Three coins are tossed. Describe(i)      Two events which are mutually exclusive.(ii)     Three events which are mutually exclusive and exhaustive.(iii)    Two events, which are not mutually exclusive.(iv)    Two events which are mutually exclusive but not exhaustive.(v)    Three events which are mutually exclusive but not exhaustive.

Solution: Two events are mutually exclusive when the intersection of them is $\varphi$.
Two events are mutually exhaustive when the union of them is sample space.
In the given question,Sample space, $S=\left\{\left(H,H,H\right),\left(H,H,T\right),\left(H,T,T\right),\left(T,H,H\right),\left(H,T,H\right),\left(T,H,T\right),\left(T,T,H\right),\left(T,T,T\right)\right\}$
Here, $H$ represents Head and $T$ represents Tail.
So,
(i)Two events which are mutually exclusive.
$A=$ All three Heads, $B$ = All three Tails
As, $A\cap B=\varphi$
So, $A$ and $B$ are mutually exclusive events.

(ii)Three events which are mutually exclusive and exhaustive.
$A=\left\{\left(H,H,H\right),\left(T,T,T\right)\right\}$
$B=\left\{\left(H,H,T\right),\left(H,T,H\right),\left(T,H,H\right)\right\}$
$C=\left\{\left(T,T,H\right),\left(T,H,T\right),\left(H,T,T\right)\right\}$
As, $A\cap B\cap C=\varphi$
$A\cup B\cup C=S$
Events $A,B,C$ are mutually exclusive and exhaustive.

(iii)Two events, which are not mutually exclusive.
$A=\left\{H,H,T\right\}$
$B=\left\{H,T,H\right\}$
As both these events have two heads and one tail, these events are not mutually exclusive.

(iv)Two events which are mutually exclusive but not exhaustive.
$A=$ All three Heads, $B$ = All three Tails
As, $A\cap B=\varphi$
$A\cup B\ne S$
So, $A$ and $B$ are mutually exclusive but not mutually exhaustive events.

(v)Three events which are mutually exclusive but not exhaustive.
$A=\left\{\left(H,H,H\right)\right\}$
$B=\left\{\left(H,H,T\right)\right\}$
$C=\left\{\left(H,T,T\right)\right\}$
Here, $A\cap B\cap C=\varphi$
$A\cup B\cup C\ne S$
So, all these three events are mutually exclusive bot not mutually exhaustive.