Poisson Distribution and Football Betting

Poisson Distribution, coupled with historical data, provides a simple and reliable method for calculating the most likely score in a football match which can be applied to betting (a.k.a Poisson Betting). This simple walk-through shows how to calculate the necessary Attack/Defence Strength measures along with a handy shortcut to generate the Poisson Distribution values. In no time you will be predicting football scores using the Poisson Distribution.

Poisson Distribution is a mathematical concept for translating mean averages into a probability for variable outcomes across a distribution. For example, if we know Manchester City average 1.7 goals per game, so by putting the Poisson Distribution formula tells us that this average equates to Manchester City 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.

Poisson Distribution – Calculating score-line probabilities

Before we can use Poisson to calculate the most likely score-line 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 by determining the “Attack Strength” and “Defence Strength” for each team and comparing them.

Once you know how to calculate result probabilities, you can compare your results to a bookmaker’s odds and potentially find value.
Selecting a representative data range is vital when calculating Attack Strength and Defence Strength – too long and the data will not be relevant for the team’s current strength, while too short may allow outliers to skew the data. The 38 games played by each team in the 2015/16 EPL season will provide a sufficient sample size to apply the Poisson Distribution for this particular football betting.

How to calculate Attack Strength of football teams?

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

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

Season total goals scored at home / number of games (in season)
Season total goals scored away / number of games (in season)
In 2015/16 English Premier League season, there were 567/380 at home and 459/380 away, equalling an average of 1.492 goals per game at home and 1.207 away.

Average number of goals scored at home: 1.492
Average number of goals scored away: 1.207
The ratio of a team’s average and the league average is what constitutes “Attack Strength”.

How to calculate Defence Strength of football teams?

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: 1.207
Average number of goals conceded away from home: 1.492
The ratio of a team’s average and the league average is what constitutes “Defence Strength”.

We can now use the numbers above to calculate the Attack Strength and Defence Strength of both Tottenham Hotspur and Everton (as of 1st March 2017).

Predicting Tottenham Hotspur’s goals
Calculate Tottenham’s Attack Strength:

Step – 1: Take the number of goals scored at home last season by the home team (Tottenham: 35) and divide by the number of home games (35/19): 1.842.
Step – 2: Divide this value by the season’s average home goals scored per game (1.842/1.492) to get an “Attack Strength” of 1.235.

(35/19) / (567/380) = 1.235

Calculate Everton’s Defence Strength:

Step – 1: Take the number of goals conceded away from home last season by the away team (Everton: 25) and divide by the number of away games (25/19): 1.315.
Step – 2: Divide this by the season’s average goals conceded by an away team per game (1.315/1.492) to get a “Defence Strength” of 0.881.

(25/19) / (567/380) = 0.881

We can now use the following formula to calculate the likely number of goals Tottenham might score (this is done by multiplying Tottenham’s Attack Strength by Everton’s Defence Strength and the average number of home goals in the Premier League):

1.235 x 0.881 x 1.492 = 1.623

Predicting Everton’s goals

To calculate the number of goals Everton might score, simply use the above formulas but replace the average number of home goals with the average number of away goals.

Everton’s Attack Strength:

(24/19) / (459/380) = 1.046

Tottenham’s Defence Strength:

(15/19) / (459/380) = 0.653

In the same way we predicted the number of goals Tottenham will score, we can calculate the likely number of goals Everton might score (done by multiplying Everton’s Attack Strength by Tottenham’s Defence Strength and the average number of away goals in the Premier League):

1.046 x 0.653 x 1.207 = 0.824

Poisson Distribution – Predicting multiple outcomes

Of course, no game ends 1.623 vs. 0.824 – 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.

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

All we need to do is enter the different event occurrences – in our case goals outcomes from 0-5 – and the expected occurrences which are the likelihood of each team scoring – in our example Tottenham at 1.623 is their average rate of success, and Everton 0.824; the calculator will output the probability of the score for the given outcome.

Poisson Distribution for Tottenham vs. Everton

Goals 0 1 2 3 4 5
Tottenham 19.73% 32.02% 25.99% 14.06% 5.07% 1.85%
Everton 43.86% 36.14% 14.89% 4.09% 0.84% 0.14%

This example shows that there is a 19.73% chance that Tottenham will fail to score, but a 32.02% chance they will score a single goal and a 25.99% chance they’ll score two. Everton, on the other hand, is at 43.86% not to score, 36.14% to score one and 14.89% to score two. Hoping for a side to score five? The probability is 1.85% for Tottenham or 0.14% for Everton – or 2% for either team to score 5.

