Digital Dice: Computational Solutions to Practical Probability Problems

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 to animate his prose. 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.

Let’s begin with a quick overview. Nahin presents 21 problems for you to solve. Some have closed form solutions but the point to the book (and the fun!) is to deviate from analytics and write a program to simulate the problem conditions and then with a random number generator and lots of trials, find the solution with your simulation. Some of the problems are easily stated and easily coded. Others require more thought to code, say with a flow chart, but all the problems are recreational and delightful to ponder.

After Nahin presents the problems, he gives you detailed solutions. For each solution, there is a program listed in MATLAB code. Nahin explains each line of his program and he includes line numbers for reference which are not a part of MATLAB code. Of course, you would ideally have your own code to think about but if not, well, just reading the solutions is satisfying. Ordinarily, I would find a non-open source program discouraging because not all readers may have it. However, Nahin’s code is easy to read and can be easily translated to another language. He even goes to lengths to avoid the built-in vector operations of MATLAB because other languages don’t have them. The only two exceptions that are MATLAB dependent are quite minor: the “convhull” function to find the find the convex hull of a set of points and “hist” function to plot the histogram of a data set. Everything else is straightforward to translate.

If a closed form solution is available, Nahin provides it with considerable detail and explanations along the way. He compares the analytical answer to the simulation results so that the reader can see the performance of the simulation. Naturally, he gives graphs in the solutions and even drawings to illustrate some problems.

Let’s look at a few of the problems. My favorite is “The toilet paper dilemma.” A toilet stall has 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 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 M₂₀₀(p) as p varies from 0 to 1.

Nahin does not, however, just state this problem. He 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 he has thoroughly explained the problem does he ask you to find the solution.

Here’s another beautiful, counter-intuitive problem: Parrondo’s paradox. Two people are gambling in a game, say A. A, you are asked to show, is a loser for you. So, you also play another game with your opponent, say B. B is also a loser for you. What can you do to reverse your odds? Well, you can play both A and B but randomly switch between them. The trick, beyond the details of the game that I leave for you to read in the book, is that two losing games can be combined to form a winning game. With the right probabilities one losing game can rescue the other so that together they are a winning game.

As a third and final fun problem example, there is the optimal stopping problem. Nahin expresses it as a dating problem. You want to find a suitable spouse but you can’t date everyone and you don’t want to choose the first person you meet. What strategy should you employ, on average, to find a good spouse?

That’s a sampling and there are another 18 others worth exploring, coding, and enjoying.

Finally, please note that the problems are not new so you may well find some quite familiar. Even so, 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.

David Mazel is an engineer in Washington, DC. He received his Ph.D. in electrical engineering from Georgia Tech and is interested in signal processing, billiards, and agent-based simulations. In his spare time, he enjoys mathematics with his children and playing with the family golden retriever.