The Mathematics of Tennis Tristan Barnett and Alan
The Mathematics of Tennis Tristan Barnett and Alan Brown Strategic Games www. strategicgames. com. au
Preface 183 games 11 hours and 5 minutes of playing time Isner winning 70 -68 in the advantage final set
Preface Was this long match predictable and what are the chances of this happening again? Hence the mathematics of tennis is concerned with the chances of players winning the match (who is likely to win? ) and match duration (when is the match going to finish? )
Preface Whilst the mathematics of tennis could be of interest to tennis organizations, commentators, players, coaches and spectators; it could also be applied to teaching by using the well-defined scoring structure of tennis to teach concepts to students in probability and statistics. Such concepts include summing an infinite series, Binomial theorem, backward recursion, forward recursion, generating functions, Markov chain theory and distribution theory. The mathematics of tennis applied to teaching also allows students to build their own tennis calculator using spreadsheets, which in turn could assist in the
Ch 1: Winning a game Counting Paths Binomial Theorem Markov Chain theory Backward Recursion Forward Recursion
Winning a game Backward Recursion Let p. A represent a constant probability of player A winning a point on serve Let PA (a, b) represent the probability that player A wins the game on serve when the score is (a, b) The backward recursion formula becomes: PA(a, b) = p. APA (a+1, b) + q. APA (a, b+1) The boundary values are: PA (a, b) = 1, if a = 4 and b ≤ 2 PA (a, b) = 0, if b = 4 and a ≤ 2 PA (3, 3) = p 2 A / (p 2 A+ q 2 A)
Winning a game Backward recursion Enter the text p. A in cell D 1. Enter the text q. A in cell D 2 Enter 0. 6 in cell E 1 Enter =1 -E 1 in cell E 2 Enter 1 in cells C 11, D 11 and E 11 Enter 0 in cells G 7, G 8 and G 9 Enter = E 1^2/(E 1^2+E 2^2) in cell F 10 Enter =$E$1*C 8+$E$2*D 7 in cell C 7 Copy and Paste cell C 7 in cells D 7, E 7, F 7, C 8, D 8, E 8, F 8, C 9, D 9, E 9, F 9, C 10, D 10 and E 10 Notice the absolute and relative referencing used in the formula =$E$1*C 8+$E$2*D 7.
Winning a game Forward recursion NA(g; h) = p. ANA (g -1, h), if g = 4 and 0 ≤ h ≤ 2; h = 0 and 1 ≤ g ≤ 4; g ≥ 3, h ≥ 3 and g = h + 1; g ≥ 3, h ≥ 3 and g = h + 2 NA (g; h) = q. ANA (g, h - 1), if h = 4 and 0 ≤ g ≤ 2; g = 0 and 1 ≤ h ≤ 4; g ≥ 3, h ≥ 3 and h = g + 1; g ≥ 3, h ≥ 3 and h = g + 2 NA (g; h) = p. ANA (g – 1, h) + q. ANA (g, h - 1), if 1 ≤ g ≤ 3 and 1 ≤ h ≤ 3; g ≥ 4, h ≥ 4 and g = h The initial value is NA (0, 0) = 1
Ch 2: Winning a match Backward recursion Winning a tiebreak game Winning a tiebreak set Winning an advantage set Winning an all tiebreak set match Winning a final set advantage match
Ch 3: Winning a match Forward recursion Winning a game Winning a tiebreak set Winning an advantage set Winning an all tiebreak set match Winning a final set advantage match
Ch 4: Duration of a game Binomial Theorem Generating functions Forward recursion Backward recursion
Ch 4: Duration of a game Binomial Theorem 4 points - p. A 4 +q. A 4 5 points - 4 p. A 4 q. A +4 q. A 4 p. A 6 points - 10 p. A 4 q. A 2 +10 q. A 4 p. A 2 8, 10, 12. . - 20 p. A 3 q. A 3(p. A 2 + q. A 2)(2 p. Aq. A)(x-8)/2 The distribution of the number of points played in a game from the
Ch 4: Duration of a game Generating functions ∑xetxf(x)=e 4 tf(4)+e 5 tf(5)+e 6 tf(6)+(Npg(3, 3)(1 -Npg(1, 1))e 8 t)/(1 -Npg(1, 1)e 2 t) The distribution of the number of points played in a game from the outset with p. A=0. 6
Ch 5: Duration of a match 1/2 Number of points in a tiebreak game Number of games in a tiebreak set Number of games in an advantage set Number of sets in a match
Ch 6: Duration of a match 1/4 Number of points in a game Number of points in a tiebreak game Number of games in a tiebreak set Number of games in an advantage set Number of sets in a match
