Question

In throwing a die, which of the following set of events are exhaustive ?
$A:$ the events ''an odd number turns up'', ''a number $\le 4$ comes up'' and '' the number $5$ turns up''.
$B:$ the events ''a number $\le 4$ turns up'', ''a number $>4$ turns up''.
$C:$ the events '' an even number turns up'', '' a number divisible by $3$ turns up'', '' number $1$ or $2$ turns up'' and '' the number $6$ turns up''.

• A. $A$
• B. $B$
• C. $C$
• D. All of these

Answer 1

The correct option is B $B$

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

Let $A_1$ be ''an odd number turns up''
$A_1=\{1,3,5\}$
Let $A_2$ be ''a number $\le 4$ comes up''
$A_2=\{1,2,3,4\}$
Let $A_3$ be '' the number $5$ turns up.
$A_3=\left\{5\right\}$
$\therefore A_1\cup A_2\cup A_3=A=\{1,2,3,4,5\}\ne S$

Let $B_1$ be ''a number $\le 4$ turns up''
$B_1=\{1,2,3,4\}$
Let $B_2$ be ''a number $>4$ turns up''.
$B_2=\{5,6\}$
$\therefore B_1\cup B_2=B=\{1,2,3,4,5,6\}=S$

Let $C_1$ be ''an even number turns up''
$C_1=\{2,4,6\}$
Let $C_2$ be ''a number divisible by $3$ turns up''
$C_2=\{3,6\}$
Let $C_3$ be ''number $1$ or $2$ turns up''
$C_3=\{1,2\}$
Let $C_4$ be ''the number $6$ turns up''
$C_4=\left\{6\right\}$
$\therefore C_1\cup C_2\cup C_3\cup C_4=C=\{1,2,3,4,6\}\ne S$