Question

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, giver that it is not intercepted, is $\frac{3}{4} .$ If three missiles are fired independently from the ship, then the probability that all three hit the target, is

• A. $\frac{1}{27}$
• B. $\frac{3}{4}$
• C. $\frac{1}{8}$
• D. $\frac{3}{8}$

Answer 1

Answer: (C)

$R :$ Missile is intercepted
$H$ : Missile hits the target
$ P(R)=\frac{1}{3} ; P(\bar{R})=\frac{2}{3} ; P(H \mid \bar{R})=\frac{3}{4} $
$ \begin{array}{l} P(H)=P(H \cap R )+ P ( H \cap \overline{ R }) \\ = P ( R ) \cdot P ( H \mid R )+ P (\overline{ R }) \cdot P ( H \mid \overline{ R }) \\ =\frac{1}{3} \cdot(0)+\frac{2}{3} \cdot \frac{3}{4} \\ =\frac{1}{2} \end{array} $
Hence, $P ( HHH )=\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2}=\frac{1}{8}$