Poisson Distribution: Predict the score in football betting

We can all safely agree that predicting the correct score of a football match is one of the most profitable strategies in sports betting. Unfortunately, this procedure is also considered one of the hardest ones. Many bettors prefer to avoid this special bet, as they think that confirming a correct score contains a lot of luck, or that there’s no way of rationally analyzing how many goals each team will score.

However, there is a mathematical concept, that can help you convert numbers to goal scoring probabilities and compare them to the offered odds. This is called the Poisson Distribution, after the French mathematician Simeon Denis Poisson (1781-1840).

This is a discrete probability distribution, that expresses the probability of a given number of events taking place in a fixed interval of time or space. These events have to occur within a known constant rate, independently of the time since the last event. If this sounds a bit complicated to you, then you can see this explained on the following examples.

The Poisson distribution is a reliable method of analyzing the potential correct score by taking into account a team’s past goal data within a season along with any historical data. It allows you to see the scoring probability of each team and lets you pick the highest odds according to the highest goal scoring percentage.

This formula converts the total goals average (both for and against) to the actual chances of exact goals being scored. For example, if Real Madrid’s goal average is 1.7 per game, the Poisson Distribution will allocate the goals percentages as follow: Chances for Real Madrid to score 0 goals to their next match are 18.3% | 1 goal 31% | 2 goals 26.4% | 3 goals 15%.

Calculating scores during a season

In order to use the Poisson Distribution you need to calculate the average number of goals each team has scored and conceded. These are called “Attack Strength” and “Defence Strength” respectively. It is a fairly straightforward procedure, however, you must make sure your data is accurate before you start using the Poisson.

The most usual data range concerns the current season. This may seem short, but if you include past data it will distort the actual team’s strength. Remember, you don’t calculate numbers only for statistical purposes, but to find out what will happen in a few days, if not hours.

However, it’s logical that the Poisson Distribution works better with sufficient number of data. Don’t expect to make accurate predictions at the start of a season, when the teams have played only a couple of matches. In order to better understand how the Poisson Distribution works, we will use the total results of the Spanish La Liga 2016/17.

Calculating the total Attack Strength

First of all you should find out the average number of goals scored per home game and per away game. This is a total average for the whole league. You just take the total number of goals scored last season and divide it by the number of football games:

• Season total goals scored at home / number of games (in season)
• Season total goals scored away / number of games (in season)

In 2016/17 La Liga season, there were 632/380 home goals and 486/380 away goals, equalling an average of 1.663 goals per game at home and 1.279 away.

• Average number of goals scored at home: 1.663
• Average number of goals scored away: 1.279

In order to calculate the Attack Strength, you just add the individual team’s average and the league average.

Calculating the total Defence Strength

You also need to find the Defense Strength, which is easy: You just inverse the number you’ve calculated earlier, as the Home Attack Strength is identical to the Away Defence Strength.

• Average number of goals conceded at home: 1.279
• Average number of goals conceded away: 1.663

In order to calculate the Defense Strength, you act as above, adding the individual team’s average and the league average.

After calculating the totals, let’s take an actual example: A match between Atletico Madrid and Valencia.

Predicting Atletico Madrid's goals

All you have to do is use the calculations mentioned above, this time not for the entire league, but for the individual team:

• Atletico Madrid scored 40 goals in 19 home matches in 2016/17 season, so their Attack Strength is 40/19=2,105
• Next, you divide this value by the season's average home goals scored per game (2.105/1.663) to get an “Attack Strength” of 1.212.

The same calculations apply on Valencia as well, this time for their Defence Strength:

• Valencia conceded 33 goals in 19 away matches in 2016/17 season, so their Defense Strength is 33/19=1.737
• Next, you divide this value by the season's average goals conceded by an away team per game (1.737/1.663) to get a “Defence Strength” of 1.044

The last calculation in order to find the number of Atletico Madrid’s goals is multiply Atletico’s Attack Strength by Valencia's Defence Strength and the average number of home goals in the Premier League):

1.212 x 1.044 x 1.663 = 2.104

Predicting Valencia's goals

The same calculations apply to Valencia as well to find the number of goals they might score. You just have to replace the average number of home goals with the average number of away goals.

