Consider a game which is played in 2 halves between two teams, A and B. The outcome of the 2nd half of the game is independent of the outcome of the 1st half.
Each team can score between 0 and N points in each half, and the probability of scoring exactly M is known and equal to $P_1^A(M)$ for team A and $P_1^B(M)$ for team B in the first half, and $P_2^A(M)$ and $P_2^B(M)$ respectively in the 2nd half.
The final score is the sum of each teams' scores. So, if team A scores once in the first half and twice in the 2nd, while team B score twice in the first and not at all in the 2nd half, then the final score will be 2-2. Thus the lowest possible scoring match is 0-0 and this an only occur when the first half is 0-0 and the 2nd half is 0-0. Similar logic applies where, if $N=4$ for example, the final score is 8-8
However, there are many possible combinations for final scores between 0-0 and 8-8. I wish to compute the probabilities for every possible final score.
I know that there are $(N+1)^4$ possible combinations in total, but the only way I have been able to work out the probabilities of each combination is by hand in a spreadsheet with actual numbers. For example, I know when N=4 there are 10 possible ways of making a final score of 7-4 (which is Not the same as 4-7 obviously) can occur in 10 ways, so I just compute the product of those 10 probabilities (3-0 and 4-4, 3-1 and 4-3, 3-2 and 4-2, 3-3 and 4-1, 3-4 and 4-0, 4-0 and 3-4, 4-1 and 3-3, 4-2 and 3-2, 4-3 and 3-1, 4-4 and 3-0).
I would like to find a way to do this more easily, and feel that there should be some formulas I can apply, just from knowing $N$ and each of the underlying probabilities, but so far I didn’t find anything.
BTW this isn't homework, it's just something i am interested in.