A mathematician watches tennis II

This has been a year to remember for anybody whose interest in tennis is more that of a nerd than that of a tennis player (which, given the uselessness of my serve, very much applies to me), in that it has given us two records that may well never be beaten. First we have Roger Federer’s record of 23 consecutive Grand Slam semi-finals (set at the Australian Open, and finally fixed at 23 when he lost in the quarter-finals at Roland Garros), and now, something I’ve been hoping for all my life: a seemingly endless match. At the time of writing, John Isner and Nicolas Mahut are waiting to resume a match that has gone into a third day. They will do so later today, with the score standing at 59-59 in the final set. This doesn’t just beat previous records — it utterly smashes them. This set is way more than twice as long as the previous longest set in a Grand Slam, it alone is far longer than the previous longest ever full match in professional tennis, both players have served far more aces in a single match (95 for Mahut, 98 for Isner) than anybody before, and so on. And if you also take account of the fact that the previous two sets had to be settled by tie-breaks, with no breaks of serve in either, then we have had 142 games in a row with no breaks of serve. (I can’t remember when the break occurred in the second set, but even this number 142 can probably be improved slightly.) [Update. The match is now over, with Isner winning 70-68, so the eventual number of consecutive unbroken service games was 137 in the final set, 161 if you include the previous two sets, and a few more still, I think, if you include the last few games of the second set. The number of aces for both players ended up well into triple figures.]

Isner said, with some justification, that nothing like this will ever happen again. But with how much justification? As ever, to answer this question involves choosing some kind of probabilistic model, and it is far from obvious how to choose an appropriate one. But it is possible to get some feel for the probabilities by looking at a crude model, while being fully aware that it is not realistic.

So let’s begin by assuming that two players are playing a single game of tennis, and that the returner wins each point with probability with the outcomes of the points being independent. (This independence assumption is not usually a good one, but watching the Isner-Mahut match I got the feeling that it was better than usual. Both players somehow retained their composure throughout and just played their normal games, which happened to involve remarkably good serving.)

A quick way to work out the probability that the returner wins (that is, the probability of a break of serve) is to consider what happens after the first six points. (If the game is over earlier than the end of the sixth point, we can let the players play two more meaningless points and the probabilities are unaffected.) If either player reaches four points, then that player wins, and otherwise the score is deuce. So if is the probability that the returner wins from deuce, then the probability that the returner wins the game works out, by a simple application of the binomial distribution, to be

As for it satisfies the equation

since if the returner wins the next two points then he wins (I say “he” because I’m talking about a men’s match here), if each player wins one point then it’s deuce again, and otherwise the returner loses. Thus,

I did a quick back-of-envelope calculation to see what would happen if — that is, if the probability of winning a point on the other person’s serve is 1/4. If I didn’t make a mistake, the probability of winning a game worked out to be something like 1/16. Let’s take that as the probability. Then the probability of a run of 142 games without a break of serve is which should be around But is about 9, and is around 0.0001234 (what were the chances that it would turn out to be such a nice number?), which is about 1 in 8000.

Given that each Grand Slam tournament involves 127 matches, so that in each year there are 508 Grand Slam matches, this seems to suggest that even in Grand Slams we should expect an event like this about every 16 years. But it suggests nothing of the kind. In almost all matches, this probabilistic model is ludicrously wrong: if, for example, one player is significantly better than the other, as is very often the case, or the match ebbs and flows, as is even more often the case, then it is certainly not the case that, game in and game out, the probability of the server winning the point is 3/4. This model is non-ridiculous only under very special circumstances: both players need to be excellent and very consistent servers, and they need to be evenly matched (on the day at least). And this state of affairs needs to last. And even then, the chances that there will be an extraordinarily long run of matches are only 1/8000 (though that probability is quite sensitive to my choice of , and perhaps in the Isner-Mahut match it was more like 1/5 or 1/6 — there certainly seemed to be a lot of love games — in which case the likelihood would go up somewhat).

Something like this seems to have happened with Isner and Mahut. One of the remarkable things about the match was the quality of the actual tennis. Particularly remarkable was that neither player went through phases of losing rhythm and being unable to get their first serves in. (There were perhaps mini-phases, but there was no sign that they could not be explained by pure chance.) And since both players were, obviously, pretty tired, the server’s advantage was presumably increased as the match wore on.

This last factor perhaps explains why the previous records were not just beaten but smashed, which would otherwise be rather mysterious. If both players got into a groove on their serves, and were too tired to cope with each other’s serves, then the value of would have gone down. But the match had to go on for quite a long time for this to happen. So perhaps the conditional probability of a long run given that there has been a long run up to now is higher than the probability of a long run starting from scratch. (In probability jargon, the process is not memoryless.)

After all that, is Isner right to say that this will never happen again? To answer that we would have to guess how often two very good and evenly matched servers are pitted against each other. But let’s suppose that that happens in say 50 of the 504 Grand Slam matches (it would be 100 if we added in women’s matches, but the server’s advantage in women’s tennis is much less, so I think one can confidently say that this will never happen in a women’s match). And let’s suppose that 20 of those matches that go to five sets (which is probably quite a generous estimate). And finally, let’s guess that the probability of a run of 118 games in the final set without a break of serve is somewhat higher than 1/8000 — we could go for 1/4000, say. Then we would expect a match like the Isner-Mahut match about once every 200 years.

Will tennis still be played in its current form in 200 years’ time? I don’t think that can be anything like guaranteed. So Isner’s bold statement could well be right.