Poisson Prediction

To predict how many goals a team will score in a match we can use the Poisson Distrubution.  The Poisson Distribution is a mathematical formula that describes the distribution of the number of discrete random events.  It is applicable to almost any random event - for example the number of cars arriving at a red light, the number of customers entering a shop per hour, the number of goals scored in a match.  The formula is written as

P(k) = (lambda^k) * (e ^ -lambda) / k!

Where P(k) mean the Probability of k goals, Lambda is the average goals per game.  e is eulers constant (this is a known constant), k! is k factorial (eg 3! = 3*2*1=6).  It's just mathematical notation.

So we have quite a simple way of calculating the probability that a team will score 1 goal in a game, or 2 or 3.  And we can calculate likewise for their opponent.  We are then calculating the probabilities of football results!  The only parameter of the equation that we don't know is lambda.  However we can estimate this based on the average goals per match by a team.

First to test the formula we can look at the number of goals each team scored in all games in the 2013/14 season.  I am using the usual data that I have collected on the English, Spanish, German, Italian, Turkish, Scottish, Dutch, Belgian, Greek and Portuguese top divisions.  Total number of matches used is 4232, so this should be a fine sample space - enough to give a good long run average.

Poisson distribution (lambda = 1.348) of goals per team per match vs actual observed distribution.  All top europen matches for 2013/14 season.
To understand the chart, the Poisson formula predicts that a team will score 0 goals in 26% of matches, 1 goal in 35% etc.  (These predictions are based on a Poisson distribution with average of 1.348). The actual percentage of goals per match is plotted beside the predictions.  I think it show quite a good fit.
Now to make this useful we can break it down by team.  Obviously probability of scoring at home is much larger than away, so we break it down by home and away too.  Now we have narrowed it down by team and by home/away we have a much much smaller sample space.  This means the actual and predicted outcomes don't look as close.  However because of the Law of Large Numbers we would expect these margins to become smaller if we add more data.
Poisson distribution (lambda=2.211) of goals for Man Utd per home match vs actual observed distribution.  Includes seasons 11/12, 12/13 and 13/14.

So to use this to predict a match lets take Man Utd's first match of the 2014/15 season on 2014-08-16, vs Swansea.  So we look at Man Utd's home record and Swansea's away record.  Man Utd have scored an average of 2.211 goals at home and have conceded on average 1.035.  Swansea have scored an average of 1.000 goals away and conceded an average of 1.509.  These averages are taken over the last three seasons, so that is Man Utd's last 57 home games and Swansea's last 57 away games.  There is a trade off here, look back too far and you are looking back on irrelevant data, but don't look back far enough and you won't have enough matches to use to calculate a genuine long run average.
I'll average 2.211 (average scored by Man Utd at home) and 1.509 (average conceded by Swansea away) and use this as the estimate for lambda for P(h) for the home team (Man Utd).  This is an estimates for the expected number of goals scored by the home team.  I'll generate the estimate for lambda for the away team similarly.
The table below lists all possible scoreline combinations (up to 7 goals per team).  Then all that is left to do is calculate the probability of each scoreline based on the lambda estimates.  Just plug lambda into the Poisson formula and out comes the actual probabilities for each possible score.    

