This week in Lexington, top seed Serena Williams faces her sister, Venus Williams, in the second round. They are both among the all-time greats, and they have played each other nine times in grand slam finals, so it’s always jarring to see them turn up in the same section of a draw and play on a Thursday.
Lately, their encounters seem to always happen long before the business end of a tournament. Their three matches between the 2017 Australian Open final and this week in Lexington all happened in the round of 32, including a planned 2019 Rome meeting from which Serena withdrew. Venus is usually unseeded, no longer the world-beater she once was, so it is at least possible that the Williams sisters would be bracket neighbors in any given week.
But should it happen quite so often? It is an understatement to say that Serena and Venus were not universally embraced upon arrival in the tennis world. If you’re conspiracy minded, every tournament draw is an opportunity to commit dastardly deeds. Perhaps early in the Williams era, it was the work of racist or otherwise misguided tournament officials who wanted to avoid all-Williams finals. Or nowadays, event honchos recognize that Venus is unlikely to reach the final, so they tinker with the bracket to make a headline-grabbing Williams-versus-Williams clash more likely.
I’m sure that most draws are conducted on the up-and-up, but the process is sufficiently opaque that it’s easy to get suspicious. It’s also easy to make mistaken generalizations from insufficient data. Let’s see what the numbers can tell us.
150 tournaments!
Lexington is the 150th tour event with both Serena and Venus in the field.*
* I think. My WTA data isn’t perfect for the early years of their careers, and there was an uncomfortable amount of manual tabulation involved in this post. Their TennisAbstract player pages are missing the 1999 Grand Slam Cup, but I’ve included it in all the numbers here. For the purposes of doing analytics, it doesn’t matter much if the total is 148 or 151, but if you’re printing a banner or making a cake, you should double-check.
Thursday’s match in Lexington will be their 31st, plus one withdrawal apiece. In 13 of the 150 events, the Williams sisters were either the top two seeds or the 3rd and 4th seeds, meaning that draw shenanigans were out of the question–they could not face each other until the final. 4 of those 13 times, that’s exactly what they did.
What are the odds?*
* Of me being able to use this sub-heading in any given blog post?
I went through the remaining 137 tournaments and identified the round in which they either did meet or could have met. For the purposes of analyzing draws, there isn’t really a difference. For instance, Serena and Venus have landed in the same half 73 out of a possible 137 times, a bit more than the 68 or 69 times that we would expect.
Because of their seeds, they had the chance of ending up in the same quarter 116 times, and that’s how it worked out 28 times, just under the 29 times that an exact one-in-four rate would’ve given them. The smaller the draw section, the fewer tournaments that Serena’s and Venus’s seeds made it possible for them to meet.
I counted the number of tournaments with a possible meeting on or before a certain round, and then the number of events in which the draw delivered that meeting, regardless of whether both Williamses got that far. Here are the results, along with the probability of that many or more actual meetings:
Section Possible Actual Chance Half 137 73 25% Quarter 116 28 62% Eighth 85 17 3% 16th 64 5 37% 32nd 42 1 74%
There’s a one-in-four chance that Serena and Venus would’ve landed in the same half as many times as they have throughout their entire careers. That’s a bit of bad luck, but it’s hardly a smoking gun. The same is true for the same quarters, as well as very early meetings that would pit them against each other in the round of 32 or 64.
That leaves one eyebrow-raising number to discuss. On 85 occasions, at least one of the two women was seeded outside the top eight, making possible a meeting in the round of 16 or earlier. Given random draws, we’d expect 10 or 11 brackets in which they could face each other so early. Instead, we got 17.
A 3% chance of so many early encounters isn’t quite as bad as it sounds. I’ve tried to walk you through this process in the way I approached it. While I wondered if Serena and Venus have met more often than random draws would normally deliver, I didn’t have a particular round in mind. As you’ve seen, I generated a bunch of numbers, and one of the five looked suspicious. You might be able to construct a story that explains why the round of 16 is different from the others (such as my theory that tournament directors want mid-week headlines), but because we generated so many numbers, we were that much more likely to end up with an extreme percentage simply by chance.
The smoking (nerf) gun
Thus, we’re able to raise the possibilities that some draws weren’t random, but we can hardly prove it. One problem–one that we could’ve foreseen from the get-go–is that some draws are definitely not tampered with. Probably most draws. And even if they were, most tournaments wouldn’t have any reason to mess with Serena’s or Venus’s placement in the bracket. Or if they did, they might prefer an all-Williams final, and thus alter the bracket in the opposite direction of what we’re hunting for.
If you like conspiracy hunting, I’ve got a tiny sample for you. Since the beginning of 2018, Venus and Serena have played in the same tournament 15 times, and their seedings (or lack thereof) made it possible for them to be drawn in the same eighth 14 of those times. Of the 14, they were placed in position for a round-of-16 or earlier meeting 5 times. There’s only a 2% chance of that … if you set aside the fact that I’m checking all sorts of subsets of matches looking for (probably spurious) patterns. If nothing else, the 5-of-14 figure explains why it seems like Serena and Venus keep landing in the same draw sections lately. They do!
Broadly speaking, then, this is all much ado about nothing. (I don’t even know if these conspiracy theorists exist, so maybe I just invented a conspiracy and spent my evening debunking it. Hooray?) It’s possible that a few tournament directors are producing non-random draws … but it would take a very different kind of investigative work to prove it. Worst case scenario, we get a few more Serena-Venus matches. It may not be fair to the older sister, but it’s a pretty good deal for tennis fans.