Ch 7: Duration of a match 4/4 Number of points in a game Number of points in a tiebreak set Number of points in an advantage set Number of points in a match Time duration in match
Ch 7: Duration of a match 4/4 Need to identify who is serving in the current set (s. A, c. A) Need to identify who won the previous game for each set (w. A, l. A) Who is serving first at the start of each set (n. A, n. B)
Ch 7: Duration of a match 4/4 Let WYpg(a; b|c. A; w. A)(t) and WYpg(a; b|c. A; l. A)(t) represent the weighted moment generating functions of the number of points remaining in the game from score line (a; b) given player A is serving and player A won and lost the game respectively. Theorem 7. 2. 1. WYpg(a; b|c. A; w. A)(t) =WYpp(()|c. A; w. A)(t)WYpg(a+1; b|c. A; w. A)(t)+WYpp(()|c. A; l. A)(t)WYpg (a; b+1|c. A; w. A)(t) WY pg(a; b|c. A; w. A)(t) = Ppg(a; b|c. A; w. A)MY pg(a; b|c. A; w. A)(t)
Ch 7: Duration of a match 4/4 w 1(Ypg(a; b|c. A; w. A))=p. Aw 1(Ypg(a+1; b|c. A; w. A))+q. Aw 1(Ypg(a; b+1|c. A; w. A))+p. APpg(a+1; b|c. A; w. A) + q. APpg(a; b + 1|c. A; w. A) w 2(Y 2 pga; b|c. A; w. A))=p. Aw 2(Ypg(a+1; b|c. A; w. A))+q. Aw 2(Ypg(a; b+1|c A; w. A))+2 p. Aw 1(Y pg(a+1; b|c. A; w. A))+2 q. Aw 1(Y pg(a; b+1|c. A; w. A))+p. APpg(a+1; b|c. A; w. A)+q. APpg(a; b+1|c. A; w. A) w 3(Ypg(a; b|c. A; w. A))=p. Aw 3(Ypg(a+1; b|c. A; w. A))+q. Aw 3(Ypg(a; b+1|c. A ; w. A))+3 p. Aw 2(Ypg(a+1; b|c. A; w. A))+3 q. Aw 2(Ypg(a; b+1|c. A; w. A))+3 p. Aw 1(Ypg(a+1; b|c. A; w. A))+3 q. Aw 1(Y pg(a; b+1|c. A; w. A)) + p. APpg(a + 1; b|c. A; w. A) + q. APpg(a; b + 1|c. A; w. A) w 4(Ypg(a; b|c. A; w. A))=p. Aw 4(Ypg(a+1; b|c. A; w. A))+q. Aw 4(Ypg(a; b+1|c. A ; w. A))+4 p. Aw 3(Ypg(a+1; b|c. A; w. A))+4 q. Aw 3(Ypg(a; b+1|c. A; w. A))+6 p. Aw 2(Ypg(a+1; b|c. A; w. A))+6 q. Aw 2(Ypg(a; b+1|c. A; w. A))+4 p. Aw 1(Ypg(a+1; b |c. A; w. A))+4 q. Aw 1(Ypg(a; b+1|c. A; w. A))+p. APpg(a+1;
Ch 8: Predictions Data Analysis (On. Court, Tennis Navigator) Updating Rule John Isner vs Nicholas Mahut Sports Multimedia
Ch 8: Predictions John Isner vs Nicholas Mahut Set Isner Mahut 1 76. 9% 65. 5% 2 57. 1% 76. 9% 3 81. 1% 77. 5% 4 71. 8% 66. 7% 5 77. 2% 81. 9% All 76. 2% 78. 7% Percentage of points won on serve in each set for the Isner vs Mahut match at the 2010 Wimbledon Championships
Ch 8: Predictions From the table it was shown that if the percentage of points won on serve for both players prior to the fifth set reflected their serving performance in the third set, then there is a 2. 8% chance that the match would reach 69 -69 all in the deciding fifth set.
Course: Tennis Statistics Unit Outcomes Learn concepts in Markov Chains, Binomial theorem, Recursion formulas, generating functions, probability theory, exponential smoothing and game theory Become familiarized with Excel software by hands-on experience in building your own tennis calculator, which in turn assists in the understanding of probability and statistical concepts Apply operation research techniques to addressing real-world problems in tennis Have a greater appreciation of the sport of tennis through its history and unique scoring structure Prepare students with quantitative skills for careers in tennis and sport abroad
Course: Tennis Statistics Structure of the course The course is divided into two parts. Part 1 defines a mathematical model by calculating the chances of winning and duration of a match. Part 2 focuses on applications in performance aspects of tennis data, predictions, serving strategies and resource allocation; regulation aspects by analysing scoring systems and the challenge system. Resources The textbook for the course is: Barnett T and Brown A (2012). The Mathematics of Tennis. Strategic Games. Tennis calculator
Course: Tennis Statistics References www. strategicgames. com. au strategicgames@hotmail. com
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