Question

A coin is tossed three times. Find P (A/B) in each of the following:
(i) A = Heads on third toss, B = Heads on first two tosses
(ii) A = At least two heads, B = At most two heads
(iii) A = At most two tails, B = At least one tail.

Answer 1

(i) Consider the given events.
A = Heads on third toss
B = Heads on first two tosses

Clearly,
A = {(H, H, H), (H, T, H), (T, H, H), (T, T, H)}
B = {(H, H, H), (H, H, T)}

$$\begin{gathered} \mathrm{Now}, \\ A\cap B=\left\{H,H,H\right\} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(A/B\right)=\frac{n\left(A\cap B\right)}{n\left(B\right)}=\frac{1}{2} \end{gathered}$$

(ii) Consider the given events.
A = At least two heads
B = At most two heads

Clearly,
A = {(H, H, H), (H, T, H), (T, H, H), (H, H, T)}
B = {(T, T, T), (H, T, H), (T, H, H), (H, H, T), (T, H, T), (H, H, T), (H, T, T)}

$$\begin{gathered} \mathrm{Now}, \\ A\cap B=\left\{\left(H,T,H\right),\left(T,H,H\right),\left(H,H,T\right)\right\} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(A/B\right)=\frac{n\left(A\cap B\right)}{n\left(B\right)}=\frac{3}{7} \end{gathered}$$

(iii) Consider the given events.
A = At most two tails
B = At least one tail

Clearly,
A = {(T, T, H), (T, H, H), (H, H, T), (T, H, T), (H, H, T), (H, T, T), (H, H, H)}
B = {(T, T, T), (T, T, H), (T, H, H), (H, H, T), (T, H, T), (H, H, T), (H, T, T)}

$$\begin{gathered} \mathrm{Now}, \\ A\cap B=\{(T,\; T,\; H),\; (T,\; H,\; H),\; (H,\; H,\; T),\; (T,\; H,\; T),\; (H,\; H,\; T),\; (H,\; T,\; T)\} \end{gathered}$$

$\therefore \mathrm{Required}\mathrm{probability}=P\left(A/B\right)=\frac{n\left(A\cap B\right)}{n\left(B\right)}=\frac{6}{7}$