Q.Two fair dice are rolled together. Find $n(S)$ and the probability that the sum of the numbers shown on the two dice is 7.

First find the total sample space size by the multiplication principle, then list every pair summing to 7.

Each die independently shows one of 6 faces, so by the multiplication principle (§1), $n(S) = 6 \times 6 = 36$ — every ordered pair $(a,b)$ with $a,b \in \{1,\ldots,6\}$ is equally likely.

The event $E=$ 'sum is 7' includes every pair whose two numbers add to 7: $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$. Listing exhaustively confirms there are exactly $6$ such pairs, so $n(E)=6$.

By the classical definition (§5): $P(E) = \dfrac{n(E)}{n(S)} = \dfrac{6}{36} = \dfrac{1}{6}$. …

A common error is treating $(4,3)$ and $(3,4)$ as the 'same' outcome and undercounting the favourable pairs to only 3 — but the two dice are distinguishable (first die and second die), so $ …