Probability of final outcome of a game of two halves

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.

• "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."—shouldn't the final score be 3-2? ... and ... "Similar logic applies where the final score is 8-8."—are you taking $N=4$? – Greg Martin Aug 24 '18 at 16:14
• @GregMartin yes ! Sorry, I will update the questions... – Joe King Aug 24 '18 at 17:53

Since the scoring for $A$ and $B$ are independent, we can treat them independently, and then multiply the results after.

For $P(A)$, the total probability of getting score $M$ across the board is $$\sum\limits_{n=0}^M P_1^A(n)\cdot P_2^A(M-n)$$

So the total answer is $$\bigg(\sum\limits_{n=0}^M P_1^A(n)\cdot P_2^A(M-n) \bigg)\cdot \bigg(\sum\limits_{n=0}^M P_1^B(n)\cdot P_2^B(M-n)\bigg)$$

Note that if $n>N$, then $P_1^A(n)=P_2^A(n)=P_1^B(n)=P_2^B(n)=0$. This should address @JoeKing's concerns.

• Thank you, this looks great, but I perhaps haven't understood it properly. If we let $N=2$ and set all probabilities equal for each half, and we want the probability for the final score P(0-0), doesn't this give (0.25 x 0.25) x (0.25 x 0.25) which doesn't seem right ? Also, surely the formula should include $N$ ? – Joe King Aug 24 '18 at 19:16
• @JoeKing It shouldn't matter, since we would obviously set the probability of $n>N$ to be 0, as in my edit. Additionally, where did you get the probability of 0.25 from? This was not specified! – Don Thousand Aug 24 '18 at 19:27
• I was just using the example of N=2, so that there are 4 possible outcomes in each half, so that the probability of each outcome in each half is 0.25 (same for the 2nd half) – Joe King Aug 24 '18 at 19:33
• @JoeKing There are only three possibilities...0,1,2...I don't understand – Don Thousand Aug 24 '18 at 19:34
• I'm sure it's just me beeing stupid, but I though like this. N=1, so first half results can be one of 0-0, 0-1, 1-0, and 1-1, and I am giving equal probablity (0,25 each) to them to keep it simple. I must be just mising something obvious !# – Joe King Aug 24 '18 at 19:36

Let $P_1(K,L)$ be the probability that team A scores $K$ in the the first half and team B scores $L$ in the first half; then we have the generating function identity $$\sum_{K=0}^N \sum_{L=0}^N P_1(K,L) x^K y^L = \bigg( \sum_{K=0}^N P_1^A(K) x^K \bigg) \bigg( \sum_{L=0}^N P_1^B(L) y^L \bigg)$$ (which is just a fancy way of saying that $P_1(K,L) = P_1^A(K) P_1^B(L)$). A similar formula applies for $P_2(K,L)$, the probability that team A scores $K$ in the the second half and team B scores $L$ in the second half.

Since the total scores are the sum of these two subscores, the corresponding generating functions multiply. In other words, if $P(S,T)$ is the probability that team A scores $S$ in total and team B scores $T$ in total, then \begin{align*} \sum_{S=0}^{2N} \sum_{T=0}^{2N} P(S,T) x^S y^T &= \bigg( \sum_{K=0}^N \sum_{L=0}^N P_1(K,L) x^K y^L \bigg) \bigg( \sum_{K=0}^N \sum_{L=0}^N P_2(K,L) x^K y^L \bigg) \\ &= \bigg( \sum_{K=0}^N P_1^A(K) x^K \bigg) \bigg( \sum_{K=0}^N P_2^A(K) x^K \bigg) \bigg( \sum_{L=0}^N P_1^B(L) y^L \bigg) \bigg( \sum_{L=0}^N P_2^B(L) y^L \bigg). \end{align*} One simply has to expand the polynomial on the right-hand side to determine any particular coefficient $P(S,T)$.

Of course, this gives the formula from Rushabh Mehta's answer. But this is probably an easier calculation to code in some languages, especially if you want many probabilities $P(S,T)$; and it always helps to understand where these formulas come from and how helpful generating functions can be.

• Generating Functions are the bomb in counting questions – Don Thousand Aug 24 '18 at 16:58
• Aaaaaah Generating Functions ! It seems like this is what I have been missing. Thank you ! – Joe King Aug 24 '18 at 17:57
• I am slightly confused about how to apply these formula (with x and y). I might be reminded of the binomial theorem form a previous life (is there a connection? I feel like I am on the cusp of "getting it" but there is just that breakthrough needed hah – Joe King Aug 24 '18 at 19:19