Poisson distribution for soccer betting

Poisson distribution for soccer betting

By Jack Ratcliffe Aug 9, 2013

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Poisson Distribution, coupled with historical data, can provide a method for calculating the likely number of goals that will be scored in a soccer match. Read on for a simple method to calculate the likely outcome of a soccer match using Poisson Distribution.

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 outcomes with Poisson Distribution

To calculate the possible outcomes for a match using Poisson Distribution, we first need to calculate a value for each team’s “Attack” and “Defence Strength”. These numbers – calculated from last season’s performance data – allow us to model possible outcomes for each game of the season.

Selecting the data range is vital when using Poisson Distribution, too long and the data wont be relevant for the teams' current strength, while too short may allow outliers to unfairly skew the data. For this analysis we are using the 38 games played by each team last season.

Calculate the average goals scored at home and away

To get the average number of goals scored last season, average the number of goals scored per team, per game, per season. In mathematical terms, that’s:

Season Goals Scored / Number of Teams / Number of Games

In 2012/13, that was 592/20/19 at home and 471/20/19 away, equalling an average of 1.558 goals per game at home and 1.239 away. The difference from this average is what constitutes a team’s “Attack Strength”.

Average number of goals scored at home: 1.558

Average number of goals scored away from home: 1.239

We’ll also need the average number of goals an average team concedes. This is simply the inverse of the above numbers (as the no. of goals a home team scores will equal the same number that an away team concedes.):

Average number of goals conceded at home: 1.239

Average number of goals conceded away from home: 1.558

We can now use the numbers above to calculate the Attack and Defence Strength of both Swansea City and Manchester United for their match on August 17th, 2013.

Predicting Swansea’s Goals

Calculate Swansea’s Attack Strength:

a. Take the number of goals scored at home last season by the home team (Swansea: 28) and divide by the number of home games (28/19): 1.473

b. Divide this value by the season’s average home goals scored per game (1.473/1.558), to get the “Attack Strength”: 0.946. This shows that Swansea scored 5.4% fewer goals at home than a hypothetical “average” Premier League side.

Calculate Man Utd’s Defence Strength:

a. Take the number of goals conceded away last season by the away team (Man Utd: 24) and divide by the number of away games (24/19): 1.263.

b. Divide this by the season’s average goals conceded by an away team per game (1.263/1.558) to get the “Defence Strength”: 0.81. Man Utd conceded 19% 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:

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

In this case, that’s 0.946* 0.81 * 1.558, which equates to Swansea scoring 1.194 goals.

Predicting Man Utd’s Goals

Calculate Man Utd’s Attack Strength:

1. Take the number of goals scored away last season by the away team (Man Utd: 41) and divide by the number of away games (41/19): 2.158

2. Divide this value by the season’s average away goals scored per game (2.158/1.239), to get the “Attack Strength”: 1.742. This shows that the Red Devils scored 74.2% more away goals than a hypothetical “average” Premier League side.

Calculate Swansea’s Defence Strength:

1. Take the number of goals conceded at home last season by the home team (Swansea: 26) and divide by the number of home games (26/19): 1.368.

2. Divide this by the season’s average goals conceded by a home team per game (1.368/1.239) to get the “Defence Strength”: 1.104. Swansea conceded 10.4% 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:

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

In this case, that’s 1.742 * 1.104 * 1.239, which equates to Man Utd scoring 2.383 goals.

Poisson Distribution Betting – Predicting Multiple Match Outcomes

Of course, no game ends 1.194 vs. 2.383 – 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; ) = (e-) (x) / 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, Man Utd at 2.383) in the average rate of success, and the calculator will output the probability of that score.

Poisson Distribution for Swansea vs Man Utd

Goals:

0

1

2

3

4

5

Swansea

30.3%

36.18%

21.60%

8.6%

2.57%

0.61%

Man Utd

9.23%

21.99%

26.2%

20.81%

12.4%

5.9%

This example shows that there is a 9.2% chance that Man Utd will not score, but a 22% chance they will get a single goal and a 26.2% chance they’ll score two.

Swansea, on the other hand, are at 30.3% not to score, 36.2% to score one and 21.6% to score two.

Hoping for a side to score five? The probability is 5.9% if United are the scorers, or 0.61% for Swansea to do it.

As both scores are independent (mathematically-speaking), you can see that the expected score is 2 – 1 to Man Utd. If you multiply the two probabilities together, you’ll get the probability of the 2-1 outcome – 0.095 or 9.5%.

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

For example, taking into account all possible draw combinations (0-0, 1-1, 2-2, 3-3, 4-4 and 5-5), this method gives a probability of 0.186 or 18.6%. Pinnacle Sports’ odds were 3.960 (a 25.3% implied probability).

Therefore if last season’s form was a perfect indicator of this season’s results, we should expect a 2-1 victory. Unfortunately it isn’t as simple as that, which is why pure Poisson analysis has limitations.

Poisson Distribution Betting – 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 without Sir Alex Ferguson is entirely ignored. The league is also arguably stronger this season, so United are less likely to score as many goals as they did last year.

Correlations are also ignored, such as the widely recognised pitch effect that shows that matches have some 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.

Click here for the latest Premier League odds.

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*Odds subject to change