• Valencia scored 24 goals in 19 away matches, so their Attack Strength is 24/19=1.263 along with the  total away Attack Strength 1.279. Dividing these numbers will give you a 0.987 value.
• Atletico Madrid conceded 14 goals in 19 home matches, so their Defence Strength is 14/19=0.737 along with the total home Defence Strength 1,279. By dividing these numbers you get a 0.576 value.

Now, you have to multiply Valencia’s Attack Strength by Atletico Madrid's Defence Strength and the average number of away goals in La Liga):

0.987 x 0.76 x 1,279 = 0.727

Poisson Distribution in practice

This is where the Poisson Distribution works, converting the aforementioned values to actual percentages for goals for each team. The values you’ve found so far (2.104 for Atletico Madrid and 0.727 for Valencia) are simply the average.

How are percentages distributed from these numbers? The correct score predicting formula is as follows:

P(x; μ) = (e-μ) (μx) / x!

In absolute maths terms it would mean that:

• e: A constant equal to approximately 2.71828. (e is the base of the natural logarithm system.)
• μ: The league's total attack strength.
• x: Both team's total attack strength for home and away games respectively.
• P(x; μ): The Poisson probability that exactly x goals where scored, when the mean number of goals is μ.

If you don't want to calculate the formula yourself (and risk making a mistake) you can use one of the free Poisson Distribution Calculator available in a lot of websites. These are few:

http://stattrek.com/online-calculator/poisson.aspx

http://keisan.casio.com/exec/system/1180573181

http://stats.areppim.com/calc/calc_poisson.php

You have to enter the different event occurrences - goals outcomes within a range of zero (0) to five (5) and the expected occurrences which are the likelihood of each team scoring - 2.104 for Atletico Madrid and 0.727 for Valencia. The calculator will output the probability of the score for the given outcome.

Poisson Distribution for Atletico Madrid vs. Valencia

Goals	0	1	2	3	4	5
Atletico Madrid	12.20%	26.70 %	28.00%	18.90%	10.00%	4.20%
Valencia	48.30%	35.10%	12.80%	3.10%	0.60%	0.10%

This example shows that there is a 12.20% chance that Atletico Madrid will fail to score, but a 26.70% chance they will score a single goal and a 28.00% chance they'll score two. The visitors Valencia is at 48.30% not to score, 35.10% to score one and 12.80% to score two.

In order to find the correct score that has the more chances to be confirmed, you have to multiply the percentages shown above the exact number of goals. For example, the chances of a 1-0 correct score for Atletico Madrid are

0.267 x 0.483 = 12.89%

For a 3-1 correct score the chances are:

0.189 x 0.351 = 6.33%

This way you can calculate either all the possible correct scores or focus only on those that interest you. To better understand how this works, you’re recommended to calculate all score chances and then compare your measures to a bookmaker's odds to see if there are discrepancies you could take advantage of.

Converting estimated chance into odds

The example of the Atletico Madrid vs. Valencia match showed us that a 1-1 draw has an 9.37% chance (0.267 x 0.351) of occurring when the Poisson Distribution formula is applied. But what if you wanted to know the predicted odds on the “draw”, rather than on individual draw 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.

Once you calculate the chances of each outcome, you convert them into odds and compare them to a bookmaker's odds in order to find potential value bets. To do this, simply calculate the probability of all possible draw combinations and add them together. In Atletico Madrid vs. Valencia match the chances of a 0-0 draw are 5.89%, the 1-1 draw 9.37%, the 2-2 draw 3.58%, the 3-3 draw 0.6%, the 4-4 draw 0.1% and so on. By adding all these numbers we count the possible draw as having a total 19.54% to appear.

The limits of Poisson Distribution

Needless to say, this formula, as any other mathematical formula, is a simple a predictive model that doesn't take into account numerous factors. A manager change, a key absence, bad weather, fatigue or squad rotation etc. are completely ignored. These are particularly important areas in lower league games, which can give bettors an edge against bookmakers. It is obviously harder to gain an edge in major leagues such as the La Liga.