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

• A. $A$ and $B$
• B. $B$ and $C$
• C. $C$ and $D$
• D. $A$ and $C$

Answer 1

Answer: (C)

There are $36$ elements in the sample space
$S=\{(x,y):x$, $y=1,2,3,4,5,6\}$.
Then,
$A=\{(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3)$,
$(3,5),(4,2),(4,4),(4,6),(5,1),(5,3),(5,5),(6,2),(6,4)$,
$(6,6)\}$
$B=\{(1,2),(2,1),(1,5),(5,1),(3,3),(2,4),(4,2),(3,6)$,
$(6,3),(4,5),(5,4),(6,6)\}$
$C=\{(1,1),(2,1),(1,2)\}$ and $D=\{(6,6)\}$
We find that
$A\cap B=\{(1,5),(2,4),(3,3),(4,2),(5,1),(6,6)\}\ne \varphi$ Therefore, $A$ and $B$ are not mutually exclusive events.
Similarly, $A\cap C\ne \varphi$, $A\cap D\ne \varphi$, $B\cap C\ne \varphi$ and $B\cap D\ne \varphi$,
Thus, the pairs, $(A,C)$, $(A,D)$, $(B,C)$, $(B,D)$ are not mutually exclusive events.
Also, $C\cap D=\varphi$ and so $C$ and $D$ are mutually exclusive events.