The draw for the
UEFA Champions League knockout stage took place earlier today, but it wasn't just the prospect of
some truly mouthwatering ties making the headlines.
Observant eyes spotted a seemingly spectacular coincidence:
today's draw was a near-identical repeat of that produced during Wednesday's rehearsal. While the order the ties came out was different, every single match featured the same pair of teams. So what were the chances of that?
It seems almost unbelievably unlikely, and an '
ESPN statistician' apparently gets an answer of roughly 1 in 2 million. On face value this makes sense: there are about 2 million possible ways of drawing 8 pairs of matches between 16 teams (the first team has 15 opponents to 'choose' from, then once that match is decided the next team has 13 opponents to choose from, and so on, giving 15 x 13 x 11 x 9 x 7 x 5 x 3 = 2 million (ish) possibilities). Unfortunately, this overlooks a large number of factors that drastically reduce the number of possible matches.
First of all, those teams who qualified as group winners in the previous stage of the competition can only be drawn against teams who qualified as group runners-up. With 8 teams in each 'half' of the draw (so to speak), this immediately drops us down to 8
factorial, or 40,320 possible matches. On top of this, however, no team can be drawn against the other team who qualified from their group, or even a team from the same football association. With 2 Spanish, 1 English, and 1 Italian team on each half of the draw, the number of 'valid' draws becomes even smaller.
The upshot of all of this (and an admittedly lazy
brute-force approach) is that there were just 5,463 possible draws that could have been made that satisfied all of these rules, giving chances of two identical draws in a row of about 0.02%. That's still pretty staggering, but nowhere near the 1 in 2 million we started off with.
Update: It has been pointed out to me that while there are 5,463 possible draws, due to the mechanics of the draw process itself (the specifics of which I was not aware of at the time of writing) not all draws were equally likely. However, the different draws do still have very similar probabilities, and there is nothing obviously special about this particular combination of fixtures. More on this to come.
One of the competitors - series 65 champion Graeme Cole - has produced some statistics for all those taking part. (Along with a handy diagram of the tournament structure.) One of these stats will serve as a proxy measure of ability: 'max percentage'. This is the proportion of the total possible points available players actually scored during their televised appearances. Obviously there are lots of reasons why these data aren't ideal, but they're the best we've got, and for those players who do play online they correlate pretty well with their ratings. Fitting a simple linear regression model of ranking against max percentage (which worked surprisingly well) we can then estimate ratings for those players who don't play online. The figure (click for a bigger version) summarizes the relationship: it's fairly noisy but good enough for a bit of fun.
Now that we have an estimate of every player's skill, the next challenge is estimating the probability of, say, a player with a rating of 1600 beating a player with a rating of 1500. Once again, apterous data lend a hand, with tens of thousands of recorded games giving an insight into just this sort of question. This time, a logistic regression model does the trick, allowing us to estimate the relationship between the difference in ratings between two players and the probability that each player will win. For example, a player who is rated 100 points higher than their opponent would be expected to win about 63% of the time, shooting up to 93% for a 500 point advantage.
