Playing the numbers racket

Tennis? It's mostly a game of luck, says William Hartston in this statistical guide to Wimbledon watching

Just suppose, for a moment, that we wanted to find out which of two tennis players was better at the game. What we'd do, if we had any sense, would be to give each of them 100 serves - 50 from each side of the court - and see who won more points. Instead of that, we divide the points into games and the games into sets in a manner that could hardly be better calculated to confuse the issue. Of couse just playing to see who is first to win 51 points would be rather boring, but the rules of tennis are not designed solely to provide the thrill of a game decided every few minutes: their real purpose is to ensure that the better player sometimes loses. We'll come back to that later. First, let's work out the odds.

Suppose (as is more or less the case) that the server wins twice as many points as he loses. Then if the returner manages to struggle to deuce, his chance of breaking serve by winning the next two points is only one in nine, with an additional four in nine chance of reaching another deuce. It is not difficult to work out from this that the chance of a service break is only one in five. When the crowd become excited, just because their hero has reached deuce and has, in the breathless words of the commentator, "a real chance here to break serve", they ought to realise it's only 20 per cent and calm down a little. Even at love-30, the odds are slightly in favour of the server to win the game.

If you go back to the beginning of the game, it turns out that the server has an 86 per cent chance of winning the game, with only a 14 per cent chance of a break. This means that we should expect serve to be broken about once every seven games. Which means that each player may expect to have his serve broken less often than once a set. And that is why most sets end 6-3 or 6-4.

As you watch the Wimbledon finals this weekend, you should realise that the thrills come not from the brilliance of a delicately played drop volley, or an elegant lob landing right on the baseline. The true excitement lies in the interaction of random fluctuations of different variables. Even those "Oh I say" great shots are no more than statistical quirks. The lob that lands half an inch outside the baseline is no worse a shot than the line-clipping winner. It just happened to turn out well. If one player's fine play gives him a slightly greater chance of holding serve than his opponent - say seven chances in eight compared with five out of six, we are still essentially tossing two biased coins, one a little more crooked than the other, and waiting for the less likely sides to come up.

You can try an experiment, if you like, with a dice and three coins. One player - called, for example, Stich - rolls the dice and wins the game unless he rolls a one; the other replies by tossing the coins, and holds his serve unless three tails come up. Just try it and see how often the better player manages to win in straight sets.

In fact, looking at the results from three Grand Slam tournaments (France, the United States and Wimbledon) in 1996, there were 96 straight set victories out of 185 men's singles matches - so in about half the matches the worse player managed to win at least one set.

Earlier this year, the science journal Nature reported an intriguing piece of research that compared different professional sports according to the number of games played in one season of their major league. The conclusion was that sports evolve a competitive structure that guarantees enough surprises to keep the audience excited. If the result of every game is almost random (as in baseball), you need hundreds of games in a season to provide a good chance that the best team will emerge on top; if the better team wins almost all the time (as in rugby football), a much shorter season is enough to determine a fair overall winner. For the spectators, it is just as unsatisfactory for the better team to win all the time as for every result to be determined by pure chance.

In general, it is the high-scoring sports (such as basketball) that have the greatest reliability, while low-scoring ones (such as football), produce the most upsets. And that is one of the reasons why so many more people watch football than basketball.

The genius of tennis is that the rules have evolved to turn a high-scoring game, in terms of the number of points played in a match, into a low-scoring game in terms of the sets which decide the issue. Occasionally a player wins a match despite losing more points and games.

Finally, for future use, here is a guide to other major sports:

Football: Two sides try to kick a ball into each other's nets. To do so, they must create "scoring opportunities" each of which has a slim chance of being converted into a goal. One good side may score about once every 45 minutes; another poor side may score once every 80 minutes. In a 90-minute random sample, the better side will probably score one, two or three goals; the worse side will score 0, one or two. Sometimes the worse side will win 2-1.

Cricket: Batsmen sometimes make mistakes. When they do their innings is over. A good batsman may make a fatal error once every 100 deliveries. His score will fluctuate wildly between 0 and 100 or more. The team's score is the sum of 11 numbers picked at random from various distributions in this range. Even 11 good batsmen may occasionally produce a total adding to less than 100. The opposing bowlers then congratulate themselves.

Golf: People try to hit a ball into a small hole. after taking two or three shots to get it near the hole. Once it is near enough to the hole, it takes sometimes one and sometimes two shots to hit it in. A round of golf is thus roughly equivalent to tossing a coin 18 times and counting one for every head and two for every tail. Tiger Woods's coin is biased towards heads, but not enough for him to win all the time.