Game and Set Win Probability Calculator
Game and Set Win Probability Calculator
If a player wins each point with probability p, the chance of winning the game is a finite polynomial in p and the chance from deuce is p² ÷ (p² + q²). Both are exact, both are published, and the assumption behind them is false in a way the page states.
Game, tie-break, set and match from one point probability
Closed form, no simulation64% of points won, 52% of games won, a seven-point tie-break and a best of three
The closed forms, and where each one is published
Game from deuce = p² ÷ (p² + q²)
Game from love-all = p⁴(1 + 4q + 10q²) + 20p³q³ × p² ÷ (p² + q²)
Tie-break to seven = p⁷(1 + 7q + 28q² + 84q³ + 210q⁴ + 462q⁵) + 924p⁶q⁶ × p² ÷ (p² + q²)
Set from no games, with per-game probability g and r = 1 − g, and T the tie-break figure above = g⁶(1 + 6r + 21r² + 56r³ + 126r⁴) + 252g⁷r⁵ + 504g⁶r⁶T
Match, best of three, from a set probability S = S²(1 + 2(1 − S))
Match, best of five = S³(1 + 3(1 − S) + 6(1 − S)²)
- where the game formula comes from
- Newton and Keller, “Probability of Winning at Tennis I. Theory and Data”, Studies in Applied Mathematics 2005;114:241–269, equation (5). It is quoted rather than derived here, which matters: a polynomial written out from scratch is a polynomial that can be written out upside down. Their equations (51) and (52) are the two match formulas and are quoted the same way
- how the game polynomial is built
- four terms. To love, p⁴. To 15, four orders in which the lost point can fall, 4p⁴q. To 30, ten orders, 10p⁴q². Reach deuce, twenty orders, 20p³q³, and then win from there. The coefficients 1, 4, 10 and 20 are the binomial counts C(3,0), C(4,1), C(5,2) and C(6,3)
- the deuce factor
- from deuce you must win two points in a row before your opponent does, and every pair split one each returns the score to deuce. So the chance is p² plus 2pq times itself, a geometric series summing to p² ÷ (1 − 2pq) — which is the same number as the p² ÷ (p² + q²) written above. The body below shows the algebra and says how the two were checked against each other
- the set formula, and what it is NOT
- it is the standard closed form for a set in which every game is won with the same probability g, independently: six games to love through six–four, then seven–five, then a tie-break at six games all. It is NOT Newton and Keller’s set formula, their equation (6), which tracks who is serving each game and is a different and better calculation. This page does not implement equation (6) and does not claim to
- why there are two probability inputs and not one
- because they describe different things and chaining them silently would be worse than asking twice: the point figure describes one player’s service points, the game figure has to cover both players’ service games. Enter a pair that do not belong together and the panel answers exactly for that assumption and tells you nothing about tennis. The arithmetic cannot check an assumption for plausibility and does not pretend to
- the assumption that makes all of it wrong
- independence, and a constant p. Points are not independent: a server at 40–0 plays the next point differently, a second serve follows a missed first, and nerve and fatigue move through a match. Treat every figure here as the answer to “what would the scoring system do to a constant edge?” and not as a forecast
Worked example
64% of points won, 52% of games won, a seven-point tie-break and a best of three
p = 0.64 and q = 0.36, so the deuce factor is 0.64² ÷ (0.64² + 0.36²) = 0.4096 ÷ 0.5392 = 0.7596 — 76.0% from deuce
The game from love-all is 0.64⁴ × (1 + 4 × 0.36 + 10 × 0.36²) + 20 × 0.64³ × 0.36³ × 0.7596 = 81.3%
Note that 81.3 is ABOVE 76.0. A player better than even is better off starting a game at love-all than at deuce, because the four chances to win it outright come first
From forty–love the same player wins 98.9%: losing from there needs three straight points to reach deuce and then the deuce game as well
The seven-point tie-break gives 87.0%, and switching to ten points gives 90.7% — a longer tie-break is less of a lottery
With a per-game probability of 52% and that tie-break, the set is 59.9% and the best of three is 64.7%; a best of five is 68.1%
Set both probabilities to exactly 50 and the game, the deuce game, both tie-breaks, the set and the match all return exactly 50.0% — the check that none of these polynomials is inverted
Set the point probability to 36 — the complement of 64 — and the game returns 18.7%, which is 100 minus 81.3 to the last digit the panel can show
What the scoring system does to a per-point edge
| Points won | Game from love-all | Game from deuce | From forty–love | Tie-break to seven | Tie-break to ten |
|---|---|---|---|---|---|
| 35% | 17.0% | 22.5% | 78.7% | 11.3% | 7.8% |
| 40% | 26.4% | 30.8% | 85.0% | 21.3% | 17.4% |
| 50% | 50.0% | 50.0% | 93.8% | 50.0% | 50.0% |
| 55% | 62.3% | 59.9% | 96.3% | 65.4% | 67.9% |
| 60% | 73.6% | 69.2% | 98.0% | 78.7% | 82.6% |
| 64% | 81.3% | 76.0% | 98.9% | 87.0% | 90.7% |
| 70% | 90.1% | 84.5% | 99.6% | 94.9% | 97.3% |
| 75% | 94.9% | 90.0% | 99.8% | 98.2% | 99.3% |
From games to sets to matches, with the tie-break held at the 64% figure
| Games won | Set | Best of three | Best of five |
|---|---|---|---|
| 40% | 28.2% | 19.4% | 14.0% |
| 45% | 41.0% | 36.6% | 33.4% |
| 50% | 54.5% | 56.8% | 58.5% |
| 52% | 59.9% | 64.7% | 68.1% |
| 55% | 67.6% | 75.3% | 80.4% |
| 60% | 78.9% | 88.5% | 93.3% |
| 65% | 87.7% | 95.8% | 98.5% |
Exact arithmetic on an assumption that is not true
Tennis has an unusual property among sports: the probability of winning a game, given a probability of winning a point, has an exact closed form. There is nothing to simulate and nothing to fit. A game is first to four points with a margin of two, so there are four ways to win it — to love, to 15, to 30, and from deuce — each with a binomial count in front of it. Newton and Keller’s equation (5), in Studies in Applied Mathematics in 2005, is that formula, and this page quotes it rather than deriving it, because a polynomial written out from memory can come out upside down.
