Question

In each of the following experiments write the sample space S, number of sample points n(S), events P, Q, R using set form and n(P), n(Q) and n(R).
Find the events among the events defined above which are:
complementary events, mutually exclusive events and exhaustive events

(i) Three coins are tossed simultaneously:
P is the event of getting at least two heads.
Q is the event of getting no head.
Q is the event of getting head on second coin.

Answer 1

Given:
Three coins are tossed.
Sample space (S) = {HHH, THH, HTH, HHT, THT, TTH, HTT, TTT}
Or,
n(S) = 8

Now,
P is the event of getting at least two heads (two or more than two).
Thus, we have:
P = {HHH, THH, HTH, HHT}
∴ n(P) = 4

Q is the event of getting no head.
Thus, we have:
Q = {TTT}
∴ n(Q) =1

R is the event of getting a head on the second coin.
Thus, we have:
R = {HHH, THH, HHT, THT}
∴ n(R) = 4

$$\begin{gathered} \mathrm{Here}, \\ \mathrm{P}\cap \mathrm{Q}=\mathrm{\varphi }\mathrm{but}\mathrm{P}\cup \mathrm{Q}\ne \mathrm{S}. \end{gathered}$$
Thus, we can say that events P and Q are mutually exclusive but not complementary.
Also,
$\mathrm{Q}\cap \mathrm{R}=\mathrm{\varphi }\mathrm{but}\mathrm{Q}\cup \mathrm{R}\ne \mathrm{S}.$
Thus, we can say that events Q and R are mutually exclusive but not complementary.