A is targeting to B, B and C are targeting to A. probability of hitting the target by A, B and C are 2/3, 1.2 and 1/3, respectively. If A is hit, then find the Probability that B hits the target and C does not.

Answer

To solve the problem, we need to find the probability that B hits the target while C does not, given that A is hit. We will use conditional probability to achieve this.

### Step-by-step Solution:

1. **Identify the Probabilities**:
- Probability that A hits B, \( P(A) = \frac{2}{3} \)
- Probability that B hits A, \( P(B) = \frac{1}{2} \)
- Probability that C hits A, \( P(C) = \frac{1}{3} \)

2. **Find the Probability that A is Hit**:
- To find the probability that A is hit, we can use the complement rule. The probability that A is not hit (i.e., both B and C miss) is:
\[ P(\text{A not hit}) = P(B \text{ misses}) \times P(C \text{ misses}) = \left(1 - P(B)\right) \times \left(1 - P(C)\right) \]
- Calculating this:
\[ P(B \text{ misses}) = 1 - \frac{1}{2} = \frac{1}{2} \]
\[ P(C \text{ misses}) = 1 - \frac{1}{3} = \frac{2}{3} \]
\[ P(\text{A not hit}) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \]
- Therefore, the probability that A is hit is:
\[ P(A \text{ hit}) = 1 - P(\text{A not hit}) = 1 - \frac{1}{3} = \frac{2}{3} \]

3. **Find the Probability that B Hits and C Does Not Hit A**:
- We need to find \( P(B \text{ hits} \cap C \text{ does not hit} | A \text{ is hit}) \).
- Using the formula for conditional probability:
\[ P(B \text{ hits} \cap C \text{ does not hit} | A \text{ is hit}) = \frac{P(B \text{ hits} \cap C \text{ does not hit} \cap A \text{ is hit})}{P(A \text{ is hit})} \]
- The probability that B hits A and C does not hit A can be expressed as:
\[ P(B) \times P(C \text{ misses}) = P(B) \times (1 - P(C)) \]
- Substituting the values:
\[ P(B \text{ hits}) = \frac{1}{2}, \quad P(C \text{ misses}) = 1 - \frac{1}{3} = \frac{2}{3} \]
\[ P(B \text{ hits} \cap C \text{ does not hit}) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \]

4. **Calculate the Final Probability**:
- Now substituting back into the conditional probability formula:
\[ P(B \text{ hits} \cap C \text{ does not hit} | A \text{ is hit}) = \frac{P(B \text{ hits} \cap C \text{ does not hit})}{P(A \text{ is hit})} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]

### Final Answer:
The probability that B hits the target and C does not, given that A is hit, is \( \frac{1}{2} \).