Question

An experiment involves rolling a pair of dice and recording the numbers that come up. The following are the 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 event are not mutually exclusive:

• A. $(A,C)$
• B. $(A,B)$
• C. $(B,C)$
• D. None of these.

Answer 1

The correct option is A $(A,C)$

Let us assume that $1,2,3,4,5$ and $6$ are the possible outcomes when the die is thrown.
In the question is given that pair of die is thrown, so sample space will be,
$S=\left\{\begin{array}{c}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),\\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$
$A$: the sum is greater than $8$,
$\therefore A=\left\{\begin{array}{c}(3,6),(4,5),(5,4),(6,3),(4,6),\\ (5,5),(6,4),(5,6)(6,5),(6,6),\end{array}\right\}$
$B$: $2$ occurs on either die
$B=\left\{\begin{array}{c}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\\ (1,2),(3,2),(4,2),(5,2),(6,2),\end{array}\right\}$
$C$: the sum is at least $7$ and a multiple of $3$.
So the sum can be only $9$ or $12$
$C=\left\{\begin{array}{c}(3,6),(4,5),(5,4),(6,3),(6,6)\end{array}\right\}$

Now, we shall check pairs of these events if they are mutually exclusive or not.
$\left(i\right)$ $A\cap B=\varphi$
Since there is no common element between the two sets.
Therefore $A$ & $B$ are mutually exclusive.

$\left(ii\right)$ $B\cap C=\varphi$
Since there is no common element between
Therefore $B$ and $C$ are mutually exclusive.

$\left(iii\right)$ $A\cap C=\left\{\right(3,6),(4,5),(5,4),(6,3),(6,6),\}\Rightarrow \left\{\right(3,6),(4,5),(5,4),(6,3),(6,6),\}\ne \varphi$
Since $A$ and $C$ has common elements.
Therefore $A$ and $C$ are not mutually exclusive.