Q.Two dice are thrown together. Write the event $E =$ 'the sum of the numbers is $8$' as a subset of the sample space, and find $n(E)$.

[!TLDR] List every ordered pair of die-faces summing to $8$. [!ANSWER] $E = \{(2,6),(3,5),(4,4),(5,3),(6,2)\}$, $n(E)=5$.

An outcome of the two-dice experiment is an ordered pair $(x,y)$ with $x, y \in \{1,\dots,6\}$. We need every pair with $x + y = 8$: if $x=2$, $y=6$; if $x=3$, $y=5$; if $x=4$, $y=4$; if $x=5$, $y=3$; if $x=6$, $y=2$. ($x=1$ would need $y=7$, and $x=7$ is impossible, since $y$ must lie between $1$ and $6$; checking all $x$ from $1$ to $6$ confirms exactly these $5$ pairs work.) [!ANSWER] $E = \{(2,6),(3,5),(4,4),(5,3),(6,2)\}$, $n(E) = 5$.

For each value of the first die $x$ from $1$ to $6$, check whether $y = 8-x$ is a valid die face (between $1$ and $6$); keep only the valid pairs.

A common mistake is missing the pair $(4,4)$ (thinking the two dice must show different numbers) or double-listing a pair — always check systematically for $x=1$ through $6$, not just a few cases.