Using Poisson Distribution to predict a Soccer Betting Winner By SYED AHMER RIZVI 1511060 Section A Quantitative Methods - I
APPLICATION OF DESCRIPTIVE STATISTICS AND PROBABILITY IN SOCCER Concept This article was published by the popular sports bookmaker Pinnacle Sports. It details the use of Poisson distribution to work with data sets from past events i.e. Average Number of goals scored in a match by a team during the past and current English Premier League Seasons to calculate the likely number of goals that will be scored by the same team in the upcoming matches. This concept forms the basic model behind the football sportsbook rates offered by online betting giants such as 365.com and wbx.com. For example Manchester United might average 1.7 goals per game in the last season. Entering this data as Expected Value/Mean into a Poisson formula would show that this average equates to Manchester United 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. Application methodology Let s assume Team 1 is playing the match at its home stadium The method used to come up with the likely number of goals for a particular game is as follows: For Team 1 Team 1 s Goals = {Team 1 s Offence} X {Team 2 s Defense} X {Average Goals/Game by any club} For Team 2 Team 2 s Goals = {Team 2 s Offence) X {Team 1 s Defense} X {Average Goals/Game by any club} Where Team 1 s Offence = {Number of Goals Scored at Home Last Season / (Number of Home Matches Last Season) X (Average Goals scored/game at Home last Season by any club)} Team 2 s Offence = {Number of Goals Scored Away Last Season / (Number of Away Matches Last Season) X (Average Goals scored/game Away last Season by any club)} Team 1 s Defense = {Number of Goals conceded at Home Last Season / (Number of Home Matches Last Season) X (Average Goals conceded/game at Home last Season by any club)} Team 2 s Defense = {Number of Goals conceded Away Last Season / (Number of Away Matches Last Season) X (Average Goals conceded/game at Home last Season by any club)} The Last Step is to use the Poisson distribution Formula to calculate the Betting/Goals Table. P(x; μ) = (e-μ) (μx) / x! Where μ = Average Goals / Game X = Different goals outcomes (0-5) in the Random Variable (x) category 1
Let s assume Team 1 s (Expected Value) = 1.654 Goals/Game and Team 2 s Goals (Expected value) = 1.278 Goals/Game. The below embedded excel sheet can be used to find the Probability of the number of goals scored by each team. For example, we want to look at chances of the match being a 2 2 Draw, we can do that as: Probability (2 2 Draw) = Probability (Team 1 s Goals = 2) X Probability (Team 2 s Goals = 2) Since we are assuming Team 1 s Goals and Team 2 s Goals are independent events = 0.2616 X 0.2275 = 5.95 % This also implies that in-case you place a bet on the final score line being 2-2, you have a probability of 5.95% of winning the bet. Similarly probabilities of all possible score-lines can be calculated. Please click on the excel sheet to check the formulas. Goal Likelihood Table Team 1's Goals 1.654 Team 2's Goals 1.278 Goals 0 1 2 3 4 5 Team 1 19.13% 31.64% 26.16% 14.43% 5.96% 1.97% Team 2 27.86% 35.60% 22.75% 9.69% 3.10% 0.79% Note: The values of Mean 1 and Mean 2 can be changed in respective cells. 2
CHOICE OF TOPIC The two main reasons for choosing the topic Using Poisson Distribution to predict Soccer Betting Winner are: Personal Interest Being a football fanatic and having followed the English Premier League religiously for the past 8 or 9 years, I was always aware that statistics plays a major part in the opinions shared by football pundits, but had never looked into the topic in detail. Therefore it was quite interesting for me to look into the nuances of how Poisson s Distribution can be used to predict matches on the basis of a single parameter μ. Non Routine Application of Probability/Statistics The concepts used to demonstrate probability in undergraduate / school level courses usually involve dices, cards and colored balls. Although these ideas help in developing a basic grasp of the concepts, the application of Probability to real life situations/industries is a new concept for most of the PGP I students at IIM Bangalore. In the recent case study discussions in QM I classes, we have looked at several sectors such as manufacturing, healthcare and others to understand the role of descriptive statistics in business. One such sector that is usually cordoned off and not brought up for discussion because of its gray nature is Betting, although the illegal betting/gambling industry in India is worth 60 Billion USD and is growing exponentially. One major subdivision