Question

When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}.$ The probability that the missile hits the target, given 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. $\frac{1}{12}$
• B. $\frac{1}{8}$
• C. $\frac{3}{8}$
• D. $\frac{3}{4}$

Answer 1

The correct option is B $\frac{1}{8}$

$R:$ Missile is intercepted
$H:$ Missile hits the target
$P\left(R\right)=\frac{1}{3};P\left(\overline{R}\right)=\frac{2}{3};P\left(H|\overline{R}\right)=\frac{3}{4}$

$P\left(H\right)=P(H\cap R)+P(H\cap \overline{R})=P\left(R\right)\cdot P\left(H|R\right)+P\left(\overline{R}\right)\cdot P\left(H|\overline{R}\right)=\frac{1}{3}\cdot \left(0\right)+\frac{2}{3}\cdot \frac{3}{4}=\frac{1}{2}$

Hence, $P\left(HHH\right)=\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{8}$