Serve math
Real 2025 ATP hold rates. See how a small edge on serve points wins sets.
Serve like a real player · ATP 2025
Real hold rates from the ATP Tour, 2025 season. Each button sets the serve-point rate that reproduces that hold rate in this model, so the sliders can land between whole numbers.
- You hold serve
- 73.6%
- You win the set
- 50.0%
Even on serve, so the set is a coin flip: exactly 50%. Nudge your slider up two points and watch it swing.
What’s going on
Treat every point as an independent trial the server wins with probability p (and loses with q = 1 − p). To hold, you need 4 points before your opponent gets 3, or you win from deuce.
P(hold) = p⁴(1 + 4q + 10q²) + 20p³q³ · p² / (1 − 2pq)
The 1, 4 and 10 count the orders in which you can win 4–0, 4–1 or 4–2 (your fourth point always comes last), and the 20 counts the orders that reach deuce at 3–3. That is the binomial coefficient at work. From deuce you need two points in a row before they do, which works out to p² / (1 − 2pq).
The set adds alternating serve and a tiebreak at 6–6, so the page computes it exactly with a recursion over every score. At p = 0.5 you hold exactly half your games; at 0.6 you hold 73.6%.
The player buttons run the model backwards. Jannik Sinner held 92.0% of his service games in 2025 (713 of 775, per the ATP). The model says that takes winning about 71.7% of serve points. Opponents held only 67.4% against his return, which the model reads as about 57.2% of their serve points. Put the two together and the model gives Sinner about an 89% chance of winning any given set.
Real points are not perfectly independent (pressure, momentum, fatigue), so treat this as a model, not a forecast. The lesson holds anyway: games and sets magnify small edges.