of this industry is Sports betting. It involves prediction of sports results and placing wagers on the outcome with the bookmaker. This activity is legal in most parts of the western world with places in Asia such as Macau and Hong Kong following the trend. We also often read about the scale of gambling involved in IPL i.e. India s richest sports league. CRITIQUE OF THE METHODOLOGY & ALTERNATIVE METHOD The model fails to recognize the relation often seen between Score Line and Extraneous Factors such as Pitch Effect or the X Factor of the new manager. These factors play a major part in the score line and the model would be not accurate without their inclusion. For example, a densely water soaked pitch prevents many goal scoring opportunities and hence brings down the average score line. To include the effect of these factors, I would recommend the use of Conditional Probability. Let s take the case of rain. A rainy weather condition is unfavorable for long through ball strategy i.e the most utilized tactic in offense in English football and therefore hinders the attacking capabilities of a team. Let Event R represent Heavy Rain. Let Event A represent Team 1 s Goals = 2 Let Event B represent Team 2 s Goals = 2 Let us assume that Rain decreases the chances of a team scoring N goals by 20N% 3
Therefore using the original data let us look at the probability of a 2-2 draw in case there is rain. Probability (2-2 Rain ) = Probability (Team 1 s Goals = 2 Rain ) X Probability (Team 2 s Goals = 2 Rain ) = P (A R) X P (B R) Now based on our assumption we can say N = 2 and therefore, P (A R) = P(A) X [1 {(20 * 2)/100}] & P (B R) = P(B) X [1 {(20 * 2)/100}] P (A R) = 0.2616 * 0.6 = 0.1569 & P (B R) = 0.2275 * 0.6 = 0.1365 Probability (2-2 Rain ) = Probability (Team 1 s Goals = 2 Rain ) X Probability (Team 2 s Goals = 2 Rain ) = 0.1569 * 0.1365 = 2.14 % As we can see by taking external factors into consideration, the probability of a 2-2 score-line reduces considerably. This has an important implication in soccer betting. When the number of external factors in consideration are large, it is very difficult to come up with a predication of an exact score-line with any level of confidence. Therefore keep your MONEY SAFE and AVOID GAMBLING. APPENDIX 1 ARTICLE http://www.pinnaclesports.com/en/betting-articles/soccer/how-to-calculate-poisson-distribution 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. Bettors will find this simple method of how to calculate the likely outcome of a soccer match using Poisson Distribution very useful. Poisson Distribution explained Poisson Distribution is a mathematical concept for translating mean averages into a probability for variable outcomes. For example, Chelsea might average 1.7 goals per game. Entering this information into a Poisson formula would show that this average equates to Chelsea 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. How to calculate soccer outcomes with Poisson Distribution 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 an Attack and Defence Strength for each team and comparing them. Selecting a representative data range is vital when calculating Attack and Defence strengths too long and the data will not be relevant for the teams current strength, while too short may allow outliers to skew the data. For this analysis we re using the 38 games played by each team in the 2013/14 EPL season. 4
Calculating Attack and Defence strengths Calculate the average goals scored at home and away The first step in calculating Attack and Defence strengths based upon last season s results is to determine the average number of goals scored per team, per home game, and per away games. Calculate this by taking the total number of goals scored last season and dividing it by the number of games played: Season Goals Scored at Home / Number of Games (in season) Season Goals Scored Away / Number of Games (in season) In 2013/14, that was 598/380 at home and 454/380 away, equalling an average of 1.574 goals per game at home and 1.195 away. Average number of goals scored at home: 1.574 Average number of goals scored away from home: 1.195 The difference from the above average is what constitutes a team s Attack 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: 1.195 Average number of goals conceded away from home: 1.574 We can now use the numbers above to calculate the Attack and Defence Strength of both Manchester United and Swansea City for their match on August 16th, 2014. Predicting Man United s Goals Calculate Man United s Attack Strength: 1. Take the number of goals scored at home last season by the home team (Man United: 29) and divide by the number of home games (29/19): 1.526 2. Divide this value by the season s average home goals scored per game (1.526/1.574), to get the Attack Strength : 0.970. This shows that Man United scored 3.05% fewer goals at home than a hypothetical average Premier League side last season. Calculate Swansea s Defence Strength: 1. Take the number of goals conceded away last season by the away team (Swansea: 28) and divide by the number of away games (28/19): 1.474. 