Tennis matches are won by the player winning the last point.

Football matches are decided by a rare event – a goal. The team scoring the last goal may not win. Cricket uses an accumulative scoring pattern like tennis – plenty of runs are scored in each match. Again, the team scoring the final run may not win. All runs scored in cricket, whether scored by the batsman or the extras conceded by opposition are of equal value. In the final analysis, only aggregate counts. Team scoring more runs (in a completed limited overs match) wins the match.

Tennis follows a distinctive scoring system. There are two teams who take turns to start the point – one player serves and the other returns. The ball remains in play until it is not returned, hits the net or sprayed outside the playing area. A point is awarded at the end of each *play*. The corresponding call for winning a point is : 0 – “love”, 1 – “15”, 2 – “30”, 3 – “40” and 4 – “game”. Thus team winning 4 points wins the game.

Points won by the server are mentioned first. A score of 40-30 means, serving team has won 3 points and receiving team 2. A score of 0-40, read as love-40, indicates all 3 points for receiving side. A game can’t be won with the difference of a single point. This means that there are no 7 point games in tennis where one team has 4 points and the other 3. In such situations, the game continues for an even number of points – 8, 10, 12, .. until one side takes a 2 point lead.

When both sides have won 3 points, the score will be 40-40 which is read as deuce. The next point does not decide the game but the side winning the point is given an *advantage*. A score of A-40 indicates serving side has 4 points and 3 points for receiver. Thus 40-A score indicates 4 points for receiving team. Side winning the point after advantage, wins the game. Otherwise the score reverts to deuce (i.e. 40-40).

This scoring system can also be understood by treating A-40 as a 40-30 situation. If the server wins next point then it will be the first to reach 4 points with a difference of 2, thereby winning the game. Else the score will become 40-40 or deuce. If receiver wins the point, then treat it as a 30-40 situation. Receiver will win the game by claiming the next point or the game reverts to deuce.

The serving side has an advantage in winning a point. It is observed that 60-65 percent points are won on serve. Here we will use p=0.64 as the probability of player on serve to win the point. This means the receiver has a chance of q = 1-p to win it. At deuce, server needs to win the next two points – a probability of p*p or p^{2}. The probability of next 2 points shared between two teams is 2pq because either p wins and then loses (q) or loses the point first (q) and then wins (p). After 2 such points, we return to situation of deuce. Combining these two scenarios, we can determine the probability (d) of winning from the position of deuce as p^{2}+2pqd which can be simplified as d = p^{2} / ( 1 – 2pq). Substituting p =0.64 and q = 0.36, we find that d = 0.76. It means that if a server wins 64% points on serve, the likelihood of winning from a position of deuce is 76%. If player wins 60% points on serve, the chance of winning from deuce reduces to 69%.

It is obvious that the chance of winning from 30-30 is the same as the chance of winning from deuce since both situations require either winning next 2 points to finish the game (one way or the other) or reach 40-40 after splitting successive points.

From the perspective of server, winning 4 points indicates 100% probability of winning the game and losing 4 points means 0% chance of success. It is proved above that the probability of winning from 3-3, 4-4, or any other n-n is 76%. By knowing these 3 constants, it is possible to determine the probability of winning from any other score. At 40-30, a player will either win the next point(p_{a}) to win the game (100% chance of winning) or lose the point(q_{a}) to reach deuce (d). This value is calculated as p_{a}*1 + q_{a}*d which equals to 91% chance of winning the game on serve from 40-30 when the server wins 64% points on serve. The chance of winning from 30-40 can be similarly calculated as winning the next point to reach deuce or losing the next point to lose the game – p_{a}*d + q_{a}*0. This value is slightly less than 50%.

