Simple Tennis
MODELs
1) Tennis Game MODEL #1
“ You need to hit a ball into the opponent’s court, until the opponent misses to hit a ball back to your court. There’s no opponent’s service, and your service is just hit a ball
Into opponents court.”
If an opponent always misses the first ball at 100% of the time, you need to make 1 ball in at least 50% to win a match.
If an opponent always misses the 2nd ball at 100% of the time, you need to make 2 balls in at least 50% to win a match, which means that you have to make each ball in with
(1/2)-1/2 (71%).
If an opponent always misses the 3rd ball at 100% of the time, you need to make 3 balls in at least 50% to win a match, which means that you have to make each ball in with
(1/2)-1/3 (79%).
Likewise, if an opponent always misses the n-th ball at 100% of the time, you need to make n balls in at least 50% to win a match, which means that you have to make each ball in with an average fraction of (1/2)-1/n .
Figure 1 Required shot percentage as a function of
number of shots
Figure 1 shows the required shot percentage to win a match, as a function of number of shots to make. If the average shot percentage is below the curve in Figure 1, you’ll lose a match.
Opponent’s strength(consistency) is shown in how many balls he/she can hit balls back to you. If you have 80% of shot consistency(if you can make 80% of balls in), you can compete with an opponent who can hit 2 balls back. If you can make 90% of balls in, you can compete with an opponent who can hit 6 balls back.
2)
Tennis Game MODEL #2
“Suppose a player A and a player B are playing a tennis match, and player A gets a constant fraction, X (0.5 < X < 1.0) to get a point, and player B gets a fraction , (1-X) to get a point on all points throughout the match”
In this tennis game model, the expected probability(Y-axis) of the following can be calculated easily as a function of X (a player A’s point winning fraction):
1) player A’s winning a game (Figure 2)
2) player
A’s winning a 6-game set (Figure 2)
3) player
A’s winning a best 3 set match, (Figure 2)
4) player
A’s winning a deuce game
(Figure 3)
5) player
A’s winning a 7 point tie-breaker
(Figure 3)
6) Player
A’s winning a best 3 set with 2 straight sets (Figure 4)
7) Player
A’s winning a best 3 set with 3
sets (Figure 4)
8) Player
B’s winning a best 3 set with 2 straight sets (Figure 4)
9) Player
B’s winning a best 3 set with 3
sets (Figure 4)
10)Player
A’s winning a game with love
(Figure 5)
11)Player
A’s winning a game with 15 (Figure 5)
12)Player
A’s winning a game with 30
(Figure 5)
13)Player
A’s winning a game with Deuce
(Figure 5)
14)Player
A’s winning a set with 6-0 (Figure 6)
15)Player
A’s winning a set with 6-10 (Figure 6)
16)Player
A’s winning a set with 6-2 (Figure 6)
17)Player
A’s winning a set with 6-3 (Figure 6)
18)Player
A’s winning a set with 6-4 (Figure 6)
19)Player
A’s winning a set with 7-5 (Figure 6)
20)Player
A’s winning a set with a tie-breaker (Figure 6)
Figure 2 Winning percentage as a function of point winning percentage
Figure 3 Winning percentage as a function of point winning percentage
Figure 4 Winning percentage as a function of point winning percentage
Figure 5 Game winning percentage as a function of point winning percentage
Figure 6 Set winning percentage as a function of point winning percentage
If a player A gets 60% of points, Player A wins 74% of games(69% in a deuce), 96% of sets(78% in a tie-breaker), and 99.6% of matches against a players B with 40% of point winning percentage. At the same percentage, player B with 40% of point winning percentage wins 26% of games(31% in a deuce), 4% of sets(22% in a tie-breaker), and 0.4% of matches. Player A has 2 sets win 93% of time and 3 sets win 7% of time. Player A with 60% point winning percentage win a game by mostly 15 or 30 in a game, followed by deuce games, and the player A wins a set mostly with 6-1 or 6-2, for example with the point winning percentage. A weaker player(with lower point winning percentage) has more chances to win in a deuce game(so called “Break (in) Deuce”), in a tie-breaker, and in a 3 set matches (Figures 3 and 4).
Interestingly, at 50-50 point (EQUAL)winning percentages of players A and B, most games are 30-games or deuce games, and set counts are mostly 6-4 or 6-3(Figures 5 and 6).
For example, let a, and b represent events that player A or player B get a point in a match. Any tennis match is then represented completely by a sequence of a or b.
If a player gets 4 points with 2 or more points difference from the sum of opponent’s points, a player gets a game. If both a player and an opponent gets 3 points in a game, the game is called as a deice game. 0 point is called Love, 1 point is called as 15, 2 points are called as 30, and 4 points are called as 40, in a game.
If a player gets 6 games with 2 or more games difference from the sum of opponent’s games, a player gets a set. If both a player and an opponent gets 6 games in a set, a tie-breaker game is played, till a side gets 7 points with 2 or more points difference.
The best of 3 set match is won by a player by taking 2 sets.
If a player A gets a love-game, which is 4 points in a row in a game, the probability to have the case aaaa happen is:
X4 ,which is shown in “Love” curve in Figure 4.
If a player B gets 2 points(30) in A’s winning game, 2 b’s and 3 a’s must happen before A’s getting the 6th point. An example is abaaba. The number of cases for arranging a and b in that way is 5C2 =(5x4)/2=8 . So the probability of A’s 30-game is
8xX4(1-X)2 ,which is shown in “30” curve in Figure 4.
A deuce game happens if ab or ba follows after 30-all. Probability to have this happen is 2C1x4C2X3(1-X)3 . For a player A to win a deuce game, the following have to follow: aa, or (ab or ba)aa, (ab or ba)(ab or ba)aa, …. , which probability is:
X2 + 2C1 X(1-X)X2 + {2C1 X(1-X)}2 X2 + …= X2 /{1- X(1-X)} . So the A’s winning probability in a deuce game is:
2C1*4C2X3(1-X)3 * X2 /{1- X(1-X)} = 12X5 (1-X)3 /{1- X(1-X)} .
Other winning probability is calculated in the same way. Please try to figure out the other probability functions by your self.
Reference: “The science of Tennis”
by K. Miura and T. Chomabayashi,
Koubunsha Publishing Company, Japan, 1980.