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.

## Sunday, 30 December 2012

## Thursday, 20 December 2012

### Christmas games theory

Through a coincidence almost as implausible as the Champions League draw, today also sees a new article of mine go up on Significance. The idea was simple: find interesting stats/maths things to say about popular Christmas-time board games. Not making the cut was a discussion of the inevitable unfairness of dice (and an excuse to talk about Awesome Dice Blog's 20,000 roll experiment to compare two manufacturers), and how to use the Markov chain nature of a game of Snakes and Ladders to estimate how long it will take you to finish (40 rolls in the MB version, apparently).

### 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.

## Monday, 3 December 2012

### Book review: How to study for a mathematics degree

My first foray into book reviews (well, since primary school, at any rate).

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