by mattadams84
Last Updated November 17, 2017 17:19 PM

I am trying to work out the probability of teams scoring a goal in a football match. Obviously there are many factors to consider, and the probability is impossible to predict correctly as there are many factors that will change in a football match. However i want to figure out the probability using only the following information. A lot of work has been done on predicting football matches using poisson distribution, but in this case, i only want it to be based on the stats that i have.

So for this example it is **Team A** vs **Team B**.

**Team A** Scores in 70% of their home matches **(a)**
**Team B** Concedes in 50% of their away matches **(b)**

**Team B** Scores in 10% of their away matches **(c)**
**Team A** Concedes in 30% of their home matches **(d)**

I have managed to calculate the probability of a goal by using the following formula:

(a+c)-(a*c)

I believe this calculation to be correct but it only takes into account the stats for **team A** or **team B** SCORING, it does not take into account the stat of the opposite team conceding.

So basically i am after a formula that takes into consideration **team A**'s ability to score against **team B**'s ability to concede, and the same for **team B** to score and **team A** to concede.

This is a problem where mathematics can at best supply a "toy" answer, as there is so much more to sporting outcomes than simple statistics.

But this looks fun, so let's crack on with it anyway. Take everything that follows with a pinch of salt.

We have two estimates, $a$ and $b$ for **team A** scoring. Let's go with the average, $(a+b)/2=0.6$, and say that's equal to $\mathbb{P}(S_A)\ge 1$, the probability that the number of goals scored by **team A**, $S_A$, is at least 1. Let's model $S_A$ as a homogeneous Poisson process, which therefore has rate parameter $\lambda_{AB}=-\log\left(1-(a+b)/2\right)\approx 0.916$. This means we expect **team A** to score $k$ goals against **team B** with probability $e^{-\lambda_{AB}}\lambda_{AB}^k/k!$, like this:

goals | probability ------+------------ 0 | 40.0% 1 | 36.7% 2 | 16.8% 3 | 5.1% 4 | 1.2% 5 | 0.2%

Similarly, the number of goals that **team B** scores against **team A** could be modelled as Poisson with rate $\lambda_{BA}=-\log\left(1-(c+d)/2\right)\approx 0.223$, with the probabilities working out like:

goals | probability ------+------------ 0 | 80.0% 1 | 17.9% 2 | 2.0% 3 | 0.1%

Finally, we treat those independently (multiply the probabilities) to find the most likely overall scores:

score A : B | probability ------------+------------ 0 : 0 | 32.0% 1 : 0 | 29.3% 2 : 0 | 13.4% 0 : 1 | 7.1% 1 : 1 | 6.5% 3 : 0 | 4.1%

So there we are. But please don't put any money on it!

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