# Will You Be Alive 10 Years from Now?: And Numerous Other Curious Questions in Probability

PAUL J. NAHIN
Pages: 256
https://www.jstor.org/stable/j.ctt5hhpxj

1. Front Matter
(pp. i-x)
(pp. xi-xiv)
3. Preface
(pp. xv-xxviii)
4. Introduction: Classic Puzzles from the Past
(pp. 1-29)

The birth of probability theory is generally dated by historians of science to 1654. That was the year the French writer and philosopher Antoine Gombaud (1607–1684), better known today by his literary name, Chevalier de Méré, presented the French mathematician Blaise de Pascal (1623–1662) with some questions concerning games of chance.¹ This is a bit of an artificial milestone as it refers to the start of ananalyticalapproach. Probabilisticthinkingcan be traced back to well before 1654. A hundred years earlier, for example, the Italian Gerolamo Cardano (1501–1575) had pondered probability, and wrote his work...

5. Challenge Problems
(pp. 30-35)

Before starting with the formal problems in this book, here are several challenge problems for you to think about as you read. As you work your way through the formal problems you may pick up ideas and mathematical techniques that will serve you well in attacking the challenge problems. These are not necessarily easy problems (you’ll have to do some integrals), and for most of them, you will have to think long and hard. All do have exact solutions, and you will find complete discussions (including Monte Carlo simulations) of each at the back of the book. But I strongly...

6. 1 Breaking Sticks
(pp. 36-41)

For our first puzzler, let’s start by considering an easy warm-up question. Then, by making just a very slight (or so it will at first seem) alteration, we’ll have a new question that will produce what I think will be a surprising result for you. This new question will still not be all that difficult to answer theoretically, but the result will be sufficiently surprising that a computer simulation will be of real help in convincing you that we haven’t made a mistake somewhere. That said, here’s the warm-up.

Suppose you have a stick that is of unit length (in...

7. 2 The Twins
(pp. 42-46)

In February 2008 I received a very interesting e-mail from Bruce C. Taylor, a professor of biomedical engineering at the University of Akron. Bruce had just been reading my book,Duelling Idiots(Princeton 2002), and that prompted him to write to me. Here’s what Bruce wrote:

I have an interesting probability problem that I have not been able to solve and I am just curious to see if you can come up with a solution. The problem came up when in one of our classes here I was assigning lab groups using a random number generator. As it turns out...

8. 3 Steve’s Elevator Problem
(pp. 47-51)

In my bookDigital Dice(Princeton 2008), I included a puzzle question called “Steve’s elevator problem.” It was named for a reader in California, Steve Saiz, who had written to me about it after reading my bookDuelling Idiots(Princeton 2002). As I wrote inDigital Dice, here’s how Steve explained it to me in a March 2004 e-mail:

Every day I ride up to the 15th floor in an elevator. This elevator only goes to floors G, 2 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17. On average, I noticed that I usually make 2 or...

9. 4 Three Gambling Problems Newton Would “Probably” Have Liked
(pp. 52-61)

To lay the groundwork for the first problem, suppose you give a fair die to each of a large number of people. Each person tosses his or her die, over and over, until a 6 appears. Once this has occurred for every person, each then reports on how many tosses he or she made. What do you think is the average number of tosses of a fair die to get the first 6? Since the probability of a 6 on a fair die is 1/6, most people immediately reply “six tosses,” whether or not they really understand why. In fact,...

10. 5 Big Quotients—Part 1
(pp. 62-65)

This is the first part of a two-part problem and so, of course, this will be the easy part. But don’t skip it! Understanding what we do here will be a big help in understanding the second part. Here’s the question: pick a number at random (uniformly), from 0 to 1. Pick another number, also at random (uniformly), from 0 to 1. What is the probability that if you divide the larger number by the smaller one the answer will be greater than 2? Greater than 3? More generally, greater thank, wherek≥ 1?

Before I show you...

