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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?
Solution
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.
