Applying Poisson Distribution to Football Betting markets Via West Brom v Arsenal April 2013

Posted on: March 28th, 2015 by Jonny Grossmark

Above is a link to the goal expectation via Sotdoc for the game between West Brom and Arsenal which was played in April 2013.

West Brom 0.26*4.37 1.14 goals

Arsenal 0.27*4.02 1.08 goals

Total goals (1.14+1.08) 2.22 goals

Expectation of a full time score of 1-1 (7.8)

 The model that I used to derive the values above is based on two factors which are simply expected shot on target production and expected accuracy.

My model had expectation that West Brom would have 4.37 shots on target and convert 26% = 1.14 goals and that Arsenal would have 4.02 shots on target and convert 27% = 1.08 goals.

You can put the data into a poisson distribution calculator  which will show the percentage probability of each team scoring 0 1 2 3 goals… by simply putting (x) Poisson random variable which is the number of goals so ( 0) ( 1) ( 2) ( 3) etc and below is the average rate of success and you simply add the expected goal total.

If I want to calculate the expectation of West Brom  scoring 0 goals then I simply add 0 to the Poisson variable x box and below I add the value 1.14 (the average rate of success) which is the expected goal total.

The output from the poisson calculator is 0.32 which means that there is a 32% probability that West Brom would not score given the data that I have supplied.

If I repeat the process for Arsenal  then I simply add 0 to the Poisson random variable box ( x) and 1.08 to the average rate of success box as my model has expectation that Arsenal  will score 1.08 goals.

The output for the Poisson Calculator is 0.34 which means that there was 34%  probability that Arsenal  would not score against West Brom.

If you wanted expectation of 1-1 which appeared to be the most likely outcome then 0.364 x 0.367 = 0.1336> 1/0.1336 = 7.49 and before the game started 1-1 was 7.8 which means that there was no value backing 1-1 before the game started.


There is a huge flaw in terms of applying Poisson Distribution to football betting markets which the mainstream football “bloggers’ appear reluctant to discuss which is that expectation of a goal is dependent on the current score and there may be variables during the game that can impede as well as accelerate goal production.

Full Time > West Brom 1 goal and 5 shots on target > 0.2*5 which means that West Brom converted 20% of their shots on target and the SotDoc model predicted before the game that they would convert 26% but the expected shot on target production was almost bang on and I discovered that in general shot on target production is very easy to predict because it will generally be in a narrow range but accuracy i impossible to predict on a game by game basis.

Arsenal 2 goals and 6 shots on target > which means that Arsenal had a higher accuracy then expected and more shots on target then the model predicted.

West Brom 0 Arsenal 1 half time goal 20 minutes > early away goal metric = we can calculate to a much higher accuracy then any pre off model based on Poisson Distribution or any other basis the expectation of Arsenal winning as well as the expectation of West Brom Scoring.

Arsenal last 33 away games when opening the scoring 0-20 minutes in the league where they winning 1-0 at half time, Won 64%, and 70% ending with both teams scoring.

To paint a more comprehensive picture we can look at data going back to 2008-2009 and see that 0-1 half time added to the early away goal in the Premier League ( data is for teams currently in the PL) , 71% ended in an away win and 52% ended with both teams scoring.

In effect at half time in the Arsenal game you had lowered expectation of West Brom fighting back in the second half with around the toss of a coin expectation of West Brom scoring in the second half.

If you want to be even more detailed and look at Arsenal individual early away goal data and 0-1 half time since 2008-2009 > Won 82% Drawn 9% and lost 9% with 45% ending with both teams scoring.


If you backed 1-2 correct score before the game started applying your poisson distribution model then you simply guessed lucky and in the long run your pre off model will not be as lucky as the model is only as good as the input and not even the Sotdoc can predict the current score before the game starts as time decays  as well as any early goal events as this is crucial in terms of defining accuracy as time decays in a game.

The betting blogs/sites that are ramping possion distribution as a sophisticated tool to trigger bets are basically sending the people who believe the hype over a cliff.

Don’t believe the hype.