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
$A \cap B = \{(1, 5), (2,4), (3,3), (4,2), (5,1), (6,6)\} \ne \phi$ Therefore, $A$ and $B$ are not mutually exclusive events.
Similarly, $A \cap C\ne \phi$, $A \cap D \ne \phi$, $B \cap C \ne \phi$ and $B \cap D \ne \phi$,
Thus, the pairs, $(A, C)$, $(A, D)$, $(B, C)$, $(B, D)$ are not mutually exclusive events.
Also, $C \cap D = \phi$ and so $C$ and $D$ are mutually exclusive events.