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CUT THE KNOT CUT THE KNOT Probability Riddles Alexander Bogomolny Cut the Knot: Probability Riddles Copyright © 2020 Svetlana Bogomolny Mathematics / Recreations & Games Mathematics / Probability & Statistics All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means without the prior written permission of the copyright holder. Production, editing and typesetting done with the assistance of Wolfram Media. The compilation of riddles included in this book includes various user contributions to www.cuttheknot.org. Cover image design by Paige Bremner, knot image courtesy of Larvik Museum, Arts Council of Norway (CC BY-SA 3.0). Printed and distributed by Amazon.com Services. Contents Foreword by Nassim Nicholas Taleb vii Preface ix 1 | Intuitive Probability 1 Bridge Hands  Secretary’s Problem  Pencil Logo  Balls of Two Colors I  Division by 396  Fair Dice  Selecting a Medicine  Probability of Random Lines Crossing  Six Numbers, One Inequality  Metamorphosis of a Quadratic Function  Playing with Balls of Two Colors  Linus Pauling’s Argument  Galton’s Paradox  Solutions (p. 5) 2 | What Is Probability? 13 3 | Likely Surprises 15 Balls of Two Colors II  Coin Tossing Contest  Probability of Visiting Grandparents  Probability of Increasing Sequence  Probability of Two Integers Being Coprime  Overlapping Random Intervals  Random Clock Hands  Crossing River after Heavy Rain  The Most Likely Position  Coin Tossing Surprises I  Solutions (p. 18) 4 | Basic Probability 33 Admittance to a Tennis Club  Black Boxes in a Chain  A Question of Checkmate  Concerning Even Number of Heads  Getting Ahead by Two Points  Recollecting Forgotten Digit  Two Balls of the Same Color  To Bet or Not to Bet  Lights on a Christmas Tree  Drawing Numbers and Summing Them Up  Numbers in a Square  Converting Temperature from °C to °F  Probability of No Distinct Positive Roots  Playing with Integers and Limits  Given Probability, Find the Sample Space  Gladiator Games  In Praise of Odds  Probability in Scoring  Probability of 2n Beginning with Digit 1  Probability of First Digits in a Sequence of Powers  Probability of Four Random Integers Having a Common Factor  Probability of a Cube Ending with 11  Odds and Chances in Horse Race Betting  Acting As a Team  Sum of Two Outcomes of Tossing Three Dice  Chess Players Truel  Two Loaded Dice  Crossing Bridge in Crowds  Solutions (p. 39) 5 | Geometric Probability 73 Three Numbers  Three Points on a Circle I  Three Points on a Circle II  Two Friends Meeting  Hitting a Dart Board  Circle Coverage  Points in a Semicircle  Flat Probabilities on a Sphere  Four Random Points on a Sphere  Random Numbers and Obtuse Triangle  Random Intervals with One Dominant  Distributing Balls of Two Colors in Two Bags  Hemisphere Coverage  Random Points on a Segment  Probability of First Digit in Product  Birds on a Wire  Lucky Times at a Moscow Math Olympiad  Probability of a Random Inequality  Points on a Square Grid  A Triangle out of Three Broken Sticks  Probability in Dart Throwing  Probability in Triangle  Solutions (p. 78) 6 | Combinatorics 121 Shuffling Probability  Probability with Factorials  Two Varsity Divisions  Probability of Having 5 in the Numerator  Random Arithmetic Progressions  Red Faces of a Cube  Red and Green Balls in Red and Green Boxes  Probability of Equal Areas on a Chessboard  Random Sum  Probability of Equilateral Triangle  Shelving an Encyclopedia  Loaded Dice I  Loaded Dice II  Dropping Numbers into a 3 x 3 Square  Probability of Matching Socks  Numbered Balls Out of a Box  Planting Trees in a Row  Six Numbers, Two Inequalities  Six Numbers, Three Inequalities  Tying Knots in Brazil  Tying Knots in Russia  Guessing Hat Numbers  Probability of an Odd Number of Sixes  Probability of Average  Marking and Breaking Sticks  Bubbling of Sorts  Probability of Successive Integers  Solutions (p. 128) vi Cut the Knot: Probability Riddles 7 | Conditional Probability 157 Two Liars  Three Liars  Taking Turns to Toss a Die  Two Coins: One Fair, One Biased  Probability of the Second Marble  Quotient Estimates I  Quotient Estimates II  The Lost Boarding Pass  Lucky Contest Winners  Diminishing Hopes  Incidence of Breast Cancer  A Search for Heads and Its Consequences  A Three Group Split  Lewis Carroll’s Pillow Problem  A Follow Up on Lewis Carroll’s Pillow Problem  Sick Child and Doctor  Right Strategy for a Weaker Player  Chickens in Boxes  Two Chickens in Boxes  Two Chickens in Bigger Boxes  Solutions (p. 162) 8 | Expectation 191 Randomly Placed Letters in Envelopes  Expected Number of Fixed Points  An Average Number of Runs in Coin Tosses  How Long Will It Last?  