Über die Autorin bzw. den Autor

Paul J. Nahin is the best-selling author of many popular math books, including Mrs. Perkins's Electric Quilt, Digital Dice, Dr. Euler's Fabulous Formula, When Least Is Best, and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.

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"For those of us who thoroughly enjoy a good puzzle, Duelling Idiots is indeed a welcome book. What Paul Nahin offers is essentially the mathematical equivalent of a collection of Far Side cartoons: a series of quirky vignettes, each with an amusing punchline that reveals something new about an offbeat aspect of reality."--Mark Denny, Stanford University

"Duelling Idiots and Other Probability Puzzlers seeks to teach the fundamentals of elementary probability theory using topics that are familiar to most everyone. Its light-hearted way of explaining serious subjects is a refreshing approach."--Robert B. Banks, author of Towing Icebergs, Falling Dominoes and of Slicing Pizzas, Racing Turtles

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Duelling Idiots and Other Probability Puzzlers

PRINCETON UNIVERSITY PRESS

Contents

Acknowledgments....................................................................ixPreface............................................................................XXiIntroduction.......................................................................3The Problems.......................................................................151. How to Ask an Embarrassing Question.............................................152. When Idiots Duel................................................................163. Will the Light Bulb Glow?.......................................................224. The Underdog and the World Series...............................................265. The Curious Case of the Snowy Birthdays.........................................276. When Human Flesh Begins to Fail.................................................347. Baseball Again, and Mortal Flesh, Too...........................................34S. Ball Madness....................................................................369. Who Pays for the Coffee?........................................................4210. The Chess Champ versus the Gunslinger..........................................4511. A Different Slice of Probabilistic, Pi.........................................4912. When Negativity Is a No-No.....................................................5013. The Power of Randomness........................................................5114. The, Random Radio..............................................................5215. An Inconceivable Difficulty....................................................5516. The Unsinkable Tub Is Sinking! How to Find Her, Fast...........................5717. A Walk in the Garden...........................................................5515. Two Flies Stuck on a Piece of Flupper—How Far Apart?.....................6119. The Blind Spider and the fly...................................................6220. Reliably Unreliable............................................................6521. When Theory Fails, There Is Always the Computer................................71The Solutions......................................................................81Random Number Generators...........................................................176"Some Things Just Have to be Done By Hand!"........................................198MATLAB Programs....................................................................202Index..............................................................................267

Chapter One

1. How to Ask an Embarrasing Question (A Warm-up Question Requiring Only Arithmetic)

Suppose you are assigned the following task: You are to determine the fraction of the population that practices a certain private act (use your imagination). If you could gather a large number of randomly selected people together into a large room or auditorium, you could then simply have each person fill out an anonymous questionnaire. Since no one could be identified from such a form, people would presumably tell the truth. But suppose this is not possible, and your task is to be accomplished over time through individual encounters. It is clear that you cannot just ask people, because they may or may not answer truthfully when confronted with the embarrassing question (EQ); you can't even use the form, because now no one "gets lost in the crowd." So how can you gather accurate data? You can use the following clever technique that I found described in a medical journal article.

You, or a member of your staff, give each person in the survey a fair coin as he or she arrives at your office. You then ask the individual to briefly step into a private room. There, all alone, the person flips the coin. If it shows tails, then the person writes the answer (YES or NO) to the EQ on a piece of paper. If, on the other hand, the coin shows heads, then the subject flips the coin a second time and writes the answer (YES or NO) to the non-EQ, "Did the coin show heads on the second toss?" After that, the person returns to your office and gives you the paper.

Now, each such piece of paper will have a single YES or NO on it, and eventually you'll have, let's say, 10,000 such pieces of paper. You do not know which question each particular person was actually answering, and yet you can now calculate the answer to your question: What percentage of the population (or, at least, of the people in the survey) practices the "private act"? Explain why this is so, and illustrate your analysis by calculating the percentage of people that practice the private act if 6,230 write YES and 3,770 write NO.

2. When Idiots Duel

Duelling problems are of strong interest to most people, perhaps because of their "life-or-death" nature. Here is one that is also, I think, amusing. A and B decide to duel but, being poor, they have just one gun (a six-shot revolver) and only one bullet. Being dumb, as well, this does not deter them and they agree to "duel" as follows: They will insert the lone bullet into the gun's cylinder, A will then spin the cylinder and shoot at B (who, standing inches away, is impossible to miss). If the gun doesn't fire then A will give the gun to B, who will spin the cylinder and then shoot at A. This back-and-forth duel will continue until one fool shoots the other. What is the probability that A will win? Also, how many trigger pulls will occur, on average, before somebody wins? The second question has meaning, of course, only if there is a large number of pairs of duelling idiots; for any particular pair, the duel occurs just once and lasts for a well-defined, specific number of trigger pulls.

(If A and B point the gun at themselves, then this foolish business is simply the two idiots alternately playing what is usually called "Russian roulette." A less violent—but perhaps not by much—form of the idiots' duel can be found in the 'sudden death' playoff between two pro football teams that are tied at the end of regulation play. There, A is the team that gets first possession of the ball.)

The methodical way to attack our two questions is to write out what mathematicians call the sample space of the experiment; that is, to write down all the possible outcomes of the duel. So, if we define two events as

a = gun fires for A

x = gun does not fire,

then the sample points for which A wins are

a xxa xxxxa and so on

That is, A wins only on those sample points that are of odd length. So, the answer to the first question is simply the sum of the probabilities of all the odd length sample points, i.e.,

P(A) = 1/6 + (5/6)² 1/6 + (5/6)⁴ 1/6 + ...

where the sum has an infinite number of terms. Each term comes from the observation that if the duel lasts for exactly k trigger pulls, then the gun did not fire on the first k - 1 trigger-pulls and did fire on the kth one. The sum is easy to calculate. Just write P(A) = 1/6 S where

S = 1+(5/6)2 +(5/6)⁴ + ....

So, S is a geometric series, summed by the standard trick; i.e., multiply through by (5/6)² to get

(5/6)², S = (5/6)2 +(5/6)⁴ + ...

and then subtract. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

P(A) is greater than...