A somewhat belated follow-up to this post, where I looked at the probability of the draw for the Champions League knockout stage matching the preceding day's rehearsal draw. We have an article for the print magazine in the pipeline, so this is a little short on detail, but there is a surprisingly large amount of interesting stuff behind this problem.
I can (hopefully) be heard discussing a small aspect of this on the BBC's World Service program More or Less this coming Saturday, at 1350 UTC, although I make no guarantees that my segment won't end up on the cutting-room floor.
Showing posts with label sport. Show all posts
Showing posts with label sport. Show all posts
Thursday, 14 February 2013
Sunday, 30 December 2012
From the archives: goalfest in the Premier League
Yesterday evening saw one of the highest scoring English Premier League matches of all time. Arsenal's 7-3 drubbing of Newcastle joins just three other 10-goal games, although Portsmouth's 7-4 victory over Reading in 2007 remains the outright claimant for this particular accolade.
This was just one part, however, of a Saturday chock-full of goals, with a total of 35 scored across the eight matches played. This put me in mind of an article I wrote for Significance early in 2011 about an even more extraordinary day of football. Back then, Arsenal and Newcastle were at it again, with the latter's stunning four-goal comeback contributing to a whopping 41 scored across that day's eight Premier League games. In the article I used this as an excuse to show off the Poisson distribution, demonstrating how goals scored in football matches can be modeled surprisingly well by what is ultimately a (fairly simple) mathematical formula.
The remaining two matches of that particular weekend of football only produced two more goals, bringing the total for a complete 10 match 'round' of Premier League fixtures to 43. Based on the Poisson distribution (and assuming an average of 2.6 goals per game) I estimated there was a roughly 1 in 720 chance of seeing at least that many goals in a set of 10 games. This weekend's football was almost - but not quite - as remarkable, with six goals today bringing us to a total of 41 across 10 matches. Based on the same theory, this works out to a 1 in 250 occurrence.
This was just one part, however, of a Saturday chock-full of goals, with a total of 35 scored across the eight matches played. This put me in mind of an article I wrote for Significance early in 2011 about an even more extraordinary day of football. Back then, Arsenal and Newcastle were at it again, with the latter's stunning four-goal comeback contributing to a whopping 41 scored across that day's eight Premier League games. In the article I used this as an excuse to show off the Poisson distribution, demonstrating how goals scored in football matches can be modeled surprisingly well by what is ultimately a (fairly simple) mathematical formula.
The remaining two matches of that particular weekend of football only produced two more goals, bringing the total for a complete 10 match 'round' of Premier League fixtures to 43. Based on the Poisson distribution (and assuming an average of 2.6 goals per game) I estimated there was a roughly 1 in 720 chance of seeing at least that many goals in a set of 10 games. This weekend's football was almost - but not quite - as remarkable, with six goals today bringing us to a total of 41 across 10 matches. Based on the same theory, this works out to a 1 in 250 occurrence.
Thursday, 20 December 2012
What are the chances: Champions League draw exact repeat of rehearsal
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.
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.
Wednesday, 7 December 2011
Euro 2012: How deadly is the group of death?
Much (well, some) has been made of the Euro 2012 draw last Friday, where thanks in part to co-hosts Poland and Ukraine being the top two seeds a rather nasty looking 'group of death' formed. Group B sees the Netherlands, Germany, Portugal and Denmark pitted against one another, with certainly three of the four capable of winning the tournament. But how deadly are we talking?
It gets interesting if you compare the latest UEFA rankings with the latest FIFA ones which are very similar (about 95% correlation) but not identical. Highlights include Portugal, who are ranked 11th by UEFA, but are the 5th best European team according to FIFA The table below summarises the Euro 2012 groups with each team's UEFA and FIFA ranking (where I've filtered out all the non-UEFA teams). (Click for a full-resolution version.)
Based on FIFA rankings, group B is indeed the deadliest, with an average country ranking of 4.5 (and, interestingly, every team a higher rank than any in group A). According to UEFA, however, it's not group B but group C - home to the Republic of Ireland - which is the hardest, although there is very little to choose between the two.
It gets interesting if you compare the latest UEFA rankings with the latest FIFA ones which are very similar (about 95% correlation) but not identical. Highlights include Portugal, who are ranked 11th by UEFA, but are the 5th best European team according to FIFA The table below summarises the Euro 2012 groups with each team's UEFA and FIFA ranking (where I've filtered out all the non-UEFA teams). (Click for a full-resolution version.)
Based on FIFA rankings, group B is indeed the deadliest, with an average country ranking of 4.5 (and, interestingly, every team a higher rank than any in group A). According to UEFA, however, it's not group B but group C - home to the Republic of Ireland - which is the hardest, although there is very little to choose between the two.
