If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Simulating a Football Season

This is a nice example of how statistics are used in modeling – similar techniques are used when gambling companies are creating odds or when computer game designers are making football manager games.  We start with some statistics.  The soccer stats site has the data we need from the 2018-19 season, and we will use this to predict the outcome of the 2019-20 season (assuming teams stay at a similar level, and that no-one was relegated in 2018-19).

Attack and defense strength

For each team we need to calculate:

1. Home attack strength
2. Away attack strength
3. Home defense strength
4. Away defense strength.

For example for Liverpool (LFC)

LFC Home attack strength = (LFC home goals in 2018-19 season)/(average home goals in 2018-19 season)

LFC Away attack strength = (LFC away goals in 2018-19 season)/(average away goals in 2018-19 season)

LFC Home defense strength = (LFC home goals conceded in 2018-19 season)/(average home goals conceded in 2018-19 season)

LFC Away defense strength = (LFC away goals conceded in 2018-19 season)/(average away goals conceded in 2018-19 season)

Calculating lamda

We can then use a Poisson model to work out some probabilities.  First though we need to find our lamda value.  To make life easier we can also use the fact that the lamda value for a Poisson gives the mean value – and use this to give an approximate answer.

So, for example if Liverpool are playing at home to Arsenal we work out Liverpool’s lamda value as:

LFC home lamda = league average home goals per game x LFC home attack strength x Arsenal away defense strength.

We would work out Arsenal’s away lamda as:

Arsenal away lamda = league average away goals per game x Arsenal away attack strength x Liverpool home defense strength.

Putting in some values gives a home lamda for Liverpool as 3.38 and an away lamda for Arsenal as 0.69.  So we would expect Liverpool to win 3-1 (rounding to the nearest integer).

Using Excel

I then used an Excel spreadsheet to work out the home goals in each fixture in the league season (green column represents the home teams)

and then used the same method to work out the away goals in each fixture in the league (yellow column represents the away team)

I could then round these numbers to the nearest integer and fill in the scores for each match in the table:

Then I was able to work out the point totals to produce a predicted table:

Here we had both Liverpool and Manchester City on 104 points, but with Manchester City having a better goal difference, so winning the league again.

Using a Poisson model.

The poisson model allows us to calculate probabilities.  The mode is:

P(k goals) = (eλk)/k!

λ is the symbol lamda which we calculated before.

So, for example with Liverpool at home to Arsenal we calculate

Liverpool’s home lamda = league average home goals per game x LFC home attack strength x Arsenal away defense strength.

Liverpool’s home lamda = 1.57 x 1.84 x 1.17 = 3.38

Therefore

P(Liverpool score 0 goals) = (e-3.383.380)/0! = 0.034

P(Liverpool score 1 goal) = (e-3.383.381)/1! = 0.12

P(Liverpool score 2 goals) = (e-3.383.382)/2! = 0.19

P(Liverpool score 3 goals) = (e-3.383.383)/3! = 0.22

P(Liverpool score 4 goals) = (e-3.383.381)/1! = 0.19

P(Liverpool score 5 goals) = (e-3.383.385)/5! = 0.13 etc.

Arsenal’s away lamda = 1.25 x 1.30 x 0.42 = 0.68

P(Arsenal score 0 goals) = (e-0.680.680)/0! = 0.51

P(Arsenal score 1 goal) = (e-0.680.681)/1! = 0.34

P(Arsenal score 2 goals) = (e-0.680.682)/2! = 0.12

P(Arsenal score 3 goals) = (e-0.680.683)/3! = 0.03 etc.

Probability that Arsenal win

Arsenal can win if:

Liverpool score 0 goals and Arsenal score 1 or more

Liverpool score 1 goal and Arsenal score 2 or more

Liverpool score 2 goals and Arsenal score 3 or more etc.

i.e the approximate probability of Arsenal winning is:

0.034 x 0.49 + 0.12 x 0.15 + 0.19 x 0.03 = 0.04.

Using the same method we could work out the probability of a draw and a Liverpool win.  This is the sort of method that bookmakers will use to calculate the probabilities that ensure they make a profit when offering odds.

Essential Resources for IB Teachers

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.