Expectation of Interval Length on Circle  Waiting for a Train  Average Number of Runs in a Sequence of Random Numbers  Training Bicyclists on a Mountain Road  Number of Trials to First Success  Waiting for an Ace  Two in a Row  Waiting for Multiple Heads  Probability of Consecutive Heads  Two Dice Repetition  Expected Number of Happy Passengers  Expectation of the Largest Number  Waiting for a Larger Number  Waiting to Exceed 1  Waiting for All Six Outcomes  Walking Randomly— How Far?  Expectation of Pairings  Making Spaghetti Loops in the Kitchen  Repeating Suit When Dealing Cards  Family Size  Averages of Terms in Increasing Sequence  Solutions (p. 197) 9 | Recurrences and Markov Chains 237 A Fair Game of Chance  Amoeba’s Survival  Average Visibility of Moviegoers  Book Index Range  Probability of No Two-Tail Runs  Artificially Unintelligent  Two Sixes in a Row  Matching Socks in Dark Room  Number of Wire Loops  Determinants in ℤ  Gambling in a Company  Probability in the World Series  Probability of a Meet in 2 an Elimination Tournament  Rolling a Die  Fair Duel  Solutions (p. 240) 10 | Sampling of American Mathematics Competition Problems 261 Five-Digit Numbers Divisible by Eleven  The Odds for Two Teams  Cutting a String into Unequal Pieces  Getting an Arithmetic Progression with Three Dice  Probability of No Distinct Positive Roots  Rolling a Defective Die  Counting Coins  The Roads We Take  Marbles of Four Colors  Two Modified Dice  Exchanging Balls of Random Colors  Solutions (p. 264) 11 | Elementary Statistics 273 A Question about Median  Family Statistics  How to Ask an Embarrassing Question  An Integer Sequence with Given Statistical Parameters  Mean, Mode, Range and Median  Solutions (p. 274) Appendix A | Dependent and Independent Events 281 Appendix B | Principle of Symmetry 285 Bibliography 287 Index of Riddles 293 Foreword by Nassim Nicholas Taleb Maestro Bogomolny How do you learn a language? There are two routes; the first is to memorize imperfect verbs, grammatical rules, future vs. past tenses, recite boring context-free sentences, and pass an exam. The second approach is by going to a bar, struggling a little bit and, out of the need to blend-in and integrate with a fun group of people, then suddenly find yourself able to communicate.Inotherwords,byplaying,bybeingaliveasahumanbeing.Ipersonally have never seen anyone learn to speak a language properly by the first route. Also, I have never seen anyone fail to do so by the second one. It is a not well-known fact that mathematics can also be learned by playing— just watch the private correspondence, discussions and pranks of the members of the augustBourbakicircle.Someofus(anditincludesthisauthor)donotperformwellon tasksvia“cold” approaches,unabletomusterthemotivationtodoboringthings.But, somehow we upregulate when stimulated or when there is play (or money) involved. This may disturb many people married to cookie-cutter pedagogical methods that require things to be drab, boring, and bureaucratic for them to be effective—but that’s reality. It is thanks to Maestro Alexander B. that numerous people have learned mathematics by the second route, by playing, just for the sake of entertainment. He helped many to make it their hobby. His mathematical website cut-the-knot has trained a generation—manyseeminglyapproachedtheproblemashobbyiststhengotstuckwith it.For,ifyoulikedmathematicsjustalittlebit,MaestroBogomolnymadeitimpossible for you to not love it. Mathematics was turned into a frolic. I discovered his riddles on social media. (Alexander B. does not like the word “problems”. I now understand why.) Social media brings out the best and the worst in people. He was rigorous yet open-minded, allowing people like me who did some mathematical economics to cheat withinequalitiesbyusingthevariouscannedmethodsforfindingminimaandmaxima. He even tolerated computerized mathematics, provided of course there was some rigor in the process. I initially knew nothing about him but could observe rare attributes: an extraordinary amount of patience and a remarkable sense of humor. One summer, as he was in Israel, I informed him that I was vacationing in Lebanon. His answer: “Walking distance”. He always had a short comment that makes you smile, not laugh, which is a social art. Alexander