Question

Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment
$A:$ 'the sum is even'.
$B:$ 'the sum is a multiple of $3^{\prime }$.
$C:$ 'the sum is less than $4^{\prime }$.
$D:$ 'the sum is greater than ${11}^{\prime }$.
Which pairs of these events are mutually exclusive?

Answer 1

If two dice are thrown then 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 even'
So, the sum can be $\{2,4,6,8,10,12\}$
$A=\left\{\right(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5),(4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4),(6,6\left)\right\}$

$B:$ 'the Sum is multiple of $3$'
So, the multiple of $3$ are $\{3,6,9,12\}$
$B=\left\{\right(1,2),(2,1),(1,5),(5,1),(3,3),(2,4),(4,2),(3,6),(6,3),(4,5),(5,4),(6,6\left)\right\}$

$C:$ 'the sum is less than $4^{\prime }$
So, the sum possible are $\{1,2,3\}$
$C=\left\{\right(1,1),(2,1),(1,2\left)\right\}$

$D:$ 'the sum is greater than ${11}^{\prime }$
So, sum must be $12$ only
$D=\left\{\right(6,6\left)\right\}$

$A=\left\{\right(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5),(4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4),(6,6\left)\right\}$
$B=\left\{\right(1,2),(2,1),(1,5),(5,1),(3,3),(2,4),(4,2),(3,6),(6,3),(4,5),(5,4),(6,6\left)\right\}$
$C=\left\{\right(1,1),(2,1),(1,2\left)\right\}$
$D=\left\{\right(6,6\left)\right\}$

Mutually exclusive.
If $2$ elements are Mutually exclusive, then there should not be any common element.
$A\cap B\ne \varphi ,A\cap C\ne \varphi ,A\cap D\ne \varphi$
$B\cap C\ne \varphi ,B\cap D\ne \varphi$
$C\cap D=\varphi$
So, $C$ and $D$ are mutually exclusive events.