An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
$A:$ the sum is greater than $8$.
$B:$ $2$ occurs on either die.
$C:$ the sum is at least $7$ and a multiple of $3$.
Which pairs of these events are mutually exclusive?

(A) When a pair of dice is rolled,the sample space $S$ contains $36$ outcomes.
$A = \{(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)\}$
$B = \{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (1,2), (3,2), (4,2), (5,2), (6,2)\}$
$C = \{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (3,6), (4,5), (5,4), (6,3), (6,6)\}$
Two events are mutually exclusive if their intersection is empty $(\phi)$.
$A \cap B = \phi$ (No common elements).
$B \cap C = \{(2,5), (5,2)\} \neq \phi$.
$A \cap C = \{(3,6), (4,5), (5,4), (6,3), (6,6)\} \neq \phi$.
Therefore,only events $A$ and $B$ are mutually exclusive.