We can extend this logic recursively. At 40-15, a player will either win the point to win the game or lose the point to reach 40-30 which can be calculated using the formula p_{a}*1 + q_{a}* (Probability of winning from 40-30 calculated above). We can successively move backwards to the beginning of the game with both players at 0-0. There is more than 81% chance for the player serving to win the game. In other words, a player is likely to hold his serve in 4 out of 5 situations with the advantage of winning 64% points on serve. The conditional probabilities of a player winning the game on serve from various score lines for p_{a} = 0.64 :

0 | 15 | 30 | 40 | Game | |

0 | 0.813 | 0.667 | 0.455 | 0.199 | 0 |

15 | 0.894 | 0.787 | 0.598 | 0.311 | 0 |

30 | 0.954 | 0.894 | 0.760 | 0.486 | 0 |

40 | 0.989 | 0.969 | 0.913 | 0.760 | |

Game | 1 | 1 | 1 |

For further details about ‘THE MATHEMATICS OF TENNIS’, please refer to this link by Tristan Barnett and Alan Brown.

At the end of a game, winner will get 100% credit and loser gets none irrespective of the number of points scored. A player may win all the 4 points played, or 14 of the 26 points in a game with 10 occurrences of deuce. In both these situations, the net result is 1 game to winner and nothing for loser. Due to these peculiarities, it is possible to lose a match despite winning more points.

At the end of first game, the opponents will serve. The above process is repeated till someone wins 4 or more points with a difference of two. The probability of winning point on serve for the other team will be different. Here, we will treat both sides as equal – it means that both teams have a 64% chance to win a point on serve. Both teams will alternate serves until the set is won. A set is played until one player or team has won six games. Once again, a team must win a set by a difference of two games. It is possible to win a set 6-0, 6-1, 6-2, 6-3 or 6-4. If the scores are 6-5, then 12th game is played. Either one side will win 7-5 or the set is decided by a tiebreaker at 6-6.

Just as we calculated the probability of winning a game from various scores, it is possible to determine the probability of winning a set. If the score is 7-5 in favour of the server, then team has 100% chance of winning the set. If the score is 5-7 then the set is lost. At 6-6, the probability of winning the set is equal to the probability of winning the tiebreaker (t). At 6-5, the server will either win the game to take the set or lose it to start the tiebreaker. Like before we can calculate the probability of winning a set at 6-5 on serve as – p_{ga}*1 + q_{ga}*t. At 5-6, the probability of winning the set will be p_{ga}*t + q_{ga}*0 because winning the game will start the tiebreaker and losing the game will result in losing the set. The probability of winning at 5-4 can be calculated by knowing the probability for the scores at 6-4 and 5-5. Similarly, we can continue the recursive logic to determine the probability of winning a set after any other game score.

There is no advantage in winning a set whether a player serves first or second in the set. If both players have the same advantage in winning a point on serve, the probability of winning a set from 0-0, 1-1, 2-2, .. is always 50%. Assuming player 1 wins 62% points on serve and concedes 60% while receiving, this nominal advantage will translate to 56.8% chance of winning the set at 0-0, 56.3% at 1-1, 55.7% at 2-2, 55.2% at 3-3, 54.6% at 4-4 & 54.4% at 5-5 irrespective of who serves first.

A tiebreaker game is played at the score of 6-6. The rules of a tiebreaker game are different from a standard service game. First player to reach 7 points with a difference of at least 2 points wins the tiebreaker game and thereby the set. Unlike standard games, server changes frequently during the tiebreaker. Player serving the opening game of the set serves the first point. After that players alternate serving every two points. At the end of 12 points, if both players are still locked at 6-6, two more points are played. Either one player will win both points to claim the tiebreaker 8-6 or two more points will be added with each player on 7 points. Similar to deuce, the process will be repeated until we have the winner. Unlike standard service games, no special calls are assigned to designate points won in a tiebreaker.

The order of serve offers no advantage in winning a tiebreaker. Assuming equal strength of serve, both players will have a 50% chance of winning the tiebreaker at 0-0, 1-1, .. 6-6 etc. A nominal differential of 2% over a base of 60% will result in winning the tiebreak 53.3% at 0-0. It progressively reduces to 53%, 52.8%, 52.6% ultimately down to 52.1% at 6-6 irrespective of who serves first.

