How can I convert expected goals for two sports teams into win/lose/draw probabilities?

Let's assume we have a model to find the expected goals scored by two teams facing each other -

Team A expect goals: 2.2

Team B expected goals: 4.6

So we can see that Team B has a higher probability of winning the game. But just what is this win percentage and how do we work it out? And how do we work out the probability that both teams will score the same amount of goals and tie the game?

Thank you very much for any help! I'mm sure there are some concepts/models out there that I'm not familiar with and I'd be very grateful for a point in the right direction.

If we model the number of goals as independent Poisson distributions, e.g., $A\sim\text{Pois}(\lambda=2.2)$ and $B\sim\text{Pois}(\lambda=4.6)$, then we are interested in
$$P(A<B), \quad P(A=B), \quad P(A>B).$$
$$P(A-B<0), \quad P(A-B=0), \quad P(A-B>0).$$