The sample space $S$ can be written as

As we see there are $|S|=36$ elements in $S$. To find probability of $A_{1}$ all we need to do is find $M=|A|$. In particular, $A$ is defined as

$$

\begin{aligned}

A &=\left\{\left(X_{1}, X_{2}\right) \mid X_{1}+X_{2}=8, X_{1}, X_{2} \in\{1,2, \cdots, 6\}\right\} \\

&=\{(2,6),(3,5),(4,4),(5,3),(6,2)\} .

\end{aligned}

$$

Thus, $|A|=5$, which means that

$$

P(A)=\frac{|A|}{|S|}=\frac{5}{36}

$$

A very common mistake is not distinguishing between, say $(2,6)$ and $(6,2) .$ It is important to note that these are two different outcomes: $(2,6)$ means that the first roll is a 2 and the second roll is a 6, while $(6,2)$ means that the first roll is a 6 and the second roll is a $2 .$ Note that it is very common to write $P\left(X_{1}+X_{2}=8\right)$ when referring to $P(A)$ as defined above. In fact, $X_{1}$ and $X_{2}$ are examples of random variables.