Three coins are tossed once Find the probability of getting (i) $3$ heads (ii) $2$ heads (iii) at least $2$ heads (iv) at most $2$ heads (v) no head (vi) $3$ tails (vii) exactly two tails (viii) no tail (ix) at most two tails

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Solution

When three coins are tossed once the sample space is given by
$S = H H H, H H T, H T H, T H H, H T T, T H T, T T H, T T T$
$∴$ Accordingly $n (S) = 8$
It is known that the probability of an event $A$ is given by
$P (A) =$ $\frac{N u m b e r o f o u t c o m e s f a v o u r a b l e t o A}{T o t a l n u m b e r o f p o s s i b l e o u t c o m e s} = \frac{n (A)}{n (S)}$
(i) Let $B$ be the event of the occurrence of $3$ heads Accordingly $B = H H H$
$∴ P (B) = \frac{n (B)}{n (S)} = \frac{1}{8}$
(ii) Let $C$ be the event of the occurrence of $2$ heads Accordingly $C = H H T, H T H, T H H$
$∴ P (C) = \frac{n (C)}{n (S)} = \frac{3}{8}$
(iii) Let $D$ be the event of the occurrence of at least $2$ heads
Accordingly $D = H H H, H H T, H T H, T H H$
$∴ P (D) = \frac{n (D)}{n (S)} = \frac{4}{8} = \frac{1}{2}$
(iv) Let $E$ be the event of the occurrence of at most $2$ heads
Accordingly $E = H H T, H T H, T H H, H T T, T H T, T T H, T T T$
$∴ P (E) = \frac{n (E)}{n (S)} = \frac{7}{8}$
(v) Let $F$ be the event of the occurrence of no head
Accordingly $F = T T T$
$∴ P (F) = \frac{n (F)}{n (S)} = \frac{1}{8}$
(vi) Let $G$ be the event of the occurrence of $3$ tails
Accordingly $G = T T T$
$∴ P (G) = \frac{n (G)}{n (S)} = \frac{1}{8}$
(vii) Let $H$ be the event of the occurrence of exactly $2$ tails
Accordingly $H = H T T, T H T, T T H$
$∴ P (H) = \frac{n (H)}{n (S)} = \frac{3}{8}$
(viii) Let $I$ be the event of the occurrence of no tail
Accordingly $I = H H H$
$∴ P (I) = \frac{n (I)}{n (S)} = \frac{1}{8}$
(ix) Let $J$ be the event of the occurrence of at most $2$ tails
Accordingly $I = H H H, H H T, H T H, T H H, H T T, T H T, T T H$
$∴ P (J) = \frac{n (J)}{n (S)} = \frac{7}{8}$

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