Question:

Let $I$ be the event that a missile is intercepted, and $I^c$ be the event that a missile is not intercepted. Let $H$ be the event that a missile hits the target. Step 1: Determine the probability that a single missile is not intercepted. The probability that a missile is intercepted is given as $P(I) = \frac{1}{3}$. The probability that a missile is not intercepted is therefore: $$P(I^c) = 1 - P(I) = 1 - \frac{1}{3} = \frac{2}{3}$$ Step 2: Calculate the probability that a single missile hits the target. The probability that the missile hits the target, given that it is not intercepted, is $P(H | I^c) = \frac{3}{4}$. For a missile to hit the target, it must not be intercepted AND it must hit the target given it was not intercepted. Thus, the probability that a single missile hits the target is: $$P(H) = P(H \cap I^c) = P(H | I^c) \times P(I^c)$$ $$P(H) = \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$$ Step 3: Calculate the probability that all three missiles hit the target. Since the three missiles are fired independently, the probability that all three hit the target is the product of their individual probabilities of hitting the target: $$P(\text{all three hit target}) = (P(H))^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$$