Question

A coin is tossed three times, consider the following events.
$A:$ 'No head appears'.
$B:$ 'Exactly one head appears'.
$C:$ 'At least two heads appear'.
Do they form a set of mutually exclusive and exhaustive events?

Answer 1

Given: A coin is tossed three times.
$A:$ 'No head appears'
$B:$ 'Exactly one head appears'
$C:$ 'At least two heads appear'
The sample space is
$S=\left\{\begin{array}{c}HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\end{array}\right\}$
$A=\left\{TTT\right\}$
$B=\{HTT,THT,TTH\}$
$C=\{HHT,HTH,THH,HHH\}$

We need to find $A\cup B\cup C$ to check for exhaustive events
$A\cup B\cup C=\left\{\begin{array}{c}HHH,HHT,HTH,THH,\\ HTT,THT,TTH,TTT\end{array}\right\}=S$
Hence $A,B$ and $C$ are exhaustive events.

We need to find $A\cap B,B\cap C$ and $A\cap C$ for
checking mutually exclusive events.
$A\cap B=\varphi$.
So, $A\text{and}B$ are mutually exclusive.
$B\cap C=\varphi$.
So, $B\text{and}C$ are mutually exclusive.
$A\cap C=\varphi$.
So, $A\text{and}C$ are mutually exclusive.
Hence, $A,B\text{and}C$ form a set of mutually exclusive and exhaustive events.