11. 6 Two Ways to Proofread
(pp. 66-69)

As anyone who has ever written a book can tell you, one of the most important tasks—but a tedious one—is proofreading the initial typeset pages. There arealwaysmisprints! The only way to catch and correct them is through careful reading of the entire book, a job that is all too easy to nod off on while doing. There are at least two ways to do it, where I’ll make the following two assumptions: (1) there is an unknown total ofmmisprints in the book, and (2) the probability a reader spots a misprint, given that he...

12. 7 Chain Letters That Never End
(pp. 70-73)

Just to be sure you don’t think I am claiming computer simulation is a complete substitution for theory, here’s an example of a random process that would be nontrivial to simulate, and we’ll count ourselves lucky to have a theoretical solution. Suppose somebody decides to create a chain letter, and starts things off by sending a copy of it toCpeople. Each of thoseCpeople is asked to then send offCcopies of their own, and so on. With probabilityp, any person who receives such a letter decides to ignore it. What is the probability that...

13. 8 Bingo Befuddlement
(pp. 74-78)

In the introduction (I.7), I discussed nontransitive dice. Nontransitivity can also occur in many other situations, far removed from dice. A game that kids love to play (some adults, too!), rock, paper, scissors, is nontransitive because while the rock breaks the scissors and the scissors cuts the paper, the paper covers the rock. For a more analytical example, imagine that we have nine tennis players ranked 1, 2, 3, 4, 5, 6, 7, 8, and 9 in increasing order of skill (8alwaysbeats players 1 through 7, and always loses to player 9). Imagine further that these nine players...

14. 9 Is Dreidel Fair?
(pp. 79-82)

The ancient Jewish game of dreidel, dating back to at least the early Middle Ages (and perhaps to the time of Christ), is a game of chance played during Hanukkah by two or more players (usually children) with a four-sided top. Each side of the top is marked with one of the four Hebrew letters Nun, Gimel, Hay, and Shin, which we’ll call N, G, H, and S, respectively. At the start of a game each ofp≥ 2 players contributes one unit of money (let’s say a dollar) to form an initial pot. (A more traditional pot might...

15. 10 Hollywood Thrills
(pp. 83-86)

Shortly after his death in Biloxi, Mississippi, in a tragic bungee-jumping accident, the following preliminary movie workup was found in the papers of famed Hollywood director Irving Nutso. Nutso, who as a teenager flunked out of Caltech because he spent all his time watching movies at virtually every theater in the Pasadena, Cucamonga, and Azusa areas around the institute instead of attending classes, was at the time of his death planning his comeback film. With the working title ofRevenge of the Ducks, it would have been his return to the silver screen after being released from federal prison, having...

16. 11 The Problem of the n-Liars
(pp. 87-89)

Imagine that we havenpeople, each of whom tells the truth with probabilitypwhen making a statement. That is, ifp= 1 a person never lies (always tells the truth) and ifp= 0 a person always lies. Each of thenpeople either lies or tells the truth independent of the others. Now, suppose we line up thesenpeople in a row, shoulder to shoulder, from left to right. Starting at the leftmost person, you whisper either “yes” or “no” into his ear, and then that person turns his head and whispers what he...

17. 12 The Inconvenience of a Law
(pp. 90-92)

In this problem we return to the undocumented resident issue I sketched in the opening of the preface. As you’ll recall, there we developed the following mathematical situation, starting with an urn initially containingbblack balls andrred balls. We randomly draw one ball at a time; if it’s black we put it back in the urn, and if it’s red we discard it. We keep doing this until we’ve removed a fractionfof the red balls. We are interested in how many times, on average, each black ball will be drawn before the process terminates. In...

18. 13 A Puzzle for When the Super Bowl Is a Blowout
(pp. 93-95)

Suppose you and some friends are watching a Super Bowl game on TV and one of the teams is ahead by six touchdowns at the end of the third quarter. Instead of turning to the Weather Channel to get some excitement, do this: challenge your buddies with the following puzzle, and watch their dull, bored eyes light up at the anticipation of working on a good math problem! Be sure to have plenty of extra beer and chips handy because (as is well known) doing real math builds a superappetite.

