# Three coins are tossed once.

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 event “three tails show” and D denote the event ‘a head shows on the first coin”. Which events are

(i) mutually exclusive?

(ii) simple?

(iii) compound?

Solution:

When three coins are tossed, the sample space is given by

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

Accordingly,

A = {HHH}

B = {HHT, HTH, THH}

C = {TTT}

D = {HHH, HHT, HTH, HTT}

We now observe that

$\mathrm{A} \cap \mathrm{B}=\Phi, \mathrm{A} \cap \mathrm{C}=\Phi, \mathrm{A} \cap \mathrm{D}=\{\mathrm{HHH}\} \neq \Phi$

$B \cap C=\Phi, B \cap D=\{H H T,\{H T H\} \neq \Phi$

$C \cap D=\Phi$

(i) Event A and B; event A and C; event B and C; and event C and D are all mutually exclusive.

(ii) If an event has only one sample point of a sample space, it is called a simple event. Thus, A and C are simple events.

(iii) If an event has more than one sample point of a sample space, it is called a compound event. Thus, B and D are compound events.