Question

Three coins are tossed once. Let $A$ denote the event ‘three heads show”, $B$ denote the event “two heads and one tail show”, $C$ denote the even ''three tails show'' and $D$ denote the event ‘a head shows on the first coin”.
$i.$ Which events are mutually exclusive ?
$ii.$ Which events are simple ?
$iii.$ Which events are compound ?

Answer 1

$\left(i\right)$ Given: Three coins are tossed once
$A:$ “three heads show”
$B:$ “two heads and one tail show”
$C:$ “three tails show”
$D:$ “a head shows on the first coin”
If three coins are tossed once, then sample space is
$S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$
Now,
$A=\left\{HHH\right\}$
$B=\{HHT,THH,HTH\}$
$C=\left\{TTT\right\}$
$D=\{HHH,HHT,HTH,HTT\}$
Now, checking for mutually exclusive events:
$A\cap B=\varphi$
$A\cap C=\varphi$
$A\cap D=\left\{HHH\right\}$
$B\cap C=\varphi$
$B\cap D=\{HHT,HTH\}$
$C\cap D=\varphi$
Hence, $(A,B),(A,C),(B,C)$ and $(C,D)$ are mutually exclusive events.


$\left(ii\right)$ Given: Three coins are tossed once
$A:$ “three heads show”
$B:$ “two heads and one tail show”
$C:$ “three tails show”
$D:$ “a head shows on the first coin”
If three coins are tossed once, then sample space is
$S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$
Now,
$A=\left\{HHH\right\}$
$B=\{HHT,THH,HTH\}$
$C=\left\{TTT\right\}$
$D=\{HHH,HHT,HTH,HTT\}$
$\because A$ and $C$ have only one element.
$\therefore A$ and $C$ are simple events.

$\left(iii\right)$ Given : Three coins are tossed once
$A:$ “three heads show”
$B:$ “two heads and one tail show”
$C:$ “three tails show”
$D:$ “a head shows on the first coin”
If three coins are tossed once, then sample space is
$S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$
Now,
$A=\left\{HHH\right\}$
$B=\{HHT,THH,HTH\}$
$C=\left\{TTT\right\}$
$D=\{HHH,HHT,HTH,HTT\}$
$\because B$ and $D$ have more than one element.
$\therefore B$ and $D$ are compound events.