Let’s imagine that the 32 teams in the NFL decide...

19. 14 Darts and Ballistic Missiles
(pp. 96-102)

Two darts players, A and B, stand in front of a square dartboard with dimensions 2Rby 2R. On the board is painted a circular target area of radiusRwith its center at the center of the board. We imagine this center point is the origin of anx,ycoordinate system with axes parallel to the edges of the board. A and B each throw a dart at the board, and although each dart does indeed land somewhere on the board, the players have distinctly different throwing techniques. When A throws her dart it lands at a point with...

20. 15 Blood Testing
(pp. 103-106)

Imagine that a very large number of people,N, are to each have their blood tested for a possible dangerous infection. Think, for example, of the millions of men inducted into the armed services during World War II, and the then not-so-uncommon sexually transmitted infection of syphilis. The obvious way to do this is simply to test each person, for a total ofNtests. During World War II a different, quite clever alternative approach was actually used, based on the fact that while syphilis wasn’t uncommon, most men were actually not infected.

The underlying idea was simple: for some...

21. 16 Big Quotients—Part 2
(pp. 107-116)

A quick recap from the end of Problem 5: we pickNnumbers uniformly from 0 to 1, divide the largest by the smallest, and ask for the probability that the result is larger thank, for anyk≥ 1. This is a significantly more difficult question to answer than was the original Problem 5 question, where we treated in detail the special case ofN= 2. We could, in theory, proceed just as before, looking now at anN-dimensional unit cube and the subvolumes in that cube that correspond to our problem (in Problem 5 the “cube”...

22. 17 To Test—Or Not to Test?
(pp. 117-125)

In our modern times, new and wonderful advances are seemingly achieved between the last time we turned the TV off and when we turn it back on. This is particularly true in medicine, where apparently there is a treatment for every ill that afflicts the human body, ranging over the entire spectrum from warts and dry eyes to erectile dysfunction and urinary incontinence to asthma and bowel disease to…. Well, you get the idea, and the message, too: drug companies have spent billions of dollars telling us to “ask your doctor” about the latest pills, patches, injections, inhalers, and drops...

23. 18 Average Distances on a Square
(pp. 126-138)

This is the longest analysis in the book (it’s actually three related problems), designed to show you that computer simulation can definitely have its place—theory is always desirable, of course, but often it is simulation that will save your sanity in the short run when you need an answer fast! Our general question is easy to state: given a unit square, and two points picked at random from that square, what’s the average distance between the two points?

Stated this way, however, the question is not well defined, as there are at least three ways we can interpret what...

24. 19 When Will the Last One Fail?
(pp. 139-146)

All the stuff we use in our everyday lives eventually stops working. Eventually,westop working. Who would deny that? To be specific but less personal, let’s consider apparently identical toasters, coming right off the production line one after the other. Suppose we give one of these toasters to each ofNpeople so they can make toast for breakfast each morning. Each of theNtoasters therefore receives the same daily use—but they all last different times. One will be the first to fail, and then sometime later a second toaster will fail, and so on until the...

25. 20 Who’s Ahead?
(pp. 147-150)

Suppose two candidates for office, P and Q, receivepandqvotes, respectively, wherep>q. That is, P wins. That final result is definitively established, however, only after the ballot-counting process is completed. During the counting process the lead can switch back and forth, depending on the particular order in which the individual ballots are processed. A famous result in nineteenth-century probability theory called theballot theoremsays that the probability P isalwaysahead of Q from the first counted ballot is given by$\frac{p-q}{p+q}.$

This simple expression has a rather surprising implication. Suppose, for example, that P...

