How football betting odds work using Poisson distribution

Toward the end of the 19th century, Russia-born Polish economist, Ladislaus Bortkiewicz came up with a strategy to predict the incidence of deaths among Prussian soldiers from horse kicks.

And he did this how? He applied the Poisson distribution. It ended becoming a famous example by the way.

Fast forward a bit…

Poisson distribution can be used in many scenarios—importantly and interestingly in betting.

Sports betting is a global phenomenon, and it is estimated that this industry is worth between $700bn and $1tn globally.

And football betting is most popular among all sports.

But how does football betting odds work? how are football betting odds calculated?

It’s difficult to believe that a simple mathematical equation – Poisson distribution is used to calculate the odds for a football match.

Betting on a team winning or losing is done based on the calculation explaining the sports betting across a globe.

What is ‘Football betting odds’ and how they define bets?

If you’ve ever tried placing a few pounds on your favorite team, you would have noticed these confusing numbers in front of you.

These numbers are called ‘odds’ and they define the probabilities of each possible outcome in an event.

The higher the value of these numbers, the less probable that particular event is.

Take the Real Madrid vs Roma match above as an example.

Since the probability of Madrid winning is higher, the odds against them winning is just 1.40. A draw, which is more unlikely, has odds of 4.75.

And the odds of a highly unlikely Roma win in 7.00.

How do these numbers impact your bet amount and their returns?

Simple. In the above example, if you bet 1 pound on a Real Madrid win and win the bet, you get back a total of 1.40 pounds (inclusive of the 1 pound that you originally bet).

Which is why winning bets on more unlikely events (like a Roma victory) gets you bigger returns.

In this case, you would have got back 7 pounds for every pound you bet on Roma if they ended up winning.

Let us now see how do bookmakers calculate football betting odds using a simple Poisson Distribution equation.

What is Poisson distribution?

“Poisson distribution is the probability of the number of events that occur in a given interval when the expected number of events is known and the events occur independently of one another.”

For instance, suppose you sit in a park for a few days and count the number of people who come to the park in a black T-shirt.

Using Poisson distribution, you can guess if the number of people coming to the park on a specific day in a black t-shirt will be 10, 11, etc.

But how does it relate to football betting odds prediction?

If you are able to calculate the average attack and defense strength of the teams in a match over a certain period and calculate the Poisson distribution, you will be able to predict the odds of one team performing over other.

But if the data is too long the data would be irrelevant, and if it’s short, outliers might skew the data.

This means that not only external factors like transfers, home, and away from ground affect the odds, but also the duration of events which need to be taken into consideration for calculation.

Let’s see how football betting odds work using Poisson distribution

Before we apply Poisson, we need to get some mathematical figures.

Let’s use this method to calculate the odds for the Manchester United vs. Manchester City matches to be played on February 26, 2017.

First,  we need to find the attack and defense strength of these teams.

Calculating Average Attack and Defense for prediction

Before we identify a particular team strength and weakness, we need to find the average strength and weakness of all teams in the last playing season.

This can be calculated by dividing the total goals scored in particular season by a total number of games played in a particular season.

• Number of Goals Scored at Home / Number of Games  = 203/380 = 0.534
• Number of Goals Scored Away / Number of Games  = 158/380 = 0.415

The difference from the values above gives the “Attack intensity” of a team.

We will also need an average of goals conceded to know the weakness, which is simply the inverse of goals scored.

The average number of goals conceded at Home = 0.415

The average number of goals conceded away    = 0.534

This gives us the “ Defense intensity” of a team.

Now that we have the average strength and weakness of the teams, let’s take a look at the stats for Manchester United and Manchester City in 2015.

Based on these stats, we can calculate the Poisson distribution for the teams playing in February 2017, where Manchester United is the away team and Manchester City is the home team.

Predicting Manchester United Goals

Let’s see the possibility of Manchester United scoring at Manchester City’s home ground.

Number of Goals scored away = 22/19 = 1.15

Manchester United Goals = Number of goals scored/Season’s average goals scored away = 1.36/0.415 = 2.77

Manchester City Defense at home

This is calculated by dividing the number of goals conceded at home in the last season by the home team by the number of away games, which is  1.105 ( (21/19).

Manchester City Goals = Number of goals conceded/ Season’s average goals conceded = 1.105/0.415 = 2.66

Manchester United’s Goals = Manchester United’s Attack? Manchester City Defense? Average Number of Goals = 3.05

Predicting Manchester City Goals

Manchester City Attack at Home

Manchester City = Number of Goals scored home – 47/19 = 2.47

Manchester City Goals = Number of goals scored/Average goals scored in the last season =2.47/0.534 = 4.62

Manchester United Defense away

Take the number of goals conceded away last season by the away team and divide by the number of away games, which is equal to  1.05 ((20/19).

Manchester United Goals = Number of goals conceded/Average goals conceded in the last season =1.05/0.534 =1.966

Manchester City’s Goals = Manchester City’s Attack? Manchester United’s Defense? Average No. Goals = 1.10

Once we have the averages of Manchester United’s goals and Manchester City’s goals, we can use them to calculate the Poisson distribution for a number of goals scored by a particular team .  (various goals possibilities).

Poisson Distribution betting – Predicting multiple match outcomes

The formula for Poisson distribution is:

P(x; ?) = (e-?) (?x) / x!

e = Euler’s constant = 2.718

! = Factorial

x = number of successes of the event

µ = mean distribution of the event

It can be coded as follows

#include <stdio.h>
#include <math.h>
double factorial(int n) {
  int fact = 1;
  for (int i = 1; i<=n; i++) fact *= i;
  return fact;
}
float poisson(int r, float mean) {
  return (exp((-1) * mean) * pow(mean,r))/factorial(r);
}
int main(int argc, char const *argv[])
{
  printf("%f\n", poisson(1, 1.10));
  return 0;
}

You can also use the Poisson Distribution Calculator for such equations.

Poisson Distribution prediction for match the between Manchester United and Manchester City

The possibility that Manchester United and Manchester City would score 1 goal each is 14.44% and 36.61% respectively.

From the above distribution table we can see that the possibility of Manchester city scoring no goal is 4.73 and that of Manchester United is 33.28.

Scoring 2 goals, each for Manchester United and Manchester City is 22.02% and 20.13% respectively.

The possibilities decrease as the number of goals increases to 4 and 5 goals by the individual team.

Taking these number into consideration, the odds of Manchester City winning is high if the number of goals in a match is 0 or 1. But if Manchester United is able to score 2 goals it’s the probability of winning the match increases. 

So how is football betting odds made?

Based on the number by distribution table and the probability of these goals in a match, it is clear that a 1-1 draw has the highest possibility of 5.28, followed by 4.43 for a 2-2 draw. But the possibility of Manchester City winning with 1-0 or 2-1 also looks great.

Based on the previous match at Old Trafford, which is the home ground to Manchester United, Manchester City won by 2-1.

Not taking any sides (I support Algorithm), I would put my bet on 1-0 or 1-1 draw in favor of Manchester City.

The disadvantage of using the Poisson equation is that it doesn’t take into consideration external factors like the players/coach changed in transfer windows, the home ground factor, and injured players.

It helps you calculate the distribution only based on the averages of previous occurrences.

But then Elihu Feustel is one such person who makes a million dollars by betting using mathematical algorithms.   

Arpit Mishra Author 49 Blogs