# Poisson Distribution: Predict a soccer betting winner

### Poisson Distribution explained

Poisson Distribution is a mathematical concept for translating mean averages into a probability for variable outcomes. For example, Chelsea might average 1.7 goals per game. Entering this information into a Poisson formula would show that this average equates to Chelsea scoring 0 goals 18.3% of the time, 1 goal 31% of the time, 2 goals 26.4% of the time and 3 goals 15% of the time.

### How to calculate soccer outcomes with Poisson Distribution

Before we can use Poisson to calculate the likely outcome of a match, we need to calculate the average number of goals each team is likely to score in that match. This can be calculated determining an “Attack” and “Defence Strength” for each team and comparing them.

Selecting a representative data range is vital when calculating Attack and Defence strengths – too long and the data will not be relevant for the teams current strength, while too short may allow outliers to skew the data. For this analysis we’re using the 38 games played by each team in the 2013/14 EPL season.

Calculating Attack and Defence strengths

Calculate the average goals scored at home and away

The first step in calculating Attack and Defence strengths based upon last season’s results is to determine the average number of goals scored per team, per home game, and per away games.

Calculate this by taking the total number of goals scored last season and dividing it by the number of games played:

Season Goals Scored at Home / Number of Games (in season)

Season Goals Scored Away / Number of Games (in season)

In 2013/14, that was 598/380 at home and 454/380 away, equalling an average of 1.574 goals per game at home and 1.195 away.

• Average number of goals scored at home: 1.574
• Average number of goals scored away from home: 1.195

The difference from the above average is what constitutes a team’s “Attack Strength”.

We’ll also need the average number of goals an average team concedes. This is simply the inverse of the above numbers (as the number of goals a home team scores will equal the same number that an away team concedes):

• Average number of goals conceded at home: 1.195
• Average number of goals conceded away from home: 1.574

We can now use the numbers above to calculate the Attack and Defence Strength of both Manchester United and Swansea City for their match on August 16th, 2014.

### Predicting Man United’s Goals

Calculate Man United’s Attack Strength:

1. Take the number of goals scored at home last season by the home team (Man United: 29) and divide by the number of home games (29/19): 1.526
2. Divide this value by the season’s average home goals scored per game (1.526/1.574), to get the “Attack Strength”: 0.970. This shows that Man United scored 3.05% fewer goals at home than a hypothetical “average” Premier League side last season.

Calculate Swansea’s Defence Strength:

1. Take the number of goals conceded away last season by the away team (Swansea: 28) and divide by the number of away games (28/19): 1.474.
2. Divide this by the season’s average goals conceded by an away team per game (1.474/1.574) to get the “Defence Strength”: 0.936. This therefore highlights Swansea conceded 6.35% fewer goals than an “average” Premier League side on the road.

We can now use the following formula to calculate the likely number of goals the home team might score:

Man United’s Goals = Man United’s Attack x Swansea’s Defence x Average No. Goals

In this case, that’s 0.970* 0.936 * 1.574, which equates to United scoring 1.429 goals.

### Predicting Swansea’s Goals

Calculate Swansea’s Attack Strength:

1. Take the number of goals scored away last season by the away team (Swansea: 21) and divide by the number of away games (21/19): 1.105
2. Divide this value by the season’s average away goals scored per game (1.105/1.195), to get the “Attack Strength”: 0.925. This shows that Swansea scored 7.53% fewer away goals than a hypothetical “average” Premier League side.

Calculate Man United’s Defence Strength:

1. Take the number of goals conceded at home last season by the home team (Man United: 21) and divide by the number of home games (21/19): 1.105.
2. Divide this by the season’s average goals conceded by a home team per game (1.105/1.195) to get the “Defence Strength”: 0.925. Man United conceded 7.53% more goals than an “average” Premier League side at home.

We can now use the following formula to calculate the likely number of goals the away team might score:

Swansea’s Goals = Swansea’s Attack x Man United’s Defence x Average No. Goals

In this case, that’s 0.925* 0.925 * 1.195, which equates to Swansea scoring 1.022 goals.

### Poisson Distribution betting – Predicting multiple match outcomes

Of course, no game ends 1.429 vs. 1.022 – this is simply the average. Poisson Distribution, a formula created by French mathematician Simeon Denis Poisson, allows us to use these figures to distribute 100% of probability across a range of goal outcomes for each side. The results are shown in the table below:

The formula itself looks like this: P(x; μ) = (e-μ) (μx) / x!, however, we can use online tools such as this Poisson Distribution Calculator to do most of the equation for us.

All we need to do is enter the different goals outcomes (0-5) in the Random Variable (x) category, and the likelihood of a team scoring (for instance, Swansea at 1.022) in the average rate of success, and the calculator will output the probability of that score.

### Poisson Distribution for Man United vs. Swansea

Goals 0 1 2 3 4 5
Man United 23.95% 34.23% 24.46% 11.65% 4.16% 1.19%
Swansea 35.99% 36.78% 18.79% 6.40% 1.64% 0.33%

This example shows that there is a 23.95% chance that Man Utd will not score, but a 34.23% chance they will get a single goal and a 24.46% chance they’ll score two.

Swansea, on the other hand, are at 35.99% not to score, 36.78% to score one and 18.79% to score two.

Hoping for a side to score five? The probability is 1.19% if United are the scorers, or 0.33% for Swansea to do it.

As both scores are independent (mathematically-speaking), you can see that the expected score is 1 – 1. If you multiply the two probabilities together, you’ll get the probability of the 1-1 outcome – 0.125 or 12.59%.

Now you know how to calculate outcomes, you should compare your result to a bookmaker’s odds to help see how they differentiate.

### Example: comparing the draw

The above example showed us that a 1-1 draw has a 12.59% chance of occurring, according to our model. But what if you wanted to bet on the “draw”, rather than on individual score outcomes? You’d need to calculate the probability for all of the different draw scorelines – 0-0, 1-1, 2-2, 3-3, 4-4, 5-5 etc.

To do this, simply calculate the probability of all possible draw combinations and add them together. This will give you the chance of a draw occurring, regardless of the score.

Of course, there are actually an infinite number of draw possibilities (both sides could score 10 goals each, for example), but the chances of a draw above 5-5 are so small that it’s safe to disregard them for this model.

For the United – Swansea game, combining all of the draws gives a probability of 0.266 or 26.6%. Pinnacle Sports’ odds were 5.530 (an 18.08% implied probability).

Therefore if last season’s form was a perfect indicator of this season’s results, there would appear to be value in backing the draw, as the model shows that it more likely to happen than the Pinnacle Sports odds suggest. Unfortunately it isn’t as simple as that, which is why pure Poisson analysis has limitations.

### The limits of Poisson Distribution

Poisson Distribution is a simple predictive model that doesn’t allow for a lot of factors. Situational factors – such as club circumstances, game status etc. – and subjective evaluation of the change of each team during the transfer window are completely ignored.

In this case, it means the huge x-factor of Manchester United’s first Premier League game with new manager Louis Van Gaal is entirely ignored.

Correlations are also ignored; such as the widely recognised pitch affect that shows certain matches have a tendency to be either high or low scoring.

These are particularly important areas in lower league games, which can give punters an edge against bookmakers, while it’s harder to gain an edge in major leagues, given the expertise that modern bookmakers like Pinnacle Sports possess.

Source: Pinnacle Sports