Step-by-step Solution:

We are given: \[\] A is being targeted by B and C \[\] B hits with probability $P(B) = \frac{1}{2}$ \[\] C hits with probability $P(C) = \frac{1}{3}$ \[\] A is hit \[\] We are to find: $$ P(B \text{ hits and } C \text{ misses} \mid A \text{ is hit}) $$ Step 1: Total probability that A is hit \[\] A is hit if either B or C or both hit A. Since their hits are independent, we use: $$ P(\text{A is hit}) = 1 - P(B \text{ misses}) \cdot P(C \text{ misses}) = 1 - \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) $$ $$ = 1 - \left(\frac{1}{2}\right)\left(\frac{2}{3}\right) = 1 - \frac{1}{3} = \frac{2}{3} $$ Step 2: Find joint probability of (B hits and C misses) \[\] This is: $$ P(B \text{ hits and } C \text{ misses}) = \frac{1}{2} \cdot \left(1 - \frac{1}{3}\right) = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3} $$ Step 3: Apply conditional probability formula $$ P(B \text{ hits and } C \text{ misses} \mid A \text{ is hit}) = \frac{P(B \text{ hits and } C \text{ misses})}{P(A \text{ is hit})} = \frac{\frac{1}{3}}{\frac{2}{3}} = \boxed{\frac{1}{2}} $$ Final Answer: $$ {\frac{1}{2}} $$