Mathematics and Sports (Dolciani Mathematical Expositions, Series Number 43)

This is an eclectic compendium of the essays solicited for the 2010 Mathematics Awareness Month Web page on the theme of 'Mathematics and Sports'. In keeping with the goal of promoting mathematics awareness to a broad audience, all of the articles are accessible to university-level mathematics students and many are accessible to the general public. The book is divided into sections by the kind of sports. The section on American football includes an article that evaluates a method for reducing the advantage of the winner to a coin flip in an NFL overtime game; the section on track and field examines the ultimate limit on how fast a human can run 100 metres; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing and an article on modelling Tiger Woods' career. cover copyright page title page Preface Contents I Baseball Sabermetrics: The Past, the Present, and the Future Jim Albert 1.1 Introduction 1.2 Measuring Batting 1.3 Measuring Pitching 1.4 Measuring Fielding 1.5 New Measurements, New Data and Measures of Performance 1.6 Further Reading About the Author Surprising Streaks and Playoff Parity: Probability Problems in a Sports Context Rick Cleary 2.1 Problem 1: Rare Events Example 1: Four homers in a row Example 2: A streak of winless opponents 2.2 Problem 2: Playoff Series Length References About the Author Did Humidifying the Baseball Decrease the Number of Homers at Coors Field? Howard Penn 3.1 Introduction 3.2 The Numbers 3.3 A Useful Statistic 3.4 Comparing the two sets of data 3.5 Summary of Conclusions 3.6 Exercises References About the Author Streaking: Finding the Probability for a Batting Streak Stanley Rothman and Quoc Le 4.1 Introduction 4.2 A recursive function to calculate the probability of a player having a 56-game hitting streak at some point in a season 4.3 A non-recursive piecewise function, NR(n), to calculate the probability of a player having a 56-game hitting streak at some point in a season 4.4 The Error = |R(n) - NR(n)| 4.5 Generalizing the concept of a streak 4.5.1 Definitions 4.5.2 Inputs and calculations 4.5.3 Each Individual plate appearance is a game 4.5.4 Each Individual At-Bat is a Game 4.6 Comparing Ted Williams’ 84-game consecutive on-base streak to Joe DiMaggio’s 56-game consecutive hitting streak 4.7 These two streaks evaluated for other great hitters 4.8 Conclusion References About the Authors II Basketball Bracketology: How can math help? Tim Chartier & Erich Kreutzer & Amy Langville & Kathryn Pedings 5.1 Introduction 5.2 Colley Method 5.3 Massey Method 5.4 Weighting Methods 5.4.1 Linear weighting and the Colley method 5.4.2 Linear weighting in the Massey method 5.4.3 Alternative weightings — when life isn’t linear 5.5 2009 Results 5.6 Concluding Remarks References About the Authors Down 4 with a Minute to Go G. Edgar Parker 6.1 Shoot the 3 6.2 Shoot the “easy” two References About the Author Jump Shot Mathematics Howard Penn 7.1 Angle of elevation 60 degrees 7.2 Angle of elevation 30 degrees 7.3 Varying the distance 7.4 Varying the height References About the Author III Football How Deep Is Your Playbook? Tricia Muldoon Brown and Eric B. Kahn 8.1 Introduction 8.2 The Game of Football and Mathematics 8.3 Counting the Formations The 3-4 Defense The 4-3 Defense The Nickel Defense The Dime Defense 8.4 Conclusion About the Authors A Look at Overtime in the NFL Chris Jones 9.1 Introduction 9.2 Game Data 9.3 Analyzing the current system 9.4 An alternative proposal 9.5 Conclusion About the Author Extending the Colley Method to Generate Predictive Football Rankings R. Drew Pasteur References Appendix Top 25 ranking, at the end of the 2008 season, by this method About the