A total of over 200k games were downloaded from Soccerway, and the probability of transitioning from one score to an other were empirically calculated.
This gives the following representation of the stochastic matrix. The size of each box is proportional to the probability of staying in place.
The results are given in the form [Home score]-[Away score].
It is interesting to note that it the system is not memory free, i.e. the probability of getting from one result to the other does depend on the path one takes.
A mere example for that would be 1-1: if the home team scores the first goal, the probability to go to 2-1 is 36.74%; but if the away team scores first, the probability raises to 38.31%.