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

When a pair of dice is rolled, the sample space will have $36$ sample points
$S=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6\left)\right(2,1),(2,2),...........,(2,6\left)\right(3,1),(3,2)..............(3,6\left)\right(4,1),(4,2).............(4,6),(5,1),(5,2).............(5,6\left)\right(6,1),(6,2)...........(6,6\left)\right\}$
$A=\left\{\right(3,6),(6,3),(4,5),(5,4),(5,5),(4,6),(6,4),(5,6),(6,5),(6,6\left)\right\}$
$B=\left\{\right(1,2),(2,1),(2,3),(3,2),(2,4),(4,2),(2,5),(5,2),(2,6),(6,2\left)\right\}$
$C=\left\{\right(4,5),(5,4),(3,6),(6,3),(6,6\left)\right\}$
Clearly, $A\cap B=\varphi$
Hence, $A$ and $B$ are mutually exclusive events.
$B\cap C=\varphi$
Hence, $B$ and $C$ are mutually exclusive events.
$A\cap C=\left\{\right(4,5),(5,4),(3,6),(6,3),(6,6\left)\right\}\ne \varphi$
Hence, $A$ and $C$ are not mutually exclusive events