Via twitter : Roger Federer and Novak Djokovic often seem to face each other in the semifinals. Too often. This article by Rob Minto tries to explain that the chance of them meeting in any one competition should be 50/50. However, they’ve met 16 times out of 21 instead. Rob asks : is this too many?

I came across this article thanks to a retweet by Tim Harford, author of a number of excellent books. The article points out that if you toss 21 coins, there’s only about a 0.97% chance of getting 16 heads. This is so low that one might question whether these two tennis players are really placed randomly.

Unfortunately, the author of the article has it all wrong. He’s worked out the wrong probability, and he’s attached too much significance to it.

First – the correct probability to work out is not the chance of getting 16 heads out of 21 tosses. He should work out the chance of getting 16 *or more* heads. After all, he’s wondering if Federer and Djokavic meet *too often*, not wondering if they meet *exactly so often*.

After all, suppose you tossed 7000 coins and got 3501 heads. The chance of getting exactly 3501 heads is even less than 0.97%. However, you wouldn’t therefore conclude that something’s wrong with your coin. The correct probability to work out is the chance of getting 3501 *or more* heads out of 7000 tosses, and that’s about 50%. No surprises when it happens.

So, back to the tennis. The chance of getting 16 *or more* heads out of 21 is 1.33%. This is not as small as 0.97%, but still, it’s small. However, the author of the article has made another mistake – selection bias.

After all, Federer and Djokavic are not the only tennis players he could have done this calculation on. Literally hundreds of players take part in Wimbledon, many of these have taken part in multiple successive years. If you include only the top 10 in the world, that’s already 45 pairs of tennis players to calculate probabilities with and look for coincidences. And then, there are other years, and other sports.

The point is that if you look for coincidences enough, you will certainly find some. A 1.33% chance is small, but not zero. An event that unlikely happens once every 75 times. Look hard enough and it’s easy to find “amazing” 1 in 75 or 1 in 100 coincidences. I wrote about this two years ago when Paul the Octopus was famous.

So in short, this “amazing” coincidence is actually quite normal. Federer and Djokavic can play, confident that when they face each other yet again, they were brought together by fate or chance, not by a finagling Wimbledon official.