mscroggs.co.uk
mscroggs.co.uk
Click here to win prizes by solving the mscroggs.co.uk puzzle Advent calendar.
Click here to win prizes by solving the mscroggs.co.uk puzzle Advent calendar.

subscribe

Blog

 2014-06-21 
With World Cup fever taking over, you may have forgotten that Wimbledon is just a few days away.

Tennis scoring

Tennis matches are split into sets (three sets for ladies' matches, five sets for men's), which are in turn split into games. The players take it in turns to serve for a game. The scoring in a game is probably best explained with a flowchart (click to enlarge):
To win a set, a player must win at least six games and two more games than their opponent. If the score reaches six games all, then a tie break is played. In this tie break, the first player to win at least seven points and two points more than their opponent wins. In the final set there is no tie break, so matches can last a long time.

Winning with the smallest share of points

Due to the way tennis is split into sets and games, the player who wins the most points will not necessarily win the match. This got me thinking: what is the smallest proportion of points which can be won while still winning the tennis match?
First, let's consider a men's match. In order to win with the lowest proportion of points, our player should let his opponent win two sets without winning a point and win the other three sets. In the two lost sets, the opponent should win 0-6 taking every point: in total the opponent will win 48 points in these sets.
Leaving the final set for now, the other two sets are won by our player. To win these with the smallest proportion of the points, they should be won 7-6 on a tie break. In the 6 lost games, the opponent should take all the points. In the won games and the tie break, our player should win by two points with the lowest total score. (Winning with more than the lowest total score will mean both players win an equal number of extra points, moving the proportion of points our player wins closer to 50%, higher than it needs to be.)
Therefore, our player will win 4 points out of 6 in the games he wins, win 0 out of 4 points in the games he loses and wins the tie break 7 points to 5. This means that in total our player will 62 points out of 144 in the two won sets.
For the same reason as above, the final set should be won with the lowest total score: 6-4. Using the same scores for each game, our player wins 24 points out of 52.
Overall, our player has won 86 points out of 244, a mere 35% of the points.
If the match is a ladies' match then the same analysis will work, but with each player winning one less set. This gives our player 55 points out of 148, 37% of the points.
This result demonstrates why tennis remains exciting through the whole match. The way tennis is split into sets and games means that our opponent can win 65% of the points but if the pressure gets to them at the most important points, our player can still win the match. This makes for a far more interesting competition than a simple race to one hundred points which could quickly become a foregone conclusion.

Comparing players with serving stats

During tennis matches, players are often compared using statistics such as the percentages of serves which are successful. Imagine a match between Player A and Player B.
In the first set, Player A and Player B are successful with 100% and 92% of their serves respectively. In the second set, these figures are 56% and 48%. Player A clearly looks to be the better server, as they have a higher percentage in each set. However if we look at the two sets in more detail:
Player APlayer B
First Set20/2067/73
Second Set45/8013/27
Total65/10080/100
Table showing successful serves/total serves.
Overall, Player B has an 80% serve success rate, while Player A only manages 65%.
This is an example of Simpson's paradox: a trend which appears in the set-by-set data disappears when the data is combined. This occurs because when we look at the set-by-set percentages, the total number of serves is not taken into account: Player A served more in the second set so their overall percentage will be closer to 56%; Player B served more in the first set so their overall percentage will be closer to 92%.
Tags: sport, tennis, news

Similar posts

World Cup stickers 2018, pt. 3
World Cup stickers 2018, pt. 2
World Cup stickers 2018
Euro 2016 stickers

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "d" then "e" then "c" then "a" then "g" then "o" then "n" in the box below (case sensitive):

Archive

Show me a random blog post
 2019 

Dec 2019

Christmas card 2019

Nov 2019

Christmas (2019) is coming!

Sep 2019

A non-converging LaTeX document
TMiP 2019 treasure punt

Jul 2019

Big Internet Math-Off stickers 2019

Jun 2019

Proving a conjecture

Apr 2019

Harriss and other spirals

Mar 2019

realhats

Jan 2019

Christmas (2018) is over
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

people maths golden ratio captain scarlet realhats pythagoras error bars rugby london harriss spiral manchester science festival rhombicuboctahedron puzzles trigonometry wool probability arithmetic sport platonic solids polynomials books hexapawn mathsjam noughts and crosses mathsteroids matt parker sorting games nine men's morris php tmip sound christmas card triangles stickers pizza cutting radio 4 advent calendar curvature manchester palindromes golden spiral reuleaux polygons electromagnetic field misleading statistics bodmas geometry dataset dragon curves flexagons statistics ternary christmas a gamut of games programming royal baby pac-man game show probability news go mathslogicbot countdown plastic ratio chebyshev weather station world cup reddit light inline code propositional calculus gerry anderson twitter binary video games bubble bobble graph theory speed logic chess the aperiodical london underground oeis final fantasy talking maths in public machine learning estimation folding tube maps javascript hats accuracy dates data football python coins fractals frobel tennis map projections asteroids national lottery big internet math-off european cup menace approximation martin gardner cambridge craft chalkdust magazine draughts game of life cross stitch interpolation latex raspberry pi folding paper braiding

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2019