Monday, 5 December 2011
Sunday, 25 September 2011
How do you solve a problem like Sebastian?
A while ago over on Significance I looked at what happened if you compared different Formula One scoring systems with that year's drivers' championship. Yesterday, Sebastian Vettel came within 1 point of snatching the title with a whopping five races to go. In 14 races he's won nine times, come second twice and fourth once, amassing a ridiculous 309 points out of a possible 350.
As it stands, Vettel is 124 points clear of his nearest rival Jenson Button, who would have to win every single remaining race and hope that Vettel scores nothing if he is to win the drivers' championship. In short, the season is as good as over, but is the scoring system to blame? A couple of years ago the FIA tweaked the scoring system to try and encourage second-placed drivers to 'race to win'. Previously you got eight points for second and ten for first, which (it was perceived) didn't offer enough incentive to try and push on for first place. Now you get 25 points for winning and 18 for second, a greater incentive that - in theory - will encourage more aggressive racing.
So what happens if we run this season's results (so far) under the older system? If you're happy to assume that the scoring system doesn't significantly affect how a driver races (quite a big assumption, I admit, but this is Just For Fun) you can enjoy this table:
As my previous dabbles with comparing scoring systems suggest, it doesn't make much difference. Admittedly, Vettel would have already won the championship by now, but only just (a 51 point lead with 50 points available), and no-one in their right mind thinks he won't win this year anyway.
If you ask me, we need something more radical than a tweaked scoring system to make things exciting. My proposal: do away with qualifying and have the cars line up in reverse finishing order from the previous race. It's simple, it would certainly increase overtaking, and no-one likes qualifying anyway.
Addendum: have a bonus table, including the results under the old(er) system of ten points for first, six for second. Perhaps unsurprisingly, Vettel is even further ahead on this one.
As it stands, Vettel is 124 points clear of his nearest rival Jenson Button, who would have to win every single remaining race and hope that Vettel scores nothing if he is to win the drivers' championship. In short, the season is as good as over, but is the scoring system to blame? A couple of years ago the FIA tweaked the scoring system to try and encourage second-placed drivers to 'race to win'. Previously you got eight points for second and ten for first, which (it was perceived) didn't offer enough incentive to try and push on for first place. Now you get 25 points for winning and 18 for second, a greater incentive that - in theory - will encourage more aggressive racing.
So what happens if we run this season's results (so far) under the older system? If you're happy to assume that the scoring system doesn't significantly affect how a driver races (quite a big assumption, I admit, but this is Just For Fun) you can enjoy this table:
As my previous dabbles with comparing scoring systems suggest, it doesn't make much difference. Admittedly, Vettel would have already won the championship by now, but only just (a 51 point lead with 50 points available), and no-one in their right mind thinks he won't win this year anyway.If you ask me, we need something more radical than a tweaked scoring system to make things exciting. My proposal: do away with qualifying and have the cars line up in reverse finishing order from the previous race. It's simple, it would certainly increase overtaking, and no-one likes qualifying anyway.
Addendum: have a bonus table, including the results under the old(er) system of ten points for first, six for second. Perhaps unsurprisingly, Vettel is even further ahead on this one.
Saturday, 30 October 2010
The Sound of Silence
It's all been a bit quiet here lately. Too quiet, if you ask me. Alas, my output has been somewhat reduced by an exciting new website from the Royal Statistical Society, namely the online home of their quarterly magazine, Significance. (Do you see what they did there?)
So yes, what would have been content for Statscream has turned into content for Significance, and so whilst I have various things in the works that I'm pretty sure will only be suitable for this corner of the Internet, there will be numerous posts where I merely link to my pieces elsewhere.
My opening gambits, therefore, are:
Anyone Fore Golf - a look at whether there is evidence of home advantage in the Ryder Cup; and
Putting the Man in Man Booker - investigating the gender gap in literary prizes.
I have even become dangerously web 2.0 and set up a Twitter account - statacake - thus increasing my statistical Internetular outlets to 3. The hypothesis "do more means of communication result in less content" will be tested in due course...
So yes, what would have been content for Statscream has turned into content for Significance, and so whilst I have various things in the works that I'm pretty sure will only be suitable for this corner of the Internet, there will be numerous posts where I merely link to my pieces elsewhere.
My opening gambits, therefore, are:
Anyone Fore Golf - a look at whether there is evidence of home advantage in the Ryder Cup; and
Putting the Man in Man Booker - investigating the gender gap in literary prizes.
I have even become dangerously web 2.0 and set up a Twitter account - statacake - thus increasing my statistical Internetular outlets to 3. The hypothesis "do more means of communication result in less content" will be tested in due course...
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