Question

Two die are thrown, $A$ is the event that the sum of the numbers on their upper face is at least nine. $B$ is the event that the sum of the numbers on their upper face is divisible by $8$, $C$ is the event that the same number on the upper faces of both dice. The events $A$ and $B$ are:

• A. Mutually exclusive
• B. Mutually exhaustive
• C. Either
• D. Neither

Answer 1

The correct option is A Mutually exclusive

When two dice are thrown

$S=\left\{\right(1,1),(1,2),(1,3\left)\right(1,4),(1,5),(1,6),(2,1),(2,2),(2,3\left)\right(2,4),(2,5),(2,6),(3,1),(3,2),(3,3\left)\right(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3\left)\right(5,4),(5,5),(5,6),(6,1),(6,2),(6,3\left)\right(6,4),(6,5),(6,6\left)\right\}$

Therefore, $n\left(S\right)=36$

$A$ is the event that sum of the numbers on their upper face is at least nine.

$\therefore A=\left\{\right(3,6),(4,5),(4,6),(5,4),(5,5),(5,6),(6,3),(6,4),(6,5),(6,6\left)\right\}$

$\Rightarrow n\left(A\right)=10$

$B$ is the event that sum of numbers on their upper face is divisible by $8$.

$\therefore B=\left\{\right(2,6),(3,5),(4,4),(5,3),(6,2\left)\right\}$.

$\Rightarrow n\left(B\right)=5$

$C$ is the event that same number appears an upper faces of both dice.

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

$\Rightarrow n\left(C\right)=6$.

We see that $A\cap B=\varphi$.

Hence, $A$ and $B$ are mutually exclusive events.