The principles of Poisson Distribution

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# The principles of Poisson Distribution

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The easiest and simplest way to calculate the most likely score in soccer is to use the Poisson Distribution and bettors can use this method to make a reliable bet. The calculation of the Attack/Defence strength is in detail explained below and it is used to get the Poisson Distribution values. The concept behind the Poisson Distribution is purely mathematics and it turns the mean averages into a probability. If we use this method to calculate the probability of Manchester City scoring a goal, when their average is 1.7 goals per game, we will get that City will score 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 the score-line probabilities using the Poisson Distribution

After we are done calculating the “Attack Strength” and “Defence Strength” for each team and after comparing them by calculating the average number of goals that are likely to be scored by each team during that game, we can calculate the score-line that we are most likely to get in a match.

In order to calculate the Attack Strength and Defence Strength, a representative data range is crucial. If the data range is too long, then it will be useless because it will not represent the current strength of the team. Too short data range on the other hand will skew the data. Ideally, the Poisson Distribution will make the best use of the data of the 38 games played by each team in the 2015/16 EPL season.

## Calculating the Attack Strength

We need to determine the average number of goals scored per team, per home game and per away game, using the last season’s results, and by this we can start calculating the Attack Strength.

Therefore: total number of goals scored last season divided by the number of games played, as shown below:

• Total goals in a season scored at home divided by the number of games (in one season)
• Total goals in a season scored away divided by the number of games (in one season)

For example, in the English Premier League of 2015/16, there were 567/380 at home and 459/380 away, and this equals to an average of 1.492 goals per game at home and 1.207 goals away.

Therefore, “Attack Strength” is the ratio of a team’s average and the league average.

## Calculating Defence Strength

The next value that we need is the average number of goals a team concedes. This value we already have because the number of goals a team scores is actually the number of goals the opposite team concedes.

Therefore:

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

The Defence Strength is the ration of a team’s average and the league average.

Attack Strength and Defence Strength of Tottenham Hotspur and Everton

Case 1: Tottenham’s goals

Tottenham’s Attack Strength

1. Number of goals scored at home last season by the home team (Tottenham: 35) divided by the number of home games (35/19): 1.842.
2. 842 divided by the season’s average home goals scored per game 1.492 equals 1.235 and this is the Attack Strength.

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

Everton’s Defence Strength

1. Number of goals conceded away by the away team (Everton) last season (25) divided by the number of away games (25/19): 1.315.
2. 315 divided by the season’s average goals conceded by an away team per game (1.315/1.492) equals 0.881 and this is the Defence Strength.

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

In order to calculate the likely number of goals Tottenham might score, we can use the formula as follows:

1.235 x 0.881 x 1.492 = 1.623

where:

Tottenham’s Attack Strength is multiplied by Everton’s Defence Strength and the average number of home goals in the Premier League.

Case 2: Everton’s goals

In this case, we will just 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

The likely number of goals Everton might score is done in the following way:

Everton’s Attack Strength multiplied by Tottenham’s Defence Strength and the average number of goals in the Premier League:

1.046 x 0.653 x 1.207 = 0.824

## Using Poisson Distribution to predict various outcomes

A 100 % of probability is distributed across a range of goal outcomes for each side by using this mathematical formula created by the French mathematician Simeon Denis Poisson.

Poisson Distribution Formula: P(x; μ) = (e– μ) (μx) / x!

There is an easier way to calculate by using a Poisson Distribution Calculator.

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.

## Tottenham vs. Everton – Poisson Distribution

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%

According to this, Tottenham will fail to score with a chance of 19.73, but there is a 32.02% chance they will score a single goal and a 25.99% chance they’ll score two. On the other hand, Everton, has a chance of 43.86% not to score, 36.14% to score one and 14.89% to score two. The probability one team to score five goals is 1.85% for Tottenham or 0.14% for Everton – or 2% for either team to score five goals.

In mathematical terms, both scores are independent from each other. We can see that the expected score is 1-0 – pairing the most probable outcomes for each team. If we multiply those two probabilities together, we will get the probability of the 1-0 outcome – (0.3202*0.4386) =0.1404 or 14.04%.

The above mentioned Poisson Distribution method is a way to calculate score-line probabilities. In this manner, a bettor can use it to compare his measures to a bookmaker’s odds and thus, take advantage of the discrepancies. This can be also further strengthened by his own assessment of situational factors, such as injuries, weather, etc.

## How to convert estimated chance into odds

We have seen that when the Poisson Distribution formula is applied, the chance for a 1-1 draw is 11.53 % (0.3202*0.3614). However, if we want to predict odds on the “draw”, rather than on the individual draw outcomes, then we have to calculate the probability for all of the different scorelines: 0-0, 1-1, 2-2, 3-3, 4-4, 5-5 etc. and then add them together. In this way, we will get the chance of draw occurring, regardless of the score.

The number of draw possibilities is infinite, but the chances of a draw above 5-5 are so small and therefore we can safely not take them into consideration for this model.

In the example of Tottenham vs. Everton, the probability when we combine all of the draws is 0.2472 or 24.72% – this would give true odds of 4.05 (1/0.2472).

## What are the disadvantages of Poisson Distribution

Situational factors, such as club circumstances, game status etc. and the change of each team during the transfer window are factors which are completely ignored in the Poisson Distribution method. Therefore, the new Everton’s manager, Ronald Koeman, was not taken into consideration and thus this method failed to measure the extent to which he might have influenced the team. Furthermore, as Europa League fixture was close, it also did not consider that Tottenham players might be fatigued. Other factors which are ignored when using this method are the pitch effect where certain matches tend to be either high or low scoring.

What is also important to mention is that the odds that are given with this method do not factor in the margin a bookmaker charges and they are especially important because they are used to find the value of the bet.