As both scores are independent (mathematically-speaking), you can see that the expected score is 1–0 – pairing the most probable outcomes for each team. If you multiply those two probabilities together, you’ll get the probability of the 1-0 outcome – (0.3202*0.4386) =0.1404 or 14.04%.

Now you know how to calculate score-line probabilities using Poisson Distribution for football betting, you can compare your measures to a bookmaker’s odds and see if there are discrepancies to take advantage of, especially if you factor in your own assessment of relevant situational factors such as weather, injury or HFA.

Converting estimated chance into betting odds

The above example showed us that a 1-1 draw has an 11.53% chance (0.3202*0.3614) 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. 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.

Using the Tottenham vs. Everton example, combining all of the draws gives a probability of 0.2472 or 24.72% – this would give true odds of 4.05 (1/0.2472).

The limits of Poisson Distribution

Poisson Distribution is a simple predictive model that doesn’t allow for numerous 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, the above Poisson formula calculation fails to quantify any effect Everton’s new manager (Ronald Koeman) might have had on the team. It also fails to take Tottenham’s potential fatigue into consideration now that they are playing close to a Europa League fixture.

Correlations are also ignored; such as the widely recognised pitch effect 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 bettors an edge against bookmakers. It is harder to gain an edge in major leagues such as the Premier League given the expertise and resources that modern bookmakers have at their disposal.

Last, but not least, these odds do not factor in the margin a bookmaker charges which are hugely important to the whole process of finding value.

Want to apply the Poisson Distribution to football betting?

Get the best Premier League odds and highest limits at Pinnacle.

This article was first published in Pinnacle’s Betting Resources titled: Poisson Distribution: Predict the score in soccer betting

Poisson Distribution Betting

Poisson Distribution: How to predict the score in football betting

Poisson Distribution, coupled with historical data, provides a simple and reliable method for calculating the most likely score in a football match which can be applied to betting. This simple walk-through shows how to calculate the necessary Attack/Defence Strength measures along with a handy shortcut to generate the Poisson Distribution values. In no time you can be predicting football scores using the Poisson Distribution.

Poisson Distribution is a mathematical concept for translating mean averages into a probability for variable outcomes across a distribution. For example, if we know Manchester City average 1.7 goals per game, so by putting the Poisson Distribution formula tells us that this average equates to Manchester City 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.

Poisson Distribution – Calculating score-line probabilities

Before we can use Poisson Distribution to calculate the most likely score-line 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 by determining the “Attack Strength” and “Defence Strength” for each team and comparing them.

Once you know how to calculate result probabilities, you can compare your results to a bookmaker’s odds and potentially find value.

Selecting a representative data range is vital when calculating Attack Strength and Defence Strength – too long and the data will not be relevant for the team’s current strength, while too short may allow outliers to skew the data. The 38 games played by each team in the 2015/16 EPL season will provide a sufficient sample size to apply the Poisson Distribution.

How to calculate Attack Strength

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

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

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

In 2015/16 English Premier League season, there were 567/380 at home and 459/380 away, equalling an average of 1.492 goals per game at home and 1.207 away.

  • Average number of goals scored at home: 1.492
  • Average number of goals scored away: 1.207

The ratio of a team’s average and the league average is what constitutes “Attack Strength”.

How to calculate Defence 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: 1.207
  • Average number of goals conceded away from home: 1.492

The ratio of a team’s average and the league average is what constitutes “Defence Strength”.

We can now use the numbers above to calculate the Attack Strength and Defence Strength of both Tottenham Hotspur and Everton (as of 1st March 2017).

Predicting Tottenham Hotspur’s goals

Calculate Tottenham’s Attack Strength:

  1. Step – 1: Take the number of goals scored at home last season by the home team (Tottenham: 35) and divide by the number of home games (35/19): 1.842.
  2. Step – 2: Divide this value by the season’s average home goals scored per game (1.842/1.492) to get an “Attack Strength” of 1.235.

