Sports betting with the Poisson distribution
The Poisson Distribution in Sports Betting is a distribution that is used to model discrete events. Events in which your results can only be whole numbers greater than or equal to 0. In a soccer match, 3.2 goals cannot be scored and 12.7 phone calls cannot be received in one hour. This does not mean that a team cannot score an average of 3.2 goals per game and an operator cannot receive an average of 12.7 calls per hour.
Instead, they are two different things. While one is the result of a certain event (a soccer match), which will always be a whole number greater than or equal to 0, the other event is classified as the average of individual results. Hence, people shouldn’t confuse the two events.
They have proved that, statistically, the Poisson distribution fits relatively well to predict the number of goals scored in hockey games. In the case of soccer, the adjustment worsens a bit. Many authors indicate that to improve their predictions a correction should be made for results of 0 goals and even 1 goal. For this example, we will use the distribution without any adjustment. So, let’s learn more about Poisson Distribution in Sports Betting Formula!
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Table of content:
Its formula is the following:
Prob (B = k) = exp (-m) xm ^ k / k!
Where m is the mean and k is the number of events for which we are calculating the probability. And this equation is read as the probability that k events appear in an event with mean of occurrences m.
Let’s see an example:
Let’s suppose that Botijos FC has played 20 matches and scored 28 goals. What is the probability that in the next game he will score exactly 1 goal?
The average of goals per game is 28/20 = 1.4 goals per game.
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Now we have two alternatives:
- Go to excel and place = Poisson (1; 1.4; False) = 0.345 (this function is for Excel In Spanish, for those with the English configuration, the parentheses would remain, (1,1.4, False).
- P (B = 1) = exp (-1.4) x 1.4 ^ 1/1 ! = exp (-1.4) x 1.4 = 0.345
That is, the probability that Botijos FC scores a goal is 34.5%
If what we want to know is the probability that Botijos will score at least one goal. It can also be done in two ways. Nevertheless, we’ll focus on Excel which is the fastest, meaning that we must use the complementary event. It is like this:
- P (B> = 1) = 1 – P (B = 0)
- P (B> = 1) = 1 – Poisson (0; 1,4 ; False) = 1 – 0.25 = 0.75 = 75%
Let us now see its application to the result of a match. What is the probability that the game will be under 1.5 if Botijos plays against Chinchorro FC, knowing that he scores 2 goals per game on average?
To calculate this, the first thing we must do is obtain the set of results that meet under 1.5. They are 3:
0-0, 0-1, 1-0
We now calculate the probability of each of them and the probability of the under will be the sum of all of them.
P (B = 0, C = 0) = Probability that Botijos will score 0 goals x Probability that Chinchorro will score 0 goals = Poisson (0; 1.4; False) x Poisson (0; 2; False) = 0.25 x 0.135 = 0.033
That is, the probability of 0-0 is 3.3%.
P (B = 0, C = 1) = Poisson (0; 1.4; False) x Poisson (1; 2; False)
= 0.25 x 0.41
= 0.1
· P (B = 1, C = 0) = Poisson (1; 1.4; False) x Poisson (0; 2; False)
= 0.59 x 0.135
= 0.08
P (Under 1.5) = 0.033 + 0.1 + 0.08 = 0.213 = 21.3%
Applying the same reasoning of the complementary event we would have that the probability of over 1.5 would be 1-0.213 = 0.787 = 78.7%
Following this same procedure, all possible match outcomes can be modelled to determine the odds of the home team winning. It can be a draw, the away team winning, and even calculating the probability of other Over/Underlines, such as 2.5.
In these cases, the probability of the individual result of 0, 1, 2, 3, 4, 5, and more than 5 goals is usually calculated. To calculate the latter, it’s also done by the complementary event.
P (B> 5) = 1 – P (B <= 5) and P (B <= 5) is calculated in Excel in a very simple way:
P (B <= 5) = Poisson (5; mean; TRUE)
The true of the last part of the formula indicates that it is a cumulative probability, from 0 to 5. Just what we need.
