The Betfair Prof / Leighton Vaughan Williams / 26 August 2009 / Leave a comment

What does football forecasting have to do with the number of Prussian soldiers
kicked to death by horses? Thanks to Simeon Denis Poisson, the great scientist, mathematician and statistician, it has everything to with it.

To explain why, we need to travel back to the early nineteenth century and the strange case of the Prussian soldiers and the dangerous horses they rode into battle. Official reports suggested that these horses had been a noticeable cause of death for the cavalry for a period of at least 20 years. When the Czar heard this he was so concerned that he commissioned a study to determine whether this was due to chance or to a punishment inflicted by God.

An answer was provided by Simeon Poisson, who showed that the death rate from kicks followed a particular statistical pattern, which came to be known as the Poisson pattern or more formally the 'Poisson distribution'. So what does this have to do with football forecasting? Well, there is one thing that goals in a football match and lethal kicks by horses have in common and it is this - they both occur relatively infrequently and they both occur at some overall average rate.

In order to estimate the chance of any game ending with a particular scoreline, we first need an estimate of the number of goals a team is likely to score in that match. We can then read off from the Poisson distribution, which can be provided in the form of a table, the percentage chance of that team scoring a given number of goals. The value of the table lies in the fact that the predictions of the Poisson distribution tend to conform to actual football results as surely as they did when the horses were despatching the unlucky Prussians to an early death.

Let's take, as an example, the number of goals we might expect England and Brazil to score in a mythical World Cup final. For the sake of argument, and temporarily suspending disbelief, let's allocate England an expected 1.5 goals and Brazil one goal. By accessing the Poisson distribution we learn that England would have a 22.3% chance of scoring no goals in the final, a 55.8% chance of scoring one goal or less and so on. Similarly, Brazil would have a 36.8% chance of scoring no goals, a 73.6% chance of scoring one goal or less, and so on.

From these statistics, we can provide an accurate estimate of the final itself ending with any particular scoreline. For example, the probability of a goalless final can be derived by multiplying the probability of England failing to score by the probability of Brazil failing to score, i.e. 0.223 multiplied by 0.368, or 8.2%.

That means a 91.8% chance, based on these assumptions, of at least one goal being scored in this World Cup Final between England and Brazil.

If only our assumptions were as accurate as the table!