Italian translation at settesei.it
In David Ferrer’s final grand slam, the draw gods handed him a doozy of a first-round assignment in Rafael Nadal. Ferrer has struggled all year, and no one seriously expected him to improve on his 6-24 career record against the King of Clay. In the end, he didn’t: Ferrer was forced to retire midway through the second set with a calf injury. But before his final Flushing exit, he gave Rafa a bit of a scare.
Nadal won the first set, 6-3. The second set was a bit messier: Ferrer broke to love in the opening game, Rafa broke him back in the next, and a bit later, Ferrer broke again to take a 3-2 lead. He maintained that one break advantage until he physically couldn’t continue. Leading 4-3 and serving the next game, he was been two holds away from leveling the match.
Does that mean Nadal “almost” lost the set? People on the internet argue about these things, and while I don’t understand why, I do love a good probability question. If it overlaps with semantics (yay sematics!), that’s a bonus.
Let’s forget the word choice for now and reframe the question: Ignoring the injury, what were Ferrer’s chances of winning the set? If we assume that both players were equal, it’s a simple thing to plug into my win probability model and–ta da!–we find that from 4*-3, Ferrer had a roughly 85% chance of winning the set.
But wait: I can already hear the Rafa fans screaming at me, these two players aren’t exactly equal. In the 102 points the Spanish duo played on Monday night, Ferrer won 38% on return and Nadal won 47%. For an entire five-set match, those rates work out to a 93% chance of Rafa winning. Maybe that’s not quite high enough, but it’s in the ballpark. Using those figures, Ferrer’s chance of hanging on to win the second set drop significantly, to 57.5%. When you’re winning barely half of your service points, your odds of securing a pair of holds are worse than a coin flip. Had Ferrer won the set, it’s more likely that he would’ve needed to either break Rafa again or come through in a tiebreak.
That’s a pretty big difference between our two initial estimates. 85% sounds good enough to qualify for “almost” (though one study quantifies the meaning of “almost” at above 90%), but 57.5% does not.
That doesn’t quite settle it, though. The win probability model takes all notions of streakiness out of the equation. According to the formula, there’s no patches of good or bad play, no dips in motivation, so extra energy to finish off a set, etc. I’m not convinced any of those exist in any systematic manner, but it’s tough to settle the question either way. Therefore, if we have the ability to use data from real-life matches, we should.
And here, we can. Let’s start with Nadal. Going back to late 2011, I was able to identify 69 sets in which Rafa was returning down a break at 4-3. (There are probably more; my point-by-point dataset isn’t exhaustive, but the missing matches are mostly random, so the 69 should be representative of the last several years.) Of those 69, he came back to win 21, almost exactly 30%.
Ferrer has been more solid than Nadal’s opponents. (It helps that Ferrer only had to face Rafa once, while Nadal’s opponents had face him every time.) I found 122 sets in which Ferrer served at 4-3, leading by a break. He went on to win the set 109 of those times, or about 89%.
The 89% figure is definitely too high for our purposes: Not only was Ferrer a better player, on average, between 2012 and today, than he is now, but he also had the benefit of facing weaker opponents than Nadal in almost all of those 122 sets. 89%–not far from the theoretical 85% we started with–is a grossly optimistic upper limit.
Even if we take the average of Nadal’s and Ferrer’s real-life results–roughly 90% conversions for Ferru and 70% for Rafa’s opponents–80% is still overshooting the mark. As we’ve established, Ferrer’s numbers refer to a stronger version of the Spaniard, while Rafa is still near the level of his last half-decade. Even 80%, then, is overstating the chances that Nadal would’ve lost a set.
That leaves us with a range between 57%, which assumes Nadal would keep winning nearly half of Ferrer’s service points, and 80%, which is based on the experience of both players over the last several years. Ultimately, any final figure comes down to what we think about Ferrer’s level right now–not as good as it was even a couple of years ago, but at the same time, good enough to come within two games of taking a set from the top-ranked player in the world.
It would take a lot more work to come up with a more precise estimate, and even then, we’d still be stuck not only trying to establish Ferrer’s current ability level, but also his ability level in that set. Just as the word “almost” refers to a range of probabilities, I’m happy to call it a day with my own range. Taking all of these calculations together, we might settle on a narrower field of, say, 65-70%, or about two in three. There’s a good chance a healthy Ferrer would have taken that set from his long-time tormentor, but it was far from a sure thing … or even, given the usual meaning of the word, an “almost” sure thing.