(35/19) / (567/380) = 1.235

Calculate Everton’s Defence Strength:

  1. Step – 1: Take the number of goals conceded away from home last season by the away team (Everton: 25) and divide by the number of away games (25/19): 1.315.
  2. Step – 2: Divide this by the season’s average goals conceded by an away team per game (1.315/1.492) to get a “Defence Strength” of 0.881.

(25/19) / (567/380) = 0.881

We can now use the following formula to calculate the likely number of goals Tottenham might score (this is done by multiplying Tottenham’s Attack Strength by Everton’s Defence Strength and the average number of home goals in the Premier League):

1.235 x 0.881 x 1.492 = 1.623

Predicting Everton’s goals

To calculate the number of goals Everton might score, simply use the above formulas but replace the average number of home goals with the average number of away goals.

Everton’s Attack Strength:

(24/19) / (459/380) = 1.046

Tottenham’s Defence Strength:

(15/19) / (459/380) = 0.653

In the same way we predicted the number of goals Tottenham will score, we can calculate the likely number of goals Everton might score (done by multiplying Everton’s Attack Strength by Tottenham’s Defence Strength and the average number of away goals in the Premier League):

1.046 x 0.653 x 1.207 = 0.824

Poisson Distribution – Predicting multiple outcomes

Of course, no game ends 1.623 vs. 0.824 – 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.

Poisson Distribution formula:

P(x; μ) = (e-μ) (μx) / x!

However, we can use online tools such as a Poisson Distribution Calculator to do most of the equation for us.

All we need to do is enter the different event occurrences – in our case goals outcomes from 0-5 – and the expected occurrences which are the likelihood of each team scoring – in our example Tottenham at 1.623 is their average rate of success, and Everton 0.824; the calculator will output the probability of the score for the given outcome.

Poisson Distribution for Tottenham vs. Everton

Goals 0 1 2 3 4 5
Tottenham 19.73% 32.02% 25.99% 14.06% 5.07% 1.85%
Everton 43.86% 36.14% 14.89% 4.09% 0.84% 0.14%

This example shows that there is a 19.73% chance that Tottenham will fail to score, but a 32.02% chance they will score a single goal and a 25.99% chance they’ll score two. Everton, on the other hand, is at 43.86% not to score, 36.14% to score one and 14.89% to score two. Hoping for a side to score five? The probability is 1.85% for Tottenham or 0.14% for Everton – or 2% for either team to score 5.

As both scores are independent (mathematically-speaking), you can see that the expected score is 1–0 – pairing the most probable outcomes for each team. If you multiply those two probabilities together, you’ll get the probability of the 1-0 outcome – (0.3202*0.4386) =0.1404 or 14.04%.

Now you know how to calculate score-line probabilities using Poisson Distribution for betting, you can compare your measures to a bookmaker’s odds and see if there are discrepancies to take advantage of, especially if you factor in your own assessment of relevant situational factors such as weather, injury or HFA.

Converting estimated chance into odds

The above example showed us that a 1-1 draw has an 11.53% chance (0.3202*0.3614) 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 score lines – 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. 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.

Using the Tottenham vs. Everton example, combining all of the draws gives a probability of 0.2472 or 24.72% – this would give true odds of 4.05 (1/0.2472).

The limits of Poisson Distribution

Poisson Distribution is a simple predictive model that doesn’t allow for numerous 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, the above Poisson formula calculation fails to quantify any effect Everton’s new manager might have had on the team. It also fails to take Tottenham’s potential fatigue into consideration now that they are playing close to a Europa League fixture.

Correlations are also ignored; such as the widely recognised pitch effect 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 bettors an edge against bookmakers. It is harder to gain an edge in major leagues such as the English Premier League given the expertise and resources that modern bookmakers have at their disposal.

Last, but not least, these odds do not factor in the margin a bookmaker charges which are hugely important to the whole process of finding value.

Want to apply the Poisson Distribution to football betting? Get the best Premier League odds and highest limits at Pinnacle.

Both Teams To Score betting analysis

Which statistics can inform your Both Teams To Score betting?

Can bettors profit from BTTS betting?

Both teams to score betting

‘Both Teams To Score’ betting has increased in popularity because it offers a simple binary choice, while data suggests an even distribution of either outcome. In Europe’s big five leagues over the last five seasons, 51% of matches saw both teams score at least one goal, so you might as well flip a coin for Both Teams To Score betting, right?