B. created a vibrant community around his Twitter account. He would poseaquestion,collectanswersandpatientlyexplaintopeoplewheretheywerewrong. I,formyself,startedalmosteverydaywithapuzzle,withtheexcitementofunpre- dictability, as it took from 5 minutes to 4 hours to complete—and it was usually impossible to tell from the outset. For a couple of years, it was the first thing I looked at with the morning coffee. There was some mild competition, mild enough to be entertainingbutnottoointensetoresembleanacademicratrace.Oncesomeonegota proof, we had to look for another approach so it paid to wake up early and beat those with a time zone advantage. viii Cut the Knot: Probability Riddles In the two years since he left us there has been no Saturday morning—104 of them—that I did not solve a riddle randomly selected on the web in his memory. But, without him, it is not the same. ––– How did Alexander Bogomolny get there? I met him in an Italian restaurant in New Jersey. I was surprised to see a math- ematician who looked much more like a maturing actor than someone in a technical specialty:tall,athletic,jovial,andwithacharismaticpresence.But,ashehadwarned me, he had a severe hearing problem, the result of a medical treatment for the flu. This explained to me his veering away from an academic career to get involved in computer pedagogy. His hearing was worsening with time. It is hard to imagine being a professor with reduced auditory function in one ear (in spite of a hearing aid) and none in the other. There was something fresh and entertaining about him. He was happy. One could talk and laugh with him without much communication. He was neither interested in money nor rank—something refreshing as I was only exposed to academics who, whether they admit it or not, are obsessed with both. When I asked him about commercializing his website cut-the-knot his answer was “I havetwopensions.NextyearIturnseventy”.Hewasn’tinterestedinpoisoninghislife for more money. Why did I start nicknaming him Maestro? Because it was pretty much literal: he played math like a master would with a musical instrument—and mostly to himself. He was physically bothered by a sloppy derivation or an error, as if he heard a jarring note in the middle of a sonata. It was a joy to see someone so much in sync with his subject matter—and totally uncorrupted by the academic system. ––– Now, probability. In one conversation, I mentioned to him that probability riddles would be very useful for people who want to get into the most scientifically applicable scientific subject in the world (my very, very biased opinion). What I said earlier about play is even more applicable to probability, a field that really started with gamblers, used by traders and adventurers, and perfected by finance and insurance mathematicians.Probabilityappliestoallempiricalfields:gambling,finance,medicine, engineering, social science, risk, linguistics, genetics, car accidents. Let’s play with it by adding to his feed some probability riddles. His eyes lit up. Hence this book. ––– I thank Marcos Carreira, Amit Itagi, Mike Lawler, Salil Mehta, and numerous others who supported us in this project. And a special gratitude to Stephen Wolfram and Jeremy Sykes for ensuring that Cut-the-Knot stays alive and that this book sees the day. Additional thanks to Paige Bremner, Glenn Scholebo, and other members of Wolfram Media for editing the manuscript. Preface On a superficial view, we may seem to differ very widely from each other in our reasoning, and no less in our pleasures: but notwithstanding this difference,whichIthinktoberatherapparentthanreal,itisprobable that thestandardbothofreasonandTasteisthesameinallhumancreatures. E. Burke, 1757, A Philosophical Inquiry into the Origin of Our Ideas of the Sublime and Beautiful The American Heritage Dictionary defines probability theory as the branch of mathematics that studies the likelihood of occurrence of random events in order to predict thebehaviorofdefinedsystems.