Let's roll $5$ white dice and $5$ red dice and calculate the number $S = \text{(sum of the while dice)} - \text{(sum of the red dice)}$. What is the probability for this sum to be $0$ and what is the probability for this sum to be $1$.
I found this post: Sum of $5$ dice: Number of solutions diophantine equation
So basically, I understood how to calculate the probality for the sum of the $5$ dice to be a certain number, using moments generating functions and the formula $\frac{1}{1-x} = \sum\limits_{k = 0}^{\infty} x^{k}$, for $|x| < 1$.
In theory I can solve the current problem using the formula for conditional probability. If $X$ is the sum of the white dice and $Y$ is the sum of the red dice, then $P(X - Y = 0) = P(X = 5) P(Y = 5) + P(X = 6) P(Y = 6) + \dots + P(X = 30) P(Y = 30)$. However, these are too many calculations. Is there a simpler solution, that I'm missing? Please, give me a hint! Thanks!
For the sum of five dice, a roll of $k$ is just as likely as a roll of $35-k$, for any $k\in \mathbb Z$. Therefore, \begin{aligned} P(\text{white $-$ red}=0) &=\sum_{k=5}^{30} P(\text{white }=k)P(\text{red }=k) \\&=\sum_{k=5}^{30} P(\text{white }=k)P(\text{red }=35-k) \\&=P(\text{white + red }=35) \end{aligned} So you just need to find the probability that the sum of $10$ dice is $35$, which you know how to do.
Similarly, \begin{aligned} P(\text{white $-$ red}=1) &=\sum_{k=5}^{29} P(\text{white }=k+1)P(\text{red }=k) \\&=\underline{\hspace{2cm}}? \\&=\underline{\hspace{2cm}}? \end{aligned}