A and B try to hit a target. The probability that A hits the target is 7/10 and the probability that B hits the target is 4/10. If these two events are independent, the probability that B hits the target, given that the target is hit,is :

Options

A. `17//41`
B. `20//41`
C. `19//41`
D. `18//41`

Answer

To solve the problem, we need to find the probability that B hits the target given that the target is hit. We will use the concept of conditional probability.

### Step-by-Step Solution:

1. **Define the Events**:
- Let \( A \) be the event that A hits the target.
- Let \( B \) be the event that B hits the target.
- The probability that A hits the target is given as \( P(A) = \frac{7}{10} \).
- The probability that B hits the target is given as \( P(B) = \frac{4}{10} \).

2. **Calculate the Probabilities of Missing the Target**:
- The probability that A does not hit the target is \( P(A') = 1 - P(A) = 1 - \frac{7}{10} = \frac{3}{10} \).
- The probability that B does not hit the target is \( P(B') = 1 - P(B) = 1 - \frac{4}{10} = \frac{6}{10} \).

3. **Calculate the Probability that the Target is Hit**:
- The target can be hit in three scenarios:
1. A hits and B misses: \( P(A \cap B') = P(A) \cdot P(B') = \frac{7}{10} \cdot \frac{6}{10} = \frac{42}{100} \).
2. A misses and B hits: \( P(A' \cap B) = P(A') \cdot P(B) = \frac{3}{10} \cdot \frac{4}{10} = \frac{12}{100} \).
3. Both A and B hit: \( P(A \cap B) = P(A) \cdot P(B) = \frac{7}{10} \cdot \frac{4}{10} = \frac{28}{100} \).
- Therefore, the total probability that the target is hit is:
\[ P(\text{Target is hit}) = P(A \cap B') + P(A' \cap B) + P(A \cap B) = \frac{42}{100} + \frac{12}{100} + \frac{28}{100} = \frac{82}{100} = \frac{41}{50}. \]

4. **Calculate the Probability that B Hits the Target and the Target is Hit**:
- The scenarios where B hits the target and the target is hit are:
1. A misses and B hits: \( P(A' \cap B) = \frac{12}{100} \).
2. Both A and B hit: \( P(A \cap B) = \frac{28}{100} \).
- Therefore, the total probability that B hits the target and the target is hit is:
\[ P(B \cap \text{Target is hit}) = P(A' \cap B) + P(A \cap B) = \frac{12}{100} + \frac{28}{100} = \frac{40}{100} = \frac{2}{5}. \]

5. **Apply the Conditional Probability Formula**:
- The conditional probability that B hits the target given that the target is hit is given by:
\[ P(B | \text{Target is hit}) = \frac{P(B \cap \text{Target is hit})}{P(\text{Target is hit})} = \frac{\frac{2}{5}}{\frac{41}{50}}. \]
- Simplifying this gives:
\[ P(B | \text{Target is hit}) = \frac{2}{5} \cdot \frac{50}{41} = \frac{100}{205} = \frac{20}{41}. \]

### Final Answer:
The probability that B hits the target given that the target is hit is \( \frac{20}{41} \).