Question

Three coins are tossed
$i.$ Describe two events which are mutually exclusive.
$ii.$ Describe three events which are mutually exclusive and exhaustive.
$iii.$ Describe two events, which are not mutually exclusive.
$iv.$ Describe two events which are mutually exclusive but not exhaustive.
$v.$ Describe three events which are mutually exclusive but not exhaustive.

Answer 1

$\left(i\right)$ Given: Three coins are tossed
The sample space is
$S=\left\{\begin{array}{c}HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\end{array}\right\}$
Let the two events be $A\text{and}B$, which is described as
$A:$ Getting $3$ heads.
$B:$ Getting $3$ tails.
$\Rightarrow A=\left\{HHH\right\}$
$\Rightarrow B=\left\{TTT\right\}$
Now,
$A\cap B=\varphi$
Hence, $A$ and $B$ are mutually exclusive.
NOTE: There can be different answers for this question, here $1$ possible solution is given.

$\left(ii\right)$ Given : Three coins are tossed,
The sample space is
$S=\left\{\begin{array}{c}HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\end{array}\right\}$
Let the two events be $A,B\text{and}C$, which is described as
$A:$ getting exactly two tails
$B:$ getting at least two heads
$C:$ getting exactly three tails
So,
$A=\{HTT,THT,TTH\}$
$B=\{HHT,HTH,THH,HHH\}$
$C=\left\{TTT\right\}$
Now,
$A\cap B=\varphi ,A\cap C=\varphi ,B\cap C=\varphi$
Since no element is common in $A\&B,A\&C$ and $B\&C$
Hence $A,B$ and $C$ are the mutually exclusive events.
For exhaustive events, we should prove
$A\cup B\cup C=S$
$A\cup B\cup C=\{HTT,THT,TTH\}\cup \{HHT,HTH,THH,HHH\}\cup \left\{TTT\right\}=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$
$=S$
Hence $A,B$ and $C$ are the mutually exhaustive event.
NOTE: There can be different answers for this question, here $1$ possible solution is given.


$\left(iii\right)$ Given: Three coins are tossed
The sample space is
$S=\left\{\begin{array}{c}HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\end{array}\right\}$
$A:$ getting at least two heads
$B:$ getting exactly three heads
So,
$A=\{HHH,HHT,HTH,THH\}$
$B=\left\{HHH\right\}$
Now,
$A\cap B=\left\{HHH\right\}\ne \varphi$
As there is a common element in $A$ and $B$,
so $A$ and $B$ are not mutually exclusive.
NOTE: There can be different answers for this question, here $1$ possible solution is given.

$\left(iv\right)$ Given : Three coins are tossed
The sample space is
$S=\left\{\begin{array}{c}HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\end{array}\right\}$
Let the two events be $A\text{and}B$, which is described as
$A:$ Getting $3$ heads
$B:$ Getting $3$ tails
$\Rightarrow A=\left\{HHH\right\},B=\left\{TTT\right\}$
Now,
$A\cap B=\varphi$
$A\cup B=\left\{HHH\right\}\cup \left\{TTT\right\}\ne S$
Hence, $A$ and $B$ are mutually exclusive but not exhaustive.
NOTE: There can be different answers for this question, here $1$ possible solution is given.

$\left(v\right)$Given : Three coins are tossed
The sample space is
$S=\left\{\begin{array}{c}HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\end{array}\right\}$
Let the two events be $A$ and $B$, which is described as
$A:$ Getting $3$ heads
$B:$ Getting $3$ tails
$C:$ Getting exactly $2$ heads
So,
$A=\left\{HHH\right\}$
$B=\left\{TTT\right\}$
$C=\{HHT,HTH,THH\}$
Now,
$A\cap B=\varphi ,B\cap C=\varphi ,A\cap C=\varphi$
Hence $A,B$ and $C$ are mutually exclusive events.
$A\cup B\cup C=\{HHH,TTT,HHT,HTH,THH\}\ne S$
Hence $A,B$ and $C$ are not exhaustive event.
$\therefore A,B$ and $C$ are mutually exclusive but not exhaustive event.
NOTE: There can be different answers for this question, here $1$ possible solution is given.