# Poisson Distribution Betting

In this part of the Odds Compiling Guide, we explain the Poisson distribution and the way this can be used to calculate the probabilities of a Home win, Draw or Away team win.

## The Poisson Distribution

The Poisson Distribution was first introduced by the French mathematician, Simeon Denis Poisson, and took its name after him. Poisson Distribution is a probability distribution that expresses the probability of a given number of events that occur in a fixed interval of time with a known average rate, and are independent of the time since the last event.

### Using Poisson to calculate a teams total goal scoring probability

The Poisson distribution can be easily applied to calculate the probabilities of the various outcomes of each match. Using the Poisson distribution, we can find the probability of each team to score a fixed amount of goals. The probability of a team scoring x goals in a match is given by he following formula:

Where the two inputs are:

x: The number of goals we expect the team to score (the given number of event-goals)

μ: The teams Goal Expectancy (You can find a way to calculate the Goal Expectancy of each team in the previous part of the Odds Compiling Guide here)

### Poisson formula in Microsoft Excel:

Since the previous formula may seem a bit scary to many people, the good news are that Microsoft Excel has it already there for you, and you can just call it and put the parameters of each team. To call the Poisson function in excel you just have to type in the function field:

Where x= the number of goals, μ= goal expectancy, and cumulative is a logical variable that takes on either of the values TRUE or FALSE. Instead of TRUE, you can also put ! and instead of FALSE, you can put 0. When cumulative = TRUE, the function POISSON (xμ, cumulative) returns the probability that a POISSON random variable with means μ takes on a value less than or equal to x. When cumulative = FALSE, POISSON returns the probability that such a random variable takes on a value exactly equal to x. So if for example we want to find the probability of a team to score 2 goals, we call the POISSON function as follows:

For example:

where C17 and C18 are the cells where we put the Home and Away teams Goal Expectancy respectively.

For x number of goals we have:  POISSON(x, C17, 0), and we can have the following table for each teams probability to score every number of goals:

Using Poisson to calculate outcome probability of a match

Using the Poisson formula for each number of goals (0, 1, 2, 3, …, n) for each team, we can find the probabilities for each result of the match between the two teams and fix the Correct Scores table for the match. To find the probability of Home team to score x goals, and the Away team to score y goals, we have to multiply the two probabilities according to the independence assumption: