How Tennis Points Turn Into Match Odds: Markov Chains for Tennis Betting

The question

"Player A holds serve a bit more than Player B. How do I turn that into a match price?"

How tennis points turn into match odds is a lovely example of modelling a sport from the bottom up. Statometrics itself focuses on football, and this lesson is here because the method, working out a match from its states one step at a time, is the same one that drives in-play football models. Tennis just shows it most cleanly.

The idea in one sentence

If each point depends only on who is serving, you can work out the chance of winning a game, then a set, then the match, by stepping through every score from the current one.

The picture

A tennis match is a ladder: points make games, games make sets, sets make the match. At every rung, the chance of what happens next depends only on the current score and who's serving. That property, "only the present matters, not the path", is what makes it a Markov chain.

The scoring system is an amplifier. A player winning 65% of points on serve holds about 83% of service games. Games amplify into sets, and sets into matches. That's why tennis prices can be so short: a modest point edge becomes a big match edge.

Win % of points on serve	Hold %
50%	50.0%
55%	62.3%
60%	73.6%
62%	77.6%
65%	83.0%

The calculator below climbs the whole ladder from two serve percentages.

Try it · Tennis Markov calculator

Player A wins on own serve65.0%

Player B wins on own serve62.0%

Format

Sets won by A

Sets won by B

1 · Point

A65.0%

B62.0%

2 · Hold

A82.96%

B77.59%

3 · Set

A59.90%

4 · Match

A64.66%

A breaks B

22.41%

B breaks A

17.04%

A wins tiebreak

54.95%

Fair price A

1.547

Fair price B

2.830

Sets (A-B)	A wins match	Fair A
0-0	64.66%	1.547
1-0	83.92%	1.192
0-1	35.88%	2.787
1-1	59.90%	1.669

One more point of A’s serve % (66.0%) moves the match to 69.11%.

A gap of 3.0 points on serve becomes a 59.9% set and a 64.7% match for A. Each rung of the ladder widens the gap.

Check a Betfair price: Player A to win

Model chance 64.66%, fair price 1.547.

Betfair back price

Commission

Type a price to see if it’s value. Break-even after commission is 1.558.

Simplification: every set, including the last, has a standard 7-point tiebreak at 6-6. Each point depends only on who's serving, not on the score, momentum or nerves.

Worked Betfair example

Player A wins 65% of points on their own serve. Player B wins 62% on theirs. Best of three sets, a tiebreak at 6-6 in every set. (Illustrative figures.)

Hold chances. A holds 82.96% of service games. B holds 77.59%. So A breaks B 22.41% of the time and B breaks A 17.04%.
The tiebreak. Stepping through every tiebreak score with the proper serving order, A wins 54.95% of tiebreaks.
The set. Stepping through every game score from 0-0, A wins 59.90% of sets. (Who serves first in the set makes no difference to this.)
The match. A needs two sets before B does: 0.599² × (3 − 2 × 0.599) = 64.66%. Fair prices: A 1.547, B 2.830.
Compare with Betfair. A is 1.60 to back. EV on £10 after 2% commission: 0.6466 × £6 × 0.98 − 0.3534 × £10 = +£0.27. Break-even is about 1.558.
In play. If A wins the first set, their match chance jumps to 0.599 + 0.401 × 0.599 = 83.92%. If A loses it, it falls to 0.599² = 35.88%.
Sensitivity. Change A's serve figure from 65% to 66% and the match chance goes from 64.66% to 69.11%. One point of serve percentage is worth about four and a half points of match probability.

Verdict: the model is only as good as the two serve percentages. Because the ladder amplifies small differences, a small error in those inputs becomes a big error in the price.

The formula

Winning a service game

G(p) = p^4\big(1 + 4q + 10q^2\big) + 20\,p^3q^3\,\frac{p^2}{1 - 2pq}

p is the server's chance of winning a point on serve; q = 1 − p.
The first part covers winning to 0, 15 and 30.
20p³q³ is the chance of reaching deuce, and p² ÷ (1 − 2pq) the chance of winning from deuce.

In plain English: add up the ways to win before deuce, then the chance of getting to deuce times the chance of winning from there.

Winning a set

S(a, b) = G_{\text{serve}} \cdot S(a+1, b) + \big(1 - G_{\text{serve}}\big) \cdot S(a, b+1)

S(a, b) is the chance A wins the set from a games to b.
G_serve is the hold chance of whoever serves the next game.
The chain ends at 6 games with a two-game lead, 7-5, or goes to a tiebreak at 6-6.

In plain English: from any score, the chance of winning is "win this game and go on from there" plus "lose it and go on from there". The calculator runs this backwards from the end of the set.

Winning a best-of-three match

M = s^2\,(3 - 2s)

s is the chance of winning a set.

In plain English: win the first two (s²), or split them and win the decider (2 × s × (1 − s) × s). Together that's s²(3 − 2s).

Try it

Set A to 65% and B to 62% and check the match reads about 64.7%. Then set both to the same value: the match is exactly 50%, however high or low the serve figures go.

Common mistakes

Treating points as equal all match. Real players tire, raise their level on big points, or change tactics. The model assumes every point on serve is the same.
Using raw season serve stats. Serve percentages depend on the opponent and the surface. Adjust for both before trusting a price.
Ignoring how much the ladder amplifies. A tiny error at point level becomes a big one at match level. Check your inputs before your outputs.
Forgetting the rules differ. Best of five, and different final-set tiebreaks, change the numbers. Set the format before you compare with Betfair.
Thinking this is only for tennis. The same step-by-step idea prices in-play football from the score and minute, which is where Statometrics uses it (Lesson 3.2).

Neat models with fragile inputs are a big part of why good models still lose money.

Markov chains for tennis