The deuce part is the prettiest piece of arithmetic in the sport. From deuce a player must win two points in a row, and any pair split one each returns the score to deuce exactly as it was — so the chance is p², plus 2pq times the chance from deuce again, a geometric series summing to p² ÷ (1 − 2pq). Newton and Keller write the deuce factor as p² ÷ (1 − 2pq). This page writes it as p² ÷ (p² + q²), which is the same number: p + q = 1, so (p + q)² = p² + 2pq + q² = 1 and therefore p² + q² = 1 − 2pq. The two forms were checked against each other through the site’s own evaluator across the whole range of p, and they agree to the last digit. From deuce nothing matters except the ratio of the two players’ squared point probabilities.
Two things about the love-all figure catch people out, and the table above shows both. The first is how much the scoring amplifies a small edge: 55 points in a hundred becomes 62 games, and 64 points becomes 81. A tennis match is a long series of small contests, and that converts a slight advantage into a near-certainty. The second is that the love-all figure is HIGHER than the deuce figure for any player better than even and LOWER for any worse: the four quick ways to win a game come first, and deuce has thrown them away. That crossing at exactly 50 per cent is the test that catches an inverted polynomial, and it is one of the assertions this record had to pass before it could be written.
Now the part that has to be said plainly. All of this is exact and the assumption underneath it is false: it assumes every point is independent of every other and that p never changes. Neither is true — a server at 40–0 plays the next point differently from one at 30–40, a second serve is a different shot from a first, and nerve and fatigue move through a match. Newton and Keller called their paper “Theory and Data” because the comparison with data is the interesting half. So read every figure here as the answer to a clean question, what would tennis scoring do to a constant edge, and not as a forecast. A model of this kind will report 95 per cent for a player who then loses, and when it does it has not malfunctioned: it was asked about a constant edge in a sport where the edge is not constant. What this page is not: it is not a forecasting service. It computes a probability from an assumption you supply, and it neither reads nor produces a market price. It produces no commercial prediction, and no part of this site does. For the service figure that feeds the point probability see the first serve percentage and points won calculator, for the game figure that feeds the set, the dominance and service hold calculator, and for the states this page assigns probabilities to, the match score and sets interpreter settles which of them are already decided.
Frequently asked questions
What is the probability of winning a tennis game from deuce?
p² ÷ (p² + q²), where p is the chance of winning a point and q = 1 − p — equivalently p² ÷ (1 − 2pq), which is how Newton and Keller write it, the two being the same number because p² + q² = 1 − 2pq whenever p + q = 1. The game from love-all is p⁴(1 + 4q + 10q²) + 20p³q³ × p² ÷ (p² + q²), their equation (5). At p = 0.64 the deuce game is 76.0 per cent and the game from love-all 81.3.
Is this calculator simulating matches?
No, and it could not. Every figure is a finite closed form evaluated once: the expression language this site’s calculators run has no loops and no arrays, so a probability either has a formula or it is not offered. Here that is a happy constraint, because the exact answer is shorter than the simulation would be. The page also asks for a game probability separately from the point probability, because a set contains games this player serves and games the opponent serves and chaining them silently would assume the player serves every game.
Does this page handle who is serving in a set?
No, and that is the honest limit: the set formula here assumes every game is won with the same probability, independently. Newton and Keller’s equation (6) does track the alternation of serve, and this page does not implement it and does not claim to.
Can a probability this high be wrong?
