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

Answer 1

The correct option is A

$\frac{1}{8}$

Step 1: Find the probability that the missile is not intercepted

The probability that the missile is intercepted $=\frac{1}{3}$

The probability of the missile hitting target $=\frac{3}{4}$

The probability of an event not happening is $1-P\left(E\right)$

The probability that the missile is not intercepted $=1-\frac{1}{3}$

$=\frac{2}{3}$

Step 2: Find the required probability

The probability of three independent events is the product of the probability of the events.

The probability that all three missiles hit the target is,

$P={\left(\frac{2}{3}\times \frac{3}{4}\right)}^3$

$\Rightarrow$ $P={\left(\frac{1}{2}\right)}^3$

$\Rightarrow$ $P=\frac{1}{8}$

Hence, the correct option is $A)\frac{1}{8}$.