Question

Three coins are tossed describe
(i) Two events which are mutually exclusive
(ii) Three events which are mutually exclusive and exhaustive
(iii) Two events which are not mutually exclusive
(iv) Two events which are mutually exclusive but not exhaustive
(v) Three events which are mutually exclusive but not exhaustive

Answer 1

When three coins are tossed the sample space is given by

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

(i) Two events that are mutually exclusive can be

$A:$ getting no heads and $B:$ getting no tails

This is because sets $A=\left\{TTT\right\}$ and $B=\left\{HHH\right\}$ are disjoint

(ii) Three events that are mutually exclusive and exhaustive can be

$A:$ getting no heads $\Rightarrow A=\left\{TTT\right\}$

$B:$ getting exactly one head $\Rightarrow B=\{HTT,THT,TTH\}$

$C:$ getting at least two heads $\Rightarrow C=\{HHH,HHT,HTH,THH\}$

This is because $A\cap B=B\cap C=C\cap A=\varphi$

and $A\cup B\cup C=S$

(iii) Two events that are not mutually exclusive can be

$A:$ getting three heads $\Rightarrow A=\left\{HHH\right\}$

$B:$ getting at least $2$ heads $\Rightarrow B=\{HHH,HHT,HTH,THH\}$

This is because $A\cap B=\left\{HHH\right\}\ne \varphi$

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

$A:$ getting exactly one head $\Rightarrow A=\{HTT,THT,TTH\}$

$B:$ getting exactly one tail $\Rightarrow B=\{HHT,HTH,THH\}$

This is because $A\cap B=\varphi$ but $A\cup B\ne S$

(v) Three events that are mutually exclusive but not exhaustive can be

$A:$ getting exactly three heads $\Rightarrow A=\left\{HHH\right\}$

$B:$ getting one head and two tails $\Rightarrow B=\{HTT,THT,TTH\}$

$C:$ getting one tail and two heads $\Rightarrow C=\{HHt,HTH,THH\}$

This is because $A\cap B=B\cap C=C\cap A=\varphi$ but $A\cup B\cup C\ne S$