Question

In throwing a die. Let $A$ be the event ''coming up of an odd number'', $B$ be the event ''coming up of a number divisible by $2$'', $C$ be the event ''coming up of a number $\ge 4$'' and $D$ be the event ''coming up of a number $<3,$ then which of the following is/are true?

• A. $A$ and $C$ are mutually exclusive and exhaustive events.
• B. $A$ and $B$ are mutually exclusive and exhaustive events.
• C. $A,C$ and $D$ form an exhaustive system.
• D. $B,C$ and $D$ form an exhaustive system.

Answer 1

Answer: (B), (C)

$A,C$ and $D$ form an exhaustive system.

The sample space of the experiment is
$S=\{1,2,3,4,5,6\}$
and $A=\{1,3,5\}$
$B=\{2,4,6\}$
$C=\{4,5,6\}$
$D=\{1,2\}$

$A\cup B=S$ and $A\cap B=\varphi$
$\therefore A$ and $B$ are mutually exclusive and exhaustive events.
$A\cup C\ne S$ and $A\cap C\ne \varphi$

$A\cup C\cup D=S$
$\therefore A,C$ and $D$ form an exhaustive system.
$B\cup C\cup D\ne S$