Routinely. The arithmetic is exact and the assumption is not true. A figure of 95 per cent means “a player with a permanent, unvarying edge of this size would win 95 times in 100”, and no tennis player has one. The number is a property of the scoring system, not a prediction about two people.
Does a longer tie-break favour the better player?
Yes, and the table above measures it: at a point probability of 64 per cent the seven-point tie-break is won 87.0 per cent of the time and the ten-point one 90.7, and the gap widens the further p sits from a half. That is the arithmetic behind a ten-point deciding tie-break — though the Grand Slam Board’s own stated reason for adopting one in 2022 was consistency between the four tournaments rather than this.
Related calculators
References
- Newton PK, Keller JB. “Probability of Winning at Tennis I. Theory and Data.” Studies in Applied Mathematics 2005;114:241–269, read at sites.usc.edu (served from bpb-us-w1.wpmucdn.com). Equation (5) is the closed form for holding serve from love-all, which extracted as p⁴(1 + 4q + 10q²) + 20p³q³ × p²/(1 − 2pq) with p the probability of winning a rally on serve. Equation (2) is the deuce game, (6) the set with the alternation of serve tracked, (13) the thirteen-point tie-break, and (51) and (52) the best-of-three and best-of-five matches. About 60,000–70,000 characters came back and the equations arrived as text with superscripts and subscripts flattened, which is why the identity 1 − 2pq = p² + q² was checked by algebra and through the site’s own evaluator rather than read off a typeset line.
- International Tennis Federation. Rules of Tennis 2026, Rules 5, 6, 7 and 10. itftennis.com/media/7221/2026-rules-of-tennis-english.pdf. Rule 5(a), the standard game: Love, 15, 30, 40, “Deuce” at three points each, then “Advantage”. Rule 5(b), the tie-break game: first to seven with a margin of two; “The player whose turn it is to serve shall serve the first point”; “The following two points shall be served by the opponent(s)”; then two consecutive points each; and the player due to serve first becomes “the receiver in the first game of the following set”. Rule 6, the advantage set and the tie-break set. Rule 7, best of three or five. Rule 10: “During a tie-break game, players shall change ends after every six points.”
- International Tennis Federation. Rules of Tennis 2026, Appendix VI — Alternative Procedures and Scoring Methods. No-Ad: at three points each “a deciding point shall be played”, with the receiver choosing the half of the court. Short sets: first to four games with a margin of two, a tie-break “if the score reaches four games all” or, “at the discretion of the sanctioning body”, at three games all. Match tie-break, seven or ten points: played at one set all, or two sets all in a best of five, and it “replaces the deciding final set”. Its own Change of Ends entry gives an alternative to Rule 10 — “after the first point and thereafter after every four points” — which is an approved option, not a replacement.
- Cross R. “Grand Slam Statistics”, June 2009, physics.usyd.edu.au/~cross/StatisticsPrimer.htm. A physicist’s teaching page, and the only source read that states every DENOMINATOR: first serve % is the “fraction of first serves that went in”, winning % on 1st serve the “fraction of good first serves won by the server”, winning % on 2nd serve the “fraction of second serves won by the server”, and break point conversion the “fraction or percentage of break point opportunities won (converted) by the receiver”. It gives the decomposition “p = fw + (1 – f) s” with “f = N1/(N1 + N2), w = W1/N1 and s = W2/N2”, equal to “p(A) = (W1 + W2)/(N1 + N2)”. Its worked figures are the 2009 Australian Open men’s final: 175 first serves, 112 in (64%), 74 of 112 won (66%), 30 of 63 second serves won (48%), and “16 opportunities to break Federer’s serve” with “7 occasions” successful. Its game-level figures — “they each stepped up to serve 25 times”, 18/25 and 19/25, and Federer breaking “6 times” at a stated 32% — do not reconcile with each other or with the published 51-game score, and the batch says so on the pages that use it.
- Grand Slam Tennis. 2026 Official Grand Slam Rule Book. itftennis.com/media/5986/grand-slam-rulebook-2026-f.pdf. Article I Section K: the ITF Rules of Tennis apply except where the Grand Slam rules say otherwise. Article I Section L: men’s singles main draw “shall be the best of five (5) sets”, everything else best of three “unless otherwise determined by each Grand Slam Tournament”, and the tie-breaks of the Rules of Tennis may be used “including the final set where a 10 point tie-break will be played”. Section L does NOT state the game score that starts it; that is six games all, from ITF Rule 6. Section C describes GST as a collaboration of the four tournaments.
- Tennis365. “Wimbledon and other three Grand Slams confirm uniformity as they will use 10-point tie-breaks in final set”, tennis365.com, 16 March 2022. A Grand Slam Board decision: “A 10-point tie-break will decide the final set at the Australian Open, French Open, Wimbledon and US Open”, which “will commence at the 2022 edition of Roland-Garros” and “will first run on a trial basis”, “based on a strong desire to create greater consistency in the rules of the game at the Grand Slams”. It states the four rules it replaced, one per tournament.