(Ofcourse“whatisrandom?” isaquestionthatisnot all that simple to answer.) Starting with this definition, it would (probably) be right to conclude that proba- bilitytheory,beingabranchofmathematics,isanexact,deductivesciencethatstudies uncertain quantities related to random events. This might seem to be a strange mar- riage of mathematical certainty and the uncertainty of randomness. This collection of probability problems is a collaborative effort born on the web and tested in social networks. Even though meeting over the wires is a far cry from a face-to-face encounter, it has a clear positive side. Web acquaintances are drawn to each other due to mutual interests, mostly without seeking to learn or attaching importance to unrevealed personal details. Some of the problems are original but most have been gathered from a variety of sources: books, magazine articles and online resources. The collection is reasonably comprehensive, though its value is mostly in that many if not most of the problems come with multiple solutions. So much so that quite often it was difficult to assign problems to specific chapters. Receiving solutions from people, many of whom would confess to not being mathematicians or having forgotten all the math they studied in school or college, was an exhilaratingexperience.Oneoftheideaswepursuedinwritingthisbookwastoshare that experience with a broader audience. We prefer the term riddle to problem. Chapter 1 Intuitive Probability One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it. Laplace, 1814, Philosophical Essay on Probabilities The theory of probability supplies a good deal of counterintuitive results (see Chap- ter 3, page 15). However, the theory of probability arose from practical applications and is, in essence, a formal encapsulation of the intuitive view of chance. Thischaptercollectsafewprobabilityproblemswhosesolutionsarebasedonintu- ition and common sense and do not require any theoretical knowledge. Be warned, though, that even great mathematical minds sometimes get beguiled by intuition. For example, when asked for the probability that heads appears at least 2 onceintwothrowsofafaircoin,JeanleRondd’Alembert(1717–1783),replied (the 3 3 correctansweris )becauseinhisopiniontherewasnoreasontocontinuetossingthe 4 coin after heads showed up on the first toss. Thus, he judged events H, TH, TT as equiprobable. The bracketed items under the section and subsection heads are citations to items listed in the bibliography. The page numbers refer to the exact portion of the cited item. Some riddles and solutions were created for the Cut The Knot project and are not published elsewhere. Riddles 1.1 Bridge Hands [45, p. 39] A game of bridge is played by two pairs of players, with players on a team sitting opposite each other. Which situation is more likely after four bridge hands have been dealt: you and your partner hold all the clubs or you and your partner have no clubs? 1.2 Secretary’s Problem [45, p. 39] A secretary types four letters to four people and addresses the four envelopes. If the letters are inserted at random, each in a different envelope, what is the probability that exactly three letters will go into the right envelope? 2 Cut the Knot: Probability Riddles 1.3 Pencil Logo [102, p. 1] A pencil with pentagonal cross-section has a maker’s logo imprinted on one of its five faces.Ifthepencilisrolledonthetable,whatisprobabilitythatitstopswiththelogo facing up? 1.4 Balls of Two Colors I [54, pp. 1–2], [65, Problem 319] A box contains p white balls and q black balls, and beside the box lies a large pile of black balls. Two balls are drawn at random (with equal likelihood) out of the box. If they are of the same color, a black ball from the pile is put into the box; otherwise, the white ball is put back into the box. The procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white? 1.5 Division by 396 [91, Problem 56] Find the probability that if the digits 0, 1, 2, ..., 9 are placed in random order in the blank spaces of 5_383_8_2_936_5_8_203_9_3_76 the resulting number will be divisible by 396. 1.6 Fair Dice Michel Paul Toss two dice. What is the probability of having 2 or 5 on at least one die? 1.7 Selecting a Medicine [58, pp. 97–98] Halfthepeoplewhocontractedacertaindiseasewhichisspreadingacrossthecountry have died and half got better on their own. Two medicines, A and B, have been developed but not actually tested. A was administered to three patients and all survived. B was administered to eight patients and seven survived. Intheunfortunateeventofyourcontractingthedisease,whichmedicinewouldyou choose? 1.8 Probability of Random Lines Crossing [62, Problem 8] An urn contains the numbers 1, 2, 3,...,2018. We randomly draw, without replace- ment, four numbers in order from the urn which we will denote a, b, c, d. What is the probability that the following system will have a solution strictly inside (i.e., not on the axes) the first quadrant? (cid:40) ax+by = ab cx+dy = cd.