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?

Answer 1

Given: A pair of dice are thrown.
The sample space is
$S=\left\{\begin{array}{cccccc}(1,1),& (1,2),& (1,3),& (1,4),& (1,5),& (1,6),\\ (2,1),& (2,2),& (2,3),& (2,4),& (2,5),& (2,6)\\ (3,1),& (3,2),& (3,3),& (3,4),& (3,5),& (3,6)\\ (4,1),& (4,2),& (4,3),& (4,4),& (4,5),& (4,6)\\ (5,1),& (5,2),& (5,3),& (5,4),& (5,5),& (5,6)\\ (6,1),& (6,2),& (6,3),& (6,4),& (6,5),& (6,6)\end{array}\right\}$

$A:$ The sum is greater than $8$, so
$A=\left\{\begin{array}{c}(3,6),(4,5),(5,4),(6,3),(4,6),\\ (5,5),(6,4),(5,6),(6,5),(6,6)\end{array}\right\}$

$B:2$ occur on either die , so
$B=\left\{\begin{array}{c}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ (1,2),(3,2),(4,2),(5,2),(6,2)\end{array}\right\}$

$C:$ the sum is at least $7$ and multiple of $3$, so
$C=\left\{\right(3,6),(4,5),(5,4),(6,3),(6,6\left)\right\}$

Now,
$A\cap B=\left\{\varphi \right\}$
So, $A\text{and}B$ are mutually exclusive.
$B\cap C=\left\{\varphi \right\}$
So, $B\text{and}C$ are mutually exclusive.
$A\cap C=\left\{\right(3,6),(4,5),(5,4),(6,3),(6,6\left)\right\}$
So $A\text{and}C$ are not mutually exclusive.