(iii) Two events, which are not mutually exclusive

Suppose $A$ be the event of getting at least two heads

$A = \left( {HHH,HHT,THH,HTH} \right)$

Suppose $B$ be the event of getting only head

$B = {\text{ }}\left( {HHH} \right)$

So, $A\; \cap \;B{\text{ }} = {\text{ }}\left( {HHH,{\text{ }}HHT,{\text{ }}THH,{\text{ }}HTH} \right){\text{ }} \cap {\text{ }}\left( {HHH} \right)$

$A \cap B = (HHH)$

$A \cap B = \varphi $

As there is no common element in $A$ and $B$ so these two are not mutually exclusive.

(iv) Two events which are mutually exclusive but not exhaustive

Suppose $P$ to be the event of getting only Head

$P{\text{ }} = \;\left( {HHH} \right)$

Suppose $Q$ to be the event of getting only tail

$Q{\text{ }} = \;\left( {TTT} \right)$

$P\; \cap \;Q{\text{ }} = \;\left( {HHH} \right){\text{ }} \cap {\text{ }}\left( {TTT} \right)$

$P \cap Q = {\text{ }}\varphi $

As there is no common element in $P$ and $Q$.

Thus, they are mutually exclusive.

But,

$P\; \cup \;Q{\text{ }} = {\text{ }}\left( {HHH} \right)\; \cup \;\left( {TTT} \right)$

$P \cup Q = \;\left\{ {HHH,{\text{ }}TTT} \right\}$

$P \cup Q{\text{ }} \ne {\text{ }}S$

$\;P \cup \;Q\; \ne \;S$

Thus, $P$, $Q$ and $S$ are not exhaustive events.