Question

Two dice are thrown. The events $A$, $B$ and $C$ are as follows :
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice < 5.
Then, which of the following is true ?
(i) $A$ and $B$ are mutually exclusive.
(ii) $A$ and $B$ are mutually exclusive and exhaustive.
(iii) $A=B^{\prime }$
(iv) $A$ and $C$ are mutually exclusive.
(v) $A$ and $B^{\prime }$ are mutually exclusive.
(vi) $A^{\prime }$, $B^{\prime }$ and $C$ are mutually exclusive and exhaustive.

• A. $(i)$
• B. $(ii)$, $(iv)$, $(v)$
• C. $(vi)$, $(iii)$
• D. $(i)$, $(ii)$ and $(iii)$

Answer 1

Answer: (D)

$\because A=$ Getting an even number on the first die
$A=\{(2,1),(2,2),(2,3),(2.4),(2,5),(2,6),(4,1)$,
$(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4)$,
$(6,5),(6,6)\}$
$B=$ Getting an odd number on the first die
$B=\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1)$,
$(3,2),(3,3),(3,4),(3,5),(3,6),(5,1),(5,2)$,
$(5,3),(5,4)(5,5),(5,6)\}$
$C=$ Getting the sum of numbers $\le 5$
$=1(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3)$,
$(3,1),(3,2),(4,1)\}$
(i) True
$\Rightarrow A\cap B=\varphi$
$\therefore A$ and $B$ are mutually exclusive events.
(ii) True
$\therefore A\cup B=S$ i.e., exhaustive. Also, $A\cap B=\varphi$
(iii) True
$\therefore B=$ Getting an odd number on the first die
$\Rightarrow B^{\prime }=$ Getting an even number on the first die $=A$
$\therefore A=B^{\prime }$
(iv ) False
$\because A\cap C=\{(2,1),(2,2),(2,3),(4,1)\}\ne \varphi$, so, $A$ and $C$ are not mutually exclusive.
(v ) False
$\because B^{\prime }=A$
$\therefore A\cap B^{\prime }=A\cap A=A\ne \varphi$
So, $A$ and $B^{\prime }$ are not mutually exclusive.
(vi) False
$\because A^{\prime }\cap B^{\prime }=\varphi$
$\therefore A^{\prime }\cap B^{\prime }\cap C=\varphi$ and $A^{\prime }\cup B^{\prime }\cup C=S$
But $A^{\prime }\cap C=B\cap C$
$=\{(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)\}\ne \varphi$ $(\because A^{\prime }=B)$
and $B^{\prime }\cap C=A\cap C=\{(2,1),(2,2),(2,3),(4,1)\}\ne \varphi$
$(\because B^{\prime }=A)$
$\therefore A^{\prime }$, $B^{\prime }$ and $C$ are not mutually exclusive.