Most of the matches are played as best of 3 sets. The player winning 2 sets will win the match. All sets are determined by a tiebreaker if players are locked at 6-6. Sometimes best of 5 sets are played where a player has to win 3 sets. The final set may be played as an advantage set – no tiebreaker used at 6-6. Players continue to serve for 2 additional games until one player wins both.

If both players serve at equal potential, the chance of winning the match is 50% at 0-0 sets, 1-1 sets and 2-2 sets. A 2% differential on a base of 60% translates to nearly 63% chance of winning a 5 set match and 60% chance of winning best of 3.

Once we have rules to calculate the probability of winning a game, a tiebreaker, a tiebreak set, an advantage set, a best of 3 and best of 5 set match (either with tiebreak set or an advantage set in final set) – it is then possible to determine the chance of winning from any score based on points a & b, games c & d and sets e & f won by player 1 & 2 and who serves next.

Based on the varying probability of win after each point, it is possible to derive a value for pressure on a server. If the server leads 40-0, losing next point does not change the outcome of the match significantly but at 40-30 winning the next point wins you the game while losing will lead to deuce. So the pressure increases. At 30-40, the pressure is even higher as losing the next point costs the game. Losing a game in opening set will not change the probability of winning the match as much as it does in the final set. Thus an identical scoreline mentioned above but in the final set will result in greater pressure on serve.

On 1st March, in Dubai, World #116 qualifier Evgeny Donskoy stunned Roger Federer 3-6, 7-6(7), 7-6(5). The summary scoreline indicates that Federer broke Donskoy in first set at least one more time than he got broken. The next 2 sets were decided in tiebreaker games. Donskoy won 2nd set tiebreak at 9-7 and the final one at 7-5. Now we will translate the point by point data made available at scoreboard.com into an illustrative chart for each set.

Roger Federer, recently crowned Australian Open Champion for a record extending 18th grand slam, was facing qualifier Donskoy in their first meeting. It should have been a straightforward win for Federer. To offer live odds, punters will use a huge differential between points earned on serve for Federer and points conceded to Donskoy on his serve. These charts treat all players as equal and rely solely on match situation to determine who is ahead and the pressure on serve does not vary by rankings.

Above chart shows Federer in red and his serve in pink. Donskoy is represented in shades of blue. The summary stats show Federer winning 28 points to 18. He also leads in points won on own serve ar 18 – 11. 6-3 set score line indicates no tiebreaker. Federer broke serve 2 times with 3 break opportunities. We notice Donskoy broke him back once at the first chance. We also see that there was only 1 deuce on Donskoy’s serve.

On x-axis we see points played in each game. Barring 1 game, others were over in 4, 5 or 6 points. The longest game was 6th which extended to 8 points where deuce is represented as a yellow dot.

There is no mention of scale on y-axis. This is to indicate that the actual value of points allocated is not important – only the relative difference between red and blue curve is important. We have seen earlier that tennis scoring at the end of each game (& set) wipes out the progress made by losing player and assigns entire spoils to the winner. In this illustration, the credit for points earned in losing games is retained. This will give us an idea about the ability of a player to win points irrespective of the game outcome.

We notice that after first 3 games, the difference between two curves is nominal. Federer is winning a few more points than Donskoy purely due to an extra service game. The 4th game ends swiftly in 4 points with red curve rising sharply. It indicates a love-break for Federer. While 0-15 and 0-30 are treated as nominal points, at 0-40, the first breakpoint is assigned additional value hence red curve rises above blue. When Federer breaks serve on the next point, actual points assigned to Federer increase further to convey that Federer has surged ahead in the set.

Federer holds his serve on love in 5th game.,In the 6th game, which was the longest with a deuce, Federer earned another break point which was saved by Donskoy to take the game to deuce. This is clear by the rise in red curve indicating break point and subsequent fall which shows that the break point was saved and yellow dot indicates deuce. Red curve rises on the next two points indicating Federer won next 2 points to get his second break of the set.

Next game on Federer’s serve is 6 points long where blue line rises sharply towards the end. This means that Donskoy won the last 2 points after 30-30 to reduce the deficit. This was followed by a love-hold by Donskoy.

