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 multiple of $3$'
$C:$ 'the sum is less than $4$'
$D:$ 'the sum is greater than $11$'
Which pair of these events are mutually exclusive?

• A. $(A,B)$
• B. $(B,C)$
• C. $(C,D)$
• D. $(A,D)$

Answer 1

The correct option is C $(C,D)$

There are $36$ elements in the sample space
$S=\left\{\right(x,y):x,y=1,2,3,4,5,6\}.$

$A=\left\{\right(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5),(4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4),(6,6\left)\right\}$
$B=\left\{\right(1,2),(2,1),(1,5),(5,1),(3,3),(2,4),(4,2),(3,6),(6,3),(4,5),(5,4),(6,6\left)\right\}$
$C=\left\{\right(1,1),(2,1),(1,2\left)\right\}$
$D=\left\{\right(6,6\left)\right\}$
$\therefore A\cap B\ne \varphi ,B\cap C\ne \varphi ,C\cap D=\varphi$
and $A\cap D\ne \varphi$
$\Rightarrow C$ and $D$ are mutually exclusive.