Incorporate luck in statistical modelling I was wondering if it is possible (and if yes, how is it done) to incorporate a luck component in statistical models. 

So let’s assume I’d perform a regression on the goal difference between two teams in order to model the outcome of a sports match. How could I also model a random luck component that would make the underdog draw or win in certain rare cases?

And what if I took it a step further and tried to model the match outcome using neural networks? Is is possible to include luck in this case?
Use a probabilistic model. For example, if the probability of one team beating the other is $σ(w⋅(team1−team2))$, where $σ(⋅)$ is the logistic function, then there's always a probability that the weaker team will win. 

If you wanted to use a neural network you'd use it to replace the logistic function.

Answer copied from comments.