to an analytical solution. The book is an excellent appetizer for more mathematical books combining probability and simulation such as the highly recommended books Understanding Probability: Chance Rules in Everyday Life by Henk Tijms and Intuitive Probability and Random Processes using MATLAB by Steven Kay.
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Format: Hardcover
Over the last few years, Paul Nahin has written a variety of books such as An Imaginary Tale, Dueling Idiots, When Least is Best, and Chases and Escapes. I've read most of his books and enjoyed each one. Anyone reading his writing will find it detailed, well-presented, and there is always a hint of humor and a bit of story telling that make the prose bubble along. Digital Dice is no exception to this and it has the added advantage of being participatory by definition---the reader is expected to work on the problems and not just read the book.

My favorite problem is "The toilet paper dilemma." It is a cute problem of a toilet stall with two fresh rolls of paper. Folks enter the stall at random and independently. Some folks are "big choosers" who always take paper from the big roll, and likewise some folks are "little choosers." If the rolls are even in size the each chooses to take from nearest roll. Now, let p be the probability of a big chooser entering a stall and q=1-p the probability of a little chooser entering the stall. Each enters independently and at random. Let n be the starting length of the roll and let M_n(p) be average number of portions of paper left on a roll when the other roll empties. The problem is to explore the nature of M_200(p) as p varies from 0 to 1.

Nahin tells the story of how Donald Knuth first wrote about this problem and he shows you a lovely recursion for M_n(p) with a simple, illustrative diagram. Only after Nahin thoroughly explains the problem is the reader sent to find her own solution.

Note that the problems are not new so you may well find some quite familiar. Even then, there should be a few to interest you and enjoy. If you have seen all the problems, well, the stories that pepper the text are fun, too.
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