When taking a more in depth approach to analysing Both Teams To Score (BTTS) betting, it is clear that there is more to it than the flip of a coin. Despite the almost 50/50 statistics from a large sample size, this type of bet is dependant on the two teams playing, how they play, their previous results and even when the game is taking place.

The importance of BTTS records

In the 490 team seasons covered in the aforementioned period, there is a wide range of BTTS incidence; from 26% for Bayern Munich (2014/15) to 79% for Hoffenheim (2013/14). It is clear however, when looking at the teams who were ever present in the Premier League over the last five seasons that in the long-term, clubs produce similar numbers of matches where both teams scored.

BTTS-chart

 

We can see that a team that hangs around in the English top flight for a while can be relied upon to see both teams score in between eighteen and twenty-one of their matches per season. What bettors need however, are reliable match stats that might indicate where teams are more likely to both score and concede in their games.

What to look out for with Both Teams To Score betting?

The following table uses Spearman’s rank correlation to assess the relationship between both teams scoring and various stats for the last five seasons in the Premier League:

Statistic Spearman’s Rank correlation with BTTS
Total goals 0.616
Total clean sheets -0.471
Times failed to score -0.466
Total shots on target 0.385
Total goals scored 0.325
Total goals against 0.227
Total shots 0.133
Goal difference 0.032
League position 0.021

Goal difference may correlate strongly with league position, but neither are much good at indicating whether both teams will score in a team’s matches. It’s not a surprise to see that ‘total goals’ leads the way here but what else can we consider?

Looking at the 2015/16 Premier League, there was only a slight impact on the likelihood of BTTS based on whether both teams scored in the previous game. One or both clubs failed to find the net in 48.2% of Premier League matches last season, and if both teams had not scored in a team’s previous match, then at least one didn’t score in 51.1% of that team’s next matches on average.

Do teams have Both Teams To Score form?

If both teams had scored in a team’s previous fixture, then they did so again in that team’s next match 54.8% of the time, compared to the overall average of 51.8%. This shows that knowing the outcome of a previous match can give you a slight advantage, though of course different teams will fit this trend more closely than others.

That said, every team in the division had runs of at least three games where both teams scored, and where both clubs failed to score, so you can assume that they are likely to have such a run at some point.

Despite the almost 50/50 statistics from a large sample size, there are numerous factors for bettors to consider for Both Teams To Score (BTTS) betting.

For the record, more teams (ten) had a longer run of both teams scoring than it not happening, with only six having their longest run where one side failed to score, and every side in the 2016/17 Premier League has already had a ‘both teams score’ run of at least three matches.

Over this five season period, Everton had the longest BTTS run with 16 games in the 2012/13 season. The longest spell without both teams scoring was Burnley’s run of 12 matches in 2014/15.

Does kick-off time effect BTTS betting?

The time the match kicks off may be relevant too. The sample sizes are obviously not equal, as a lot of matches still kick off at 15:00 on a Saturday, but it’s worth looking at a breakdown of last season’s Premier League matches by starting time to see whether both teams scored.

Both teams to score table

 

Early kick-offs can often be dull and uninspiring for fans and players alike, and the numbers suggest that you won’t see both teams score during such a match too often either – this could be due to away team travel arrangements, both teams having less time to prepare or any number of reasons.

In 55.7% of 3pm Saturday kick-offs, both teams scored in the match.

As there is an almost even split of matches that began at 3pm on a Saturday afternoon and those which did not, it’s interesting that 55.7% of the regular kick offs saw both sides hit the back of the net, but only 48.2% of the other matches – something that will be worth bearing in mind when placing a bet.

Optimism bias and Both Teams To Score betting

As with all betting, it is essential to not let biases cloud your judgement – particularly in a market like this where a single bounce of the ball or a questionable refereeing decision can lead to a very fortunate goal (or unfortunate, depending on what your bet is).

The danger of optimism bias in BTTS betting is evident when analysing the lowest scoring sides record against the top six in the Premier League. You might expect games that feature teams that struggle to score (Aston Villa, Norwich and WBA) and top six teams to not be BTTS. However, both teams scored in nine of the eighteen matches between these teams last season.

Get the best odds on BTTS betting across Europe’s top five leagues, the Champions League and Europa League at Pinnacle.

Originally published in Pinnacle’s Betting Resources