26. 21 Plum Pudding
(pp. 151-155)

Problem 971 on p. 333 of the fifth edition of Whitworth’sChoice and Chance(published in 1901, with the first edition appearing in 1867) is as follows: “If a spherical plum pudding containsnindefinitely small plums, [then show that] the expectation of the distance of the nearest one from the surface is one (2n+ 1)th of the radius.”

I was immediately attracted to this problem (as would be anyone, of course¹), as I could see it offered a nice illustration of how the maximum ofnindependent random variables might occur in a “natural way” (you’ll see how,...

27. 22 Ping-Pong, Squash, and Difference Equations
(pp. 156-167)

Everybody has played Ping-Pong at some time in life, usually at a summer camp or at school or at a friend’s home. The scoring rule is simple: a point is made by the winner of each rally (it doesn’t matter who serves), and the first player to reach eleven points wins the game. There is a little caveat about having to be ahead by at least two points—a score of 11 to 9 wins, but 11 to 10 doesn’t—but I’ll ignore that here. The first question in this puzzler is a simple one: if two players, called P...

28. 23 Will You Be Alive 10 Years from Now?
(pp. 168-174)

As one gets older the above question takes on ever-increasing interest. The only real answer to it is something like “Who knows?,” or “Maybe, but maybe not,” or “Yes, God willing.” Those are all a bit less than satisfying, of course; can we at least calculate the probability of being alive 10 years from now? The answer to that question is a definiteyes, and you can do it for yourself with information you can get right off the Web. All you need is your age (which I’m sure you know) and the so-calledlife-expectancy tablefor your particular situation...

29. 24 Chickens in Boxes
(pp. 175-182)

This book opened with a lot of commentary (mostly unhappy, I’m afraid¹) on Marilyn vos Savant’s mathematics and, if only for symmetry reasons, it seems appropriate to end with another of her puzzles. So, the finalsolvedproblem in this book is from herParade Magazinecolumn of August 4, 2002, in which she printed the following question from a reader:

Suppose you’re on a game show. There are four boxes in an L-shaped configuration, like this:

The host tells you:

(1) One of the vertical boxes contains a chicken; and

(2) one of the horizontal boxes contains a chicken....

30. 25 Newcomb’s Paradox
(pp. 183-188)

In 1950 the mathematicians Merill M. Flood (1908–1991) and Melvin Dresher (1911–1992), while working at the RAND Corporation in Santa Monica, California (an air force think tank), jointly created a puzzle question in game theory that has bedeviled analysts ever since. I’ll first give it to you in its best-known, nonprobabilistic form and then in the form that gives this last entry in the book its title (and in which probability makes an appearance). There is no solution section for this puzzle because, as I write, there is no known analysis that satisfies everybody. That’s why it’s the...

31. Challenge Problem Solutions
(pp. 189-212)

(1) Letxandybe the lengths of the two randomly determined sides. That is, 0 <x< 1 and 0 <y< 1. Also, by the triangle inequality (a fancy way of saying that the shortest path between two points is along a straight line), we havex+y> 1. If we plot this inequality on the unit square (defining the triangular region above the diagonal fromYtoXin Figure S1), then it is clear that all possible triangles (obtuse and acute) are associated with just the points in the upper diagonal triangleXYZ. That isXYZ...

32. Technical Note on MATLAB®’s Random Number Generator
(pp. 213-216)

Any Monte Carlo simulation code by definition uses numbers that come from some probability distribution. Some codes use a lot of random numbers; a few of the codes in this book use millions of them. MATLAB®’s random number generator provides a convenient software source for these numbers. MathWorks, the creator of MATLAB®, has gone through a series of designs for its generators, and the latest version uses a highly sophisticated combination of shift register/bit manipulation processes that, unlike earlier generators, require no multiplication or division operations. The latest MATLAB®generator, then, is very fast. At least as important, however, is...

33. Acknowledgments
(pp. 217-218)
Paul Nahin
34. Index
(pp. 219-220)
35. Back Matter
(pp. 221-221)