2. Divide this by the season s average goals conceded by an away team per game (1.474/1.574) to get the Defence Strength : 0.936. This therefore highlights Swansea conceded 6.35% fewer goals than an average Premier League side on the road. We can now use the following formula to calculate the likely number of goals the home team might score: 5
Man United s Goals = Man United s Attack x Swansea s Defence x Average No. Goals In this case, that s 0.970* 0.936 * 1.574, which equates to United scoring 1.429 goals. Predicting Swansea s Goals Calculate Swansea s Attack Strength: 1. Take the number of goals scored away last season by the away team (Swansea: 21) and divide by the number of away games (21/19): 1.105 2. Divide this value by the season s average away goals scored per game (1.105/1.195), to get the Attack Strength : 0.925. This shows that Swansea scored 7.53% fewer away goals than a hypothetical average Premier League side. Calculate Man United s Defence Strength: 1. Take the number of goals conceded at home last season by the home team (Man United: 21) and divide by the number of home games (21/19): 1.105. 2. Divide this by the season s average goals conceded by a home team per game (1.105/1.195) to get the Defence Strength : 0.925. Man United conceded 7.53% more goals than an average Premier League side at home. We can now use the following formula to calculate the likely number of goals the away team might score: Swansea s Goals = Swansea s Attack x Man United s Defence x Average No. Goals In this case, that s 0.925* 0.925 * 1.195, which equates to Swansea scoring 1.022 goals. Poisson Distribution betting Predicting multiple match outcomes Of course, no game ends 1.429 vs. 1.022 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. The results are shown in the table below: The formula itself looks like this: P(x; μ) = (e-μ) (μx) / x!, however, we can use online tools such as this Poisson Distribution Calculator to do most of the equation for us. All we need to do is enter the different goals outcomes (0-5) in the Random Variable (x) category, and the likelihood of a team scoring (for instance, Swansea at 1.022) in the average rate of success, and the calculator will output the probability of that score. Poisson Distribution for Man United vs. Swansea Goals 0 1 2 3 4 5 Man United 23.95% 34.23% 24.46% 11.65% 4.16% 1.19% Swansea 35.99% 36.78% 18.79% 6.40% 1.64% 0.33% This example shows that there is a 23.95% chance that Man Utd will not score, but a 34.23% chance they will get a single goal and a 24.46% chance they ll score two. 6
Swansea, on the other hand, are at 35.99% not to score, 36.78% to score one and 18.79% to score two. Hoping for a side to score five? The probability is 1.19% if United are the scorers, or 0.33% for Swansea to do it. 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 0.125 or 12.59%. Now you know how to calculate outcomes, you should compare your result to a bookmaker s odds to help see how they differentiate. Example: comparing the draw The above example showed us that a 1-1 draw has a 12.59% chance of occurring, according to our model. 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. 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. For the United Swansea game, combining all of the draws gives a probability of 0.266 or 26.6%. Pinnacle Sports odds were 5.530 (an 18.08% implied probability). Therefore if last season s form was a perfect indicator of this season s results, there would appear to be value in backing the draw, as the model shows that it more likely to happen than the Pinnacle Sports odds suggest. Unfortunately it isn t as simple as that, which is why pure Poisson analysis has limitations. 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, it means the huge x-factor of Manchester United s first Premier League game with new manager Louis Van Gaal is entirely ignored. Correlations are also ignored; such as the widely recognised pitch affect 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 punters an edge against bookmakers, while it s harder to gain an edge in major leagues, given the expertise that modern bookmakers like Pinnacle Sports possess. 7