home goals (h)	away goals (a)	outcome	lambda (home)	lambda (away)	Probability of h home goals	Probability of a away goals	P(h) * P(a)	Sum
0	0	Draw	1.860	1.018	0.156	0.361	0.056	0.225
1	1	Draw	1.860	1.018	0.289	0.368	0.106
2	2	Draw	1.860	1.018	0.269	0.187	0.050
3	3	Draw	1.860	1.018	0.167	0.063	0.011
4	4	Draw	1.860	1.018	0.078	0.016	0.001
5	5	Draw	1.860	1.018	0.029	0.003	0.000
6	6	Draw	1.860	1.018	0.009	0.001	0.000
7	7	Draw	1.860	1.018	0.002	0.000	0.000
1	0	Man Utd	1.860	1.018	0.289	0.361	0.105	0.570
2	0	Man Utd	1.860	1.018	0.269	0.361	0.097
2	1	Man Utd	1.860	1.018	0.269	0.368	0.099
3	0	Man Utd	1.860	1.018	0.167	0.361	0.060
3	1	Man Utd	1.860	1.018	0.167	0.368	0.061
4	0	Man Utd	1.860	1.018	0.078	0.361	0.028
3	2	Man Utd	1.860	1.018	0.167	0.187	0.031
4	1	Man Utd	1.860	1.018	0.078	0.368	0.029
5	0	Man Utd	1.860	1.018	0.029	0.361	0.010
4	2	Man Utd	1.860	1.018	0.078	0.187	0.015
5	1	Man Utd	1.860	1.018	0.029	0.368	0.011
6	0	Man Utd	1.860	1.018	0.009	0.361	0.003
4	3	Man Utd	1.860	1.018	0.078	0.063	0.005
5	2	Man Utd	1.860	1.018	0.029	0.187	0.005
6	1	Man Utd	1.860	1.018	0.009	0.368	0.003
7	0	Man Utd	1.860	1.018	0.002	0.361	0.001
5	3	Man Utd	1.860	1.018	0.029	0.063	0.002
6	2	Man Utd	1.860	1.018	0.009	0.187	0.002
7	1	Man Utd	1.860	1.018	0.002	0.368	0.001
5	4	Man Utd	1.860	1.018	0.029	0.016	0.000
6	3	Man Utd	1.860	1.018	0.009	0.063	0.001
7	2	Man Utd	1.860	1.018	0.002	0.187	0.000
6	4	Man Utd	1.860	1.018	0.009	0.016	0.000
7	3	Man Utd	1.860	1.018	0.002	0.063	0.000
6	5	Man Utd	1.860	1.018	0.009	0.003	0.000
7	4	Man Utd	1.860	1.018	0.002	0.016	0.000
7	5	Man Utd	1.860	1.018	0.002	0.003	0.000
7	6	Man Utd	1.860	1.018	0.002	0.001	0.000
0	1	Swansea	1.860	1.018	0.156	0.368	0.057	0.204
0	2	Swansea	1.860	1.018	0.156	0.187	0.029
0	3	Swansea	1.860	1.018	0.156	0.063	0.010
1	2	Swansea	1.860	1.018	0.289	0.187	0.054
0	4	Swansea	1.860	1.018	0.156	0.016	0.003
1	3	Swansea	1.860	1.018	0.289	0.063	0.018
0	5	Swansea	1.860	1.018	0.156	0.003	0.001
1	4	Swansea	1.860	1.018	0.289	0.016	0.005
2	3	Swansea	1.860	1.018	0.269	0.063	0.017
0	6	Swansea	1.860	1.018	0.156	0.001	0.000
1	5	Swansea	1.860	1.018	0.289	0.003	0.001
2	4	Swansea	1.860	1.018	0.269	0.016	0.004
0	7	Swansea	1.860	1.018	0.156	0.000	0.000
1	6	Swansea	1.860	1.018	0.289	0.001	0.000
2	5	Swansea	1.860	1.018	0.269	0.003	0.001
3	4	Swansea	1.860	1.018	0.167	0.016	0.003
1	7	Swansea	1.860	1.018	0.289	0.000	0.000
2	6	Swansea	1.860	1.018	0.269	0.001	0.000
3	5	Swansea	1.860	1.018	0.167	0.003	0.001
2	7	Swansea	1.860	1.018	0.269	0.000	0.000
3	6	Swansea	1.860	1.018	0.167	0.001	0.000
4	5	Swansea	1.860	1.018	0.078	0.003	0.000
3	7	Swansea	1.860	1.018	0.167	0.000	0.000
4	6	Swansea	1.860	1.018	0.078	0.001	0.000
4	7	Swansea	1.860	1.018	0.078	0.000	0.000
5	6	Swansea	1.860	1.018	0.029	0.001	0.000
5	7	Swansea	1.860	1.018	0.029	0.000	0.000
6	7	Swansea	1.860	1.018	0.009	0.000	0.000

The probilities in the last two columns are calculated using the below basic probability laws.
P(A and B) = P(A) * P(B)
P(A or B) = P(A) + P(B) where A and B are mutually exclusive (as they are in our case).
So in percentage terms this gives 20.4% chance Swansea will win, 22.5% for a draw and a 57% chance that Man Utd will win.  (We only calculated up to 7 goals in our table above.  The other 0.1% is the probability that either teams scores more than 7 goals).