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 is given by

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

Accordingly,
$A=\\{(3,6),(4,5),(4,6),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6)\\}$
$B=\\{(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(1,2),(3,2),(4,2),(5,2),(6,2)\\}$
$C=\\{(3,6),(4,5),(5,4),(6,3),(6,6)\\}$
It is observed that
$A ∩ B =Φ$
$B ∩ C =Φ$
$C∩A=\\{(3,6)(4,5)(5,4)(6,3)(6,6)\\}≠Φ$
Hence, events $A$ and $B$and events $B$ and $C$ are mutually exclusive.