An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
Question:

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?

Solution:

When a pair of dice is rolled, the sample space is given by

$\mathrm{S}=\{(x, y): x, y=1,2,3,4,5,6\}$

$=\left\{\begin{array}{lllll}(1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6) \\ (2,1), & (2,2), & (2,3), & (2,4), & (2,5), & (2,6) \\ (3,1), & (3,2), & (3,3), & (3,4), & (3,5), & (3,6) \\ (4,1), & (4,2), & (4,3), & (4,4), & (4,5), & (4,6) \\ (5,1), & (5,2), & (5,3), & (5,4), & (5,5), & (5,6) \\ (6,1), & (6,2), & (6,3), & (6,4), & (6,5), & (6,6)\end{array}\right\}$

Accordingly,

$\mathrm{A}=\{(3,6),(4,5),(4,6),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6)\}$

$\mathrm{B}=\{(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(1,2),(3,2),(4,2),(5,2),(6,2)\}$

$\mathrm{C}=\{(3,6),(4,5),(5,4),(6,3),(6,6)\}$

It is observed that

$\mathrm{C} \cap \mathrm{A}=\{(3,6),(4,5),(5,4),(6,3),(6,6)\} \neq \phi$

Hence, events $A$ and $B$ and events $B$ and $C$ are mutually exclusive.

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