Author When Perfect Isn't Good Enough: Retrodictive Rankingsin College Football R. Drew Pasteur References About the Author Appendix Compilation of various rankings IV Golf The Science of a Drive Douglas N. Arnold 12.1 The double-pendulum approximation of the swing 12.2 The impact of the club head and the ball 12.3 The ball’s flight References About the Author Is Tiger Woods a Winner? Scott M. Berry G.H. Hardy's Golfing Adventure Roland Minton 14.1 Hardy’s Golf Problem 14.2 Hardy’s Analysis 14.3 Two Moments 14.4 Stroke Play 14.5 Skins Game 14.6 Tournament Golf 14.7 Handicaps 14.8 Laurels to Hardy References About the Author Tigermetrics Roland Minton 15.1 How many putts do the pros make? 15.2 Is Tiger Woods the best putter on tour? 15.3 What is a reasonable system for ranking putters? 15.4 Who is the best at hitting irons from the fairway? 15.5 Is there a hidden flaw in Tiger’s game? 15.6 Who is the best golfer overall? 15.7 What else can be learned? References About the Author V NASCAR Can Mathematics Make a Difference? Exploring Tire Troubles in NASCAR Cheryll E. Crowe 16.1 Introduction 16.2 What happened? 16.3 Race Tires vs. Street Tires 16.4 Mathematics is Making a Difference 16.5 Problem Resolved? Looking Towards the Future References About the Author VI Scheduling Scheduling a Tournament Dalibor Froncek 17.1 Some small tournaments 17.2 Tournaments for any even number of teams 17.3 Some more tournament properties References About the Author VII Soccer Bending a Soccer Ball with Math Tim Chartier References About the Author VIII Tennis Teaching Mathematics and Statistics Using Tennis Reza Noubary 19.1 Introduction 19.1.1 General 19.1.2 Specific 19.2 An Illustrative Example 19.3 Activities Activity 1: Bouncing Ball Activity 2: Applying Binomial Distribution, Matrices, Markov Chain, and Derivatives Activity 3: Calculations Based on Normal Distribution Activity 4: Constructing Confidence Intervals and Testing Hypotheses Activity 5: Applying Regression and Time Series for Prediction Activity 6: Research topics About the Author Percentage Play in Tennis G. Edgar Parker 20.1 Introduction 20.2 The Model 20.3 The Calculations 20.4 Big Shot Strategies 20.5 Analyzing Serve 20.6 Afterthoughts Reference About the Author IX Track and Field The Effects of Wind and Altitude in the 400m Sprintwith Various IAAF Track Geometries Vanessa Alday and Michael Frantz 21.1 Introduction and an Early Model 21.2 Quinn’s Model 21.3 The Effects of Track Geometry on Running Performance 21.4 Computation of the Effect of Winds 21.5 Altitude and the Propulsive Force 21.6 Data Collected and Results from Quinn 21.7 Effects of Wind Direction on Overall Performance 21.8 Effects of Altitude and Air Density 21.9 The Equal Quadrant Track 21.10 Wind Effects on the Equal Quadrant Track 21.11 The Ancient Greek Olympiad Track 21.12 Summary of Results 21.13 Directions for Possible Future Work References About the Authors Mathematical Ranking of the Division III Track and Field Conferences Chris Fisette About the Author What is the Speed Limit for Men's 100 Meter Dash Reza Noubary 23.1 Introduction 23.2 Methods Based on Trend Analysis 23.3 Methods Based on Outstanding Values 23.3.1 Methods Based on Threshold Theory 23.3.2 Methods Based on Theory of Records 23.4 Ultimate Record References About the Author May the Best Team Win: Determining the Winner of a Cross Country Race Stephen Szydlik 24.1 Warming Up 24.2 Mile 1: Basic Terminology and Some Alternatives 24.3 Mile 2: Fairness Criteria and Other Scoring Methods 24.4 Mile 3: More Criteria and Alternative Scoring Methods 24.5 Mile 4: Some Social Choice Theory 24.6 Mile 5: Impossibility? 