A coin is tossed three times, consider the following events.

$A: $ ' No head appears ',  $B:$ ' Exactly one head appears ' and  $C:$ ' Atleast two heads appear '

Do they form a set of mutually exclusive and exhaustive events?

The sample space of the experiment is

$S =\{ HHH ,\, HHT ,\, HTH$ , $THH ,\, HTT , THT$, $TTH, \,TTT\}$

and $A=\{ TTT \}$,  $B =\{ HTT , \,THT, \, TTH \}$, $C =\{ HHT \,, HTH ,\, THH , \,HHH \}$

Now

$A \cup B \cup C =$ $\{ TTT , \, H T T , \, T H T $,  $T T H , \, H H T $,  $H T H , \, T H H , \, H H H \} \, = S$

Therefore, $A, \,B$ and $C$ are exhaustive events.

Also, $A \cap B=\phi, A \cap C=\phi$ and $B \cap C=\phi$

Therefore, the events are pair-wise disjoint, i.e., they are mutually exclusive.

Hence, $A,\, B$ and $C$ form a set of mutually exclusive and exhaustive events.