## - How to use Poisson Distribution to predict soccer scores

## - Using Defence Strength & Attack Strength values

## - Calculate the most likely score-line

## - Converting estimated chance into odds

Poisson Distribution, when combined with historical match data, offers a straightforward yet effective strategy for predicting the most likely final score in soccer games, making it particularly useful for betting purposes. This guide provides a step-by-step method to compute essential Attack and Defence Strength metrics, along with a convenient approach for producing Poisson Distribution figures. Mastering this technique will enable you to forecast soccer match scores through Poisson Distribution quickly.

The Poisson Distribution is a statistical method that translates mean averages into probabilities of varied outcomes within a distribution. Taking Manchester City as an example, if their average goals per match is 1.7, the Poisson Distribution formula can predict the likelihood of different scoring outcomes. According to the formula, there's an 18.3% chance Manchester City will score no goals, a 31% chance they will score one goal, a 26.4% chance for two goals, and a 15% chance they will net three goals.

# Poisson Distribution - Calculating score-line probabilities

To employ Poisson for determining the most probable score-line of a game, we first need to ascertain the average goals likely to be scored by each team. This involves calculating each team's "Attack Strength" and "Defence Strength" and then comparing these figures.

Understanding how to derive outcome probabilities allows you to contrast your findings with bookmaker odds, potentially identifying value bets. Choosing an appropriate data set is critical in calculating Attack and Defence Strengths. Opting for too long a period might make the data less reflective of the team's current form, whereas a shorter span could let anomalies distort the results. The 38 matches each team played during the 2015/16 English Premier League season offer a robust data set for applying Poisson Distribution.

# How to calculate Attack Strength

The initial step towards establishing Attack Strength based on the prior season's outcomes is to figure out the average goals scored by each team, both at home and away.

This is done by dividing the total goals scored in the previous season by the number of matches played:

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

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

For the 2015/16 English Premier League season, this resulted in 567/380 for home and 459/380 for away matches, giving an average of 1.492 goals per home game and 1.207 goals per away game.

Average number of goals scored at home: 1.492

Average number of goals scored away: 1.207

A team's "Attack Strength" is then defined by the ratio of its average to the league's average.

# How to calculate Defense Strength

Calculating the average goals conceded by an average team is necessary, essentially the reverse of the previously mentioned figures (since the goals scored by the home team equal the goals conceded by the away team):

Average number of goals conceded at home: 1.207

Average number of goals conceded away from home: 1.492

The "Defence Strength" of a team is determined by the ratio of its average to the league's average.

Using the calculated figures, we can now evaluate the Attack Strength and Defence Strength for both Tottenham Hotspur and Everton as of 1st March 2017.

# Predicting Tottenham Hotspur’s goals

To calculate Tottenham’s Attack Strength:

Step - 1: Compute the home goals scored last season by Tottenham (35) divided by the number of home matches (35/19), which equals 1.842.

Step - 2: This number divided by the season's average home goals (1.842/1.492) yields an "Attack Strength" of 1.235.

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

Calculate Everton’s Defence Strength:

Step - 1: Calculate the away goals conceded last season by Everton (25) divided by the number of away matches (25/19), giving 1.315.

Step - 2: This figure divided by the season's average away goals conceded (1.315/1.492) results in a "Defence Strength" of 0.881.

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

We can now employ the formula below to estimate the number of goals Tottenham is likely to score. This is achieved by multiplying Tottenham's Attack Strength with Everton's Defence Strength and the league's average home goals:

**1.235 x 0.881 x 1.492 = 1.623**

# Predicting Everton’s goals

To figure out Everton's potential goal tally, we simply apply the previously mentioned formulae, swapping the average home goal figures with those for away matches.

**Everton's Attack Strength:**

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

**Tottenham's Defence Strength:**

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

Similarly, as we calculated Tottenham's scoring prospects, we can ascertain Everton's expected goals (achieved by multiplying Everton's Offensive Strength by Tottenham's Defensive Robustness and the league's average for away goals):

**1.046 x 0.653 x 1.207 = 0.824**

# Poisson Distribution – Predicting multiple outcomes

It's clear that matches don't conclude with scores like 1.623 versus 0.824; these figures represent averages. The Poisson Distribution, devised by the French mathematician Simeon Denis Poisson, uses these averages to apportion probabilities across different scoring possibilities for each team.

**Poisson Distribution formula:**

**P(x; μ) = (e-μ) (μx) / x!**

Nonetheless, for simplification, one can utilize Poisson Distribution Calculators available online to perform the complex part of the computation.

Simply input the range of possible events - for our scenario, goal counts ranging from 0-5 - and the expected frequency of these events, which in our case is Tottenham's average success rate of 1.623 and Everton's 0.824. The calculator then provides the probability for each score 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 calculation indicates Tottenham has a 19.73% likelihood of not scoring, but a 32.02% chance of scoring once and a 25.99% chance of scoring twice. Conversely, Everton's probabilities are 43.86% for not scoring, 36.14% for one goal, and 14.89% for two goals.

Anticipating either team to score five times? There's a 1.85% chance for Tottenham and a 0.14% chance for Everton, equating to a 2% chance for either team to hit the five-goal mark.

Since these probabilities are independent, the expected match outcome is 1–0, matching the most likely score for each team. Combining these probabilities gives a 14.04% chance for a 1-0 outcome - (0.3202*0.4386) = 0.1404.

With the ability to calculate scoring probabilities using Poisson Distribution for betting, comparing these estimates with bookmaker odds can reveal discrepancies worth exploiting, especially when considering key factors such as weather, injuries, or home-field advantage (HFA).

# Converting estimated chance into odds

Our example calculates an 11.53% likelihood for a 1-1 draw when applying the Poisson Distribution. To find the odds for a "draw" outcome rather than specific scores, compile the probabilities for all potential draw scenarios - 0-0, 1-1, up to 5-5, etc.

Summing the probabilities of all draw outcomes provides the overall likelihood of a draw, which can then be converted into odds for comparison with those of bookmakers, identifying potential value bets. While theoretically infinite draw scenarios exist, outcomes beyond 5-5 are exceedingly rare and can be disregarded for practical purposes.

In the Tottenham vs. Everton case, adding all draw probabilities yields a 24.72% chance or true odds of 4.05 (1/0.2472).

# The limits of Poisson Distribution

While Poisson Distribution offers a basic predictive framework, it doesn't accommodate numerous influencing factors. It overlooks situational aspects such as team conditions, match status, etc., and doesn't consider changes in team dynamics during the transfer period.

For example, this model doesn't account for the impact of Everton's new manager (Ronald Koeman) or potential Tottenham fatigue from playing near a Europa League match.

It also neglects correlations, such as the known effect of pitch conditions influencing high or low scoring matches.

These considerations become even more crucial in lower league matches, where such insights can provide bettors with an advantage over bookmakers. Gaining an edge in top-tier leagues like the Premier League is challenging due to the sophistication and resources at bookmakers' disposal.

Furthermore, these calculations do not include the bookmaker's margin, which is vital for identifying true value in betting odds.

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