Federer serving for the set at 5-3 in the 9th game. We see that it is a 5 point game which means Federer does not falter – losing only 1 point on serve. There is an even sharper rise in the red curve, the first one indicating set point and the next increase shows that Federer won the set comfortably.

Now we take a look at the height of the background area charts in pink and light blue. These indicate whether it was a routine or a clutch point. Every point is assigned at least half the max value. The differential will vary as we get closer to the result. Losing the first set does not mean end of the road with a chance to recover in next two. The highest rise is noticed on the first breakpoint of the serve. A break of serve is vital in the set and one break is enough to win the set. The second break point opportunity does not significantly increase the likelihood of Federer winning the match hence the ‘pressure on serve’ does not increase as much as the first breakpoint. Same is true for the sole break by Donskoy. Losing the serve still does not cost Federer enough and we see that he manages to close out the set on his next serve. The pressure index actually drops towards the end of the set because Federer could afford to lose a point without affecting the outcome of the set.

This was a routine first set and fails to get any mention in the ATP match report.

The second set finds only one mention in the ATP match report –

Federer looked poised to claim his ninth victory of the year when he held match points at 6/4 and 7/6 in the second set tie-break, before Donskoy fought back to force a decider.

We start with summary statistics and immediately notice that Donskoy won more points (43-37) in set despite trailing in points won on serve (30-32). Unlike the first set, there were no break point opportunities for Federer. Donskoy had one chance to break in the 10th game which was the longest at 12 points and 3 instances of deuce. We notice that Donskoy had a chance to break at 40-A after first deuce. This break point was also a set point hence the differential is higher. Federer saved a set point which is a key information that does not appear in the match report.

In tennis terms, at 6-6 both players are level. Here we see that the blue curve is higher than red at the beginning of tiebreak indicating more points won by Donskoy. In the tiebreak we notice red curve jumping above blue indicating a mini break for Federer. One mini break is enough to win the tiebreaker. Next we see a sharper rise when the score reached 6-4 with Federer serving. This was the first match point where Federer was expected to finish the match. He lost that point on serve to Donskoy. Even at 6-5, he held a second match point, but this time it was on Donskoy’s serve so the chances of converting it were lower but that does not reduce the pressure on Donskoy’s serve. Donskoy saved the second match point to level the tiebreak at 6-6.

Federer won the next point on Donskoy’s serve to lead 7-6 for his second match point on serve. Donskoy won the next 3 points – two on Federer’s serve, to win the second set and level the match. While the tiebreak points increase only by 1 unit, the chart shows wilder swings in red and blue curve at the business end of the match by assigning higher weight to clutch points.

In the second set, Federer had opportunities to win the match while Donskoy had his chance to win the set to enforce a 3rd. Clearly the dynamics of second set is different from the first. The pressure index rises higher than first set when Donskoy had a set point. This index goes off the chart towards the end when the match quickly swung from a win for Federer to the set for Donskoy.

This brings us to the exciting upset in the final set.

The ATP match summary describes all the key moments of the 3rd set.

Federer looked to have regrouped when he broke in the sixth game of the third set and served for the match at 5-4, but again he failed to close out victory, losing his serve to 30.

Donskoy then turned the tables as he broke Federer in the 11

^{th}game. But, serving for the biggest win of his career against the 18-time Grand Slam champion, the Russian was broken to love as the pulsating match when (sic) to a deciding tie-break.Federer once again put himself in a commanding position as he led 5/2 with two serves to come. But in an astonishing turn of events, World No. 116 Donskoy reeled off the final five points of the match to prevail in just over two hours.

Federer broke in the sixth game – blue line representing Federer rises.

He lost serve to 30 – red line catches up.

Donskoy turned the tables – Red line takes charge.

The Russian was broken to love – Red line remains flat while the Blue line levels up.

Federer led 5/2 with two serves to come – Blue line continues to march but not high enough indicating this was not as close as the match point situation in second set.

Donskoy reeled off the final five points of the match – Blue line remains flat. Red line initially clears the deficit then takes the deciding lead to close the game.