# The Second Serve Problem

Modern tennis players have thunderous serves. Their first serve will often average 200 km/h or more, while their second serve will cross the net at 150 km/h. Their second serve — the safe serve that they are merely spinning into the court — is faster than the first serve of all but the very best club players. And their first … well that’s a weapon for a top player, a tool for engineering points, if you’re able to get it in.

Let’s look at the probabilities for a match involving two top-ranked players. If they get their first service in, itself a 65%-ish probability for a top player, they will win the point with a roughly 75% likelihood. If they get their second service in, which happens with a 90% probability for top players, they will win the point only 50%-ish of the time (when playing other top players). The slower second serve, even if spinning madly, is generally a shorter ball, one that a good player can often hit winners off, or at least get back into a longer rally.

So, the question: Why don’t players hit two first serves instead? If you do the maths, adjusting for probability of winning the points conditional on having gotten a first or second serve in, players have a 2-3% expected value edge with their first serve over their second serve. In other words, they should be doing two first serves — hitting the ball as hard as they can twice — rather than doing a hard first serve, and a spun second serve.

A recent NY Times article wondered about this. It was presented as a kind of behavioral flaw, with coaches and others saying that it was something they had been trying to convince tennis players of for a long time — that they should hit two first serves, not a first serve and then a spun second.

You can see how, psychologically, players might “wrongly” shy away from trying a second first serve, having missed the first. At the same time, the edge gained from this “two first serves” style is small enough that it would take a bit of Moneyball-esque thinking to get you to reconsider the aversion to possible double-faults.

There are even external reasons to image why the change would make sense. A top tennis player’s second serve is not just a slower version of their first serve; it is an entirely different service motion. Where the first serve is a flat ball hit with power, the second service is slower and heavily spun, trading off aces against getting the ball in the court and having it jump forward or sideways with all the spin. Coaches have made the same argument, saying that players with good first serves might be better off dumping the second serve, which is based on those different service mechanics, and focusing on hitting two first serves.

But it doesn’t happen. There is no-one in professional tennis who hits two first serves. Now and then you’ll see a top player hit a hard second service in an attempt to catch another player out (Pete Sampras was known for doing this), but there is no professional male or female tennis player that does this on every point.

While it might seem that there is an opportunity here waiting to be exploited, I’m not convinced. The trouble is that there is an unstated assumption, one of statistical independence between a player’s first and second serves. We are saying that a player, having missed their first serve, will still get their harder second serve in with that usual 65% likelihood. Or, to turn it around, we’re saying that knowing they have no safe second serve to rely on, players will still get their first serve in with a 65% probability. Both are almost certainly not true. Even top tennis players get nervous on important points when playing other top pros, and while they are more capable than most of rising to the occasion, that is not the same thing as saying that their service probabilities would stay the same if they had no bail-out serve.