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. Using a simple Poisson Distribution formula to calculate the probability of the score or result in any given soccer match will empower your betting. Read on to learn more.
Poisson Distribution is a mathematical concept for translating mean averages into a probability for variable outcomes. For example, Manchester City might average 1.7 goals per game and by putting this information into a Poisson Distribution formula, it would show 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 - Calculate the probability of a result
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 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 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, that was 564/380 at home and 459/380 away, equalling an average of 1.484 goals per game at home and 1.207 away.
- Average number of goals scored at home: 1.484
- Average number of goals scored away from home: 1.207
The difference from the relevant average above and a team's 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: 1.484
The difference from the relevant average above and a team's 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 for their upcoming match.
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 (1.842/1.484) to get an “Attack Strength” of 1.241.
(35/19) / (564/380) = 1.241
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): 1.263.
- Step - 2: Divide this by the season’s average goals conceded by an away team per game (1.315/1.484) to get a “defence strength” of 0.886.
(25/19) / (564/380) = 0.886
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):
1.241 x 0.886 x 1.484 = 1.631
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
1.046 x 0.653 x 1.207 = 0.824
Poisson Distribution – Predicting multiple outcomes
Of course, no game ends 1.631 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 goals outcomes (0-5) as a Random Variable (x) category, and the likelihood of a team scoring (for instance, Tottenham at 1.631) in the average rate of success, and the calculator will output the probability of that score.
Poisson Distribution for Tottenham vs. Everton
This example shows that there is a 19.57% chance that Tottenham will not score, but a 31.92% chance they will get a single goal and a 26.03% 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 1.88% for Tottenham or 0.14% for Everton.
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 – 0.1153 or 11.53%.
Now you know how to calculate result probabilities, you can compare your results to a bookmaker’s odds and see if there are discrepancies to take advantage of.
Converting estimated chance into odds
The above example showed us that a 1-1 draw has an 11.53% chance of occurring when the Poisson Distribution formula is applied. 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.
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 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 0.2464 or 24.64% - this would give true odds of 4.05.
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, the above Poisson formula calculation fails to quantify any effect Everton's new manager (Ronal Koeman) might have had on the team. It also fails to take Tottenham's potential fatigue into consideration now that 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.
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