In the great Coliseum of ancient Rome, Christians were thrown to the lions as sport for the amusement of the baying Roman mob.

While most of the Roman punters were drawn by the prospect of blood and gore, the great Roman statistician Optas Indexus instead observed, and recorded numbers relating to the ‘sport’ on offer in the great amphitheatre.

Optas recorded how many Christians were devoured in each session of their unequal struggle with the lions. Over time he arrived at an average number of kills per session.

But he found that within this average was a range of actual numbers of kills, which fluctuated depending upon (among other things) the hunger/speed/agility of that day’s combatants.

From his data Optas was able to draw up a chart, which told him how likely it was that a given number of Christians would be killed in any given session.

He got together with his friend, the Roman bookmaker Pinnaceles, and they started taking bets from the Coliseum crowd on the number of kills.

They made a killing. Like the lions.


Centuries later, number crunchers were applying similar statistical analysis to, among other things, charting the numbers of Prussian soldiers accidentally killed by kicks from their own horses.

The French mathematician Simeon Poisson was looking at rates of wrongful convictions in court proceedings, and formalized a chart which took his name – the Poisson Distribution.

Poisson’s chart can be used in many scenarios – in fact in any situation where an average rate is known, and the events occur independently.

So say you stood on a bridge over a busy road for an hour every weekday evening counting the number of white vans that passed. After a while you would have an average number of observed white vans.

Using the Poisson Distribution you could determine how likely it was that you would then see 0, 1, 2, 3….white vans the next time you stood on the bridge. I have no idea why you would ever actually want to do this. But you could, if you wanted to.


But far more importantly and interestingly, Poisson Distribution can be used for goals in a football match.

If you know the average number of goals that have been scored, then Poisson gives you a pretty damn good guess of how often 0, 1, 2, 3, 4 …… goals will have been scored.

For example, if we look at the number of goals scored at Half Time in English Premier League games over the last 20 years the average is 1.173.

If we ask the Poisson Distribution to show us a breakdown based on 1.173 and compare it to what has actually happened, we get the table below…….

HT Distribution

As you can see, Poisson is virtually perfect at telling us how often a given number of goals will have been scored at Half Time in the EPL, based on the average 1.173.

It’s very easy to get these Poisson values, as Excel contains a Poisson function. You simply type;

=Poisson(0, 1.172, FALSE)

….into a cell and it shows you the result – 30.9% – if you have the cell formatted as ‘percentage’.


Looking into the past is all very interesting (if you like that sort of thing) but the more intriguing application of Poisson is to use it to project about things that will happen in the future. In other words; things that we can bet on.

Any expectation for a single upcoming match can be treated the same as the average for a big sample of previous matches. An expectation is basically ‘how many goals do you think will be scored?’. It will look something like 2.87.

Understanding the term can be a bit tricky if your brain likes thinking in neat black and white, whole number terms (how can you ‘expect’ it be something that is not a round number?!).

But 1.172 is not a round number.

So the expectation is really ‘how many goals on average would get scored, if this match was played an infinite number of times?’.

An easy trick to get an expectation is to use the quotes of a spread betting company like SportingIndex. On the upcoming Champions League final for instance we can surmise that their expectation for Total Goals is 2.7, as that is the mid-point of their Total Goals quote…..

Sporting UCL
Sporting Index quotes for UEFA Champions League Final

So we can type into excel;

=Poisson(2, 2.7, true)

…..and the result – 49.4% – is a Poisson based probability of there being 0, 1 or 2 goals scored in a game with a 2.7 Total Goals expectation.

Divide the 49.4% into 1 and you get a decimal price of 2.03. This is your price for the game to have Under 2.5 goals.

Just like that, you’re doing stats based odds compiling!

If Sporting are right, and Poisson is right, and you can find a bookmaker offering odds of greater than 2.03 for Under 2.5 goals then you’ve just found yourself a ‘value’ bet.


If you’ve got the brain/instincts of a football betting analyst or a pro gambler, you’ll be imagining other ways you can extend the use of Poisson to model football scores.

But before you go any further, here’s a little test to see if you’re made of the right stuff!

We saw from the analysis of Half Time scores that Poisson was virtually perfect.

But now let’s look at the same analysis based on Full Time scores…..

FT Distribution

You can see that for Full Time scores, there’s a blip. Essentially 0 goals has happened way more often than Poisson suggests.


If you don’t know, or can’t come up with a decent theory to explain it, then sorry – you’re never going to make it as a football betting analyst I’m afraid.

Think you can figure it out?

Give us a shout on Twitter: @OddsModel and we’ll let you know if you’re right!

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