Four persons can hit a target correctly with probabilities `(1)/(2),(1)/(3),(1)/(4)` and `(1)/(8)` respectively.If all hit at the target independently, then the probability that the target would be hit, is

Options

A. `(7)/(32)`
B. `(25)/(192)`
C. `(25)/(32)`
D. `(1)/(192)`

Explanation

To solve the problem, we need to find the probability that at least one of the four persons hits the target. We will use the concept of complementary probability, which states that the probability of at least one event occurring is equal to one minus the probability of none of the events occurring.

### Step-by-Step Solution:

1. **Identify the probabilities of hitting the target:**
- Let \( P(A) = \frac{1}{2} \) (Person A)
- Let \( P(B) = \frac{1}{3} \) (Person B)
- Let \( P(C) = \frac{1}{4} \) (Person C)
- Let \( P(D) = \frac{1}{8} \) (Person D)

2. **Calculate the probabilities of missing the target:**
- The probability of A missing the target:
\[ P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \]
- The probability of B missing the target:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \]
- The probability of C missing the target:
\[ P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \]
- The probability of D missing the target:
\[ P(D') = 1 - P(D) = 1 - \frac{1}{8} = \frac{7}{8} \]

3. **Calculate the probability that all four persons miss the target:**
Since the events are independent, the probability that all four miss the target is the product of their individual probabilities of missing:
\[ P(A' \cap B' \cap C' \cap D') = P(A') \cdot P(B') \cdot P(C') \cdot P(D') \]
\[ = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{7}{8} \]

4. **Perform the multiplication:**
- First, multiply the fractions:
\[ = \frac{1 \cdot 2 \cdot 3 \cdot 7}{2 \cdot 3 \cdot 4 \cdot 8} = \frac{42}{192} \]

5. **Simplify the fraction:**
- Simplifying \( \frac{42}{192} \):
\[ = \frac{21}{96} = \frac{7}{32} \]

6. **Calculate the probability that at least one person hits the target:**
\[ P(\text{at least one hits}) = 1 - P(A' \cap B' \cap C' \cap D') = 1 - \frac{7}{32} = \frac{32 - 7}{32} = \frac{25}{32} \]

### Final Answer:
The probability that the target would be hit is \( \frac{25}{32} \).