Question

Three coins are tossed. Describe
(i) two events A and B which are mutually exclusive.

(ii) three events A, B and C which are mutually exclusive and exhaustive.

(iii) two events A and B which am not mutually exclusive.

(iv) two events A and B which are mutually exclusive but not exhaustive.

Answer 1

If three coins are tossed, then possible number of out comes $=2^3=8$
Sample space

S = {HHH, HHT, writ H1T, TI IH,THT, TTH, TTT}

(i) Let A he the event. 'three heads show'

$\Rightarrow$ A = {HHH} and B be the event 'three tails show'.

$\Rightarrow$ B= (TTT)

$\Rightarrow A\cap B=\varphi$

Hence. A and B are mutually exclusive events.

(ii) Let A be the event 'atleast one head' a

$\Rightarrow$ A= {HHH, HHT, HTH, HTT, THH,THT, TTH}
and B be the event getting three tails.

$\Rightarrow$ B = {TTT}

$\Rightarrow A\cap B=\varphi andA\cup B=S$

(Two events are exhaustive if $A\cap B=S$)

Hence, A and B are naturally exclusive and exhaustive.

(iii) Let A be the event 'three heads show' and B be the event 'atleast two heads show'

$\Rightarrow$ A = {HHH}
and B = {HHT, HTH, THH, HHH}

$\Rightarrow A\cap B=HHH\ne \varphi$

Hence, A and B are not mutually exclusive.

(iv) Same as in (iii) A and B are not mutually exclusive.
Also, $A\cup B\ne S$.