24.7 Warmdown: Some Concluding Remarks References About the Author Biomechanics of Running and Walking Anthony Tongen and Roshna E. Wunderlich 25.1 Introduction 25.2 Applications Numerically Calculating Impulse Running Model Walking Model 25.3 Conclusions References About the Authors About the Editor "Mathematics and Sports", edited by Joseph A. Gallian, gathers 25 articles that illuminate the power and role of mathematics in the worlds of professional and recreational play. Divided into sections by the kind of sports, the book offers source materials for classroom use and student projects. Readers will encounter mathematical ideas from an eclectic group of writers, including undergraduate students, graduate students, and professional mathematicians. Following a preface, this book contains: (I) Baseball: (1) Sabremetrics: The Past, the Present, and the Future (Jim Albert); (2) Surprising Streaks and Playoff Parity: Probability Problems in a Sports Context (Rick Cleary); (3) Did Humidifying the Baseball Decrease the Number of Homers at Coors Field? (Howard Penn); (4) Streaking: Finding the Probability for a Batting Streak (Stanley Rothman and Quoc Le); (II) Basketball: (5) Bracketology: How can math help? (Tim Chartier, Erich Kreutzer, Amy Langville, and Kathryn Pedings); (6) Down 4 with a Minute to Go (G. Edgar Parker); (7) Jump Shot Mathematics (Howard Penn); (III) Football: (8) How Deep Is Your Playbook? (Tricia Muldoon Brown and Eric B. Kahn); (9) A Look at Overtime in the NFL (Chris Jones); (10) Extending the Colley Method to Generate Predictive Football Rankings (R. Drew Pasteur); (11) When Perfect Isn't Good Enough: Retrodictive Rankings in College Football (R. Drew Pasteur); (IV) Golf: (12) The Science of a Drive (Douglas N. Arnold); (13) Is Tiger Woods a Winner? (Scott M. Berry); (14) G. H. Hardy's Golfing Adventure (Roland Minton); (15) Tigermetrics (Roland Minton); (V) NASCAR: (16) Can Mathematics Make a Difference? Exploring Tire Troubles in NASCAR (Cheryll E. Crowe); (VI) Scheduling: (17) Scheduling a Tournament (Dalibor Froncek); (VII) Soccer: (18) Bending a Soccer Ball with Math (Tim Chartier); (VIII) Tennis: (19) Teaching Mathematics and Statistics Using Tennis (Reza Noubary); (20) Peentage Play in Tennis (G. Edgar Parker); and (IX) Track and Field: (21) The Effects of Wind and Altitude in the 400m Sprint with Various IAAF Track Geometries (Vanessa Alday and Michael Frantz); (23) What is the Speed Limit for Men's 100 Meter Dash? (Reza Noubary); (24) May the Best Team Win: Determining the Winner of a Cross Country Race (Stephen Szydlik); (25) Biomechanics of Running and Walking (Anthony Tongen and Roshna E. Wunderlich) This book is an eclectic compendium of the essays solicited for the 2010 Mathematics Awareness Month web page on the theme of Mathematics and Sports. In keeping with the goal of promoting mathematics awareness to a broad audience, all of the articles are accessible to college level mathematics students and many are accessible to the general public. The book is divided into sections by the kind of sports. The section on football includes an article that evaluates a method for reducing the advantage of the winner of a coin flip in an NFL overtime game; the section on track and field examines the ultimate limit on how fast a human can run 100 meters; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing, and an article on modeling Tiger Wood's career. The articles provide source material for classroom use and student projects. Many students will find mathematical ideas motivated by examples taken from sports more interesting than the examples selected from traditional sources. Edited By Joseph A. Gallian. A Compendium Of Essays Solicited For The 2010 Mathematics Awareness Month Web Page (cf. P. [4] Of Cover). Includes Bibliographical References.