Four persons can hit a target correctly with probabilities `(1)/(2),(1)/(3),(1)/(4)` and `(1)/(8)` respectively. Probability that target is hit if all hit at the target at same time

Options

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

Answer

To solve the problem step by step, we will calculate the probability that at least one of the four persons hits the target when they all fire at the same time.

### Step-by-Step Solution:

1. **Identify the Probabilities of Hitting the Target:**
- Let the probabilities of hitting the target for the four persons A, B, C, and D be:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{1}{3} \)
- \( P(C) = \frac{1}{4} \)
- \( P(D) = \frac{1}{8} \)

2. **Calculate the Probabilities of Not Hitting the Target:**
- The probabilities of not hitting the target for each person are:
- \( P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \)
- \( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \)
- \( P(D') = 1 - P(D) = 1 - \frac{1}{8} = \frac{7}{8} \)

3. **Calculate the Probability that None of Them Hit the Target:**
- The probability that none of them hit the target is the product of their individual probabilities of not hitting the target:
\[ P(\text{none hit}) = P(A') \times P(B') \times P(C') \times P(D') \]
\[ = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{7}{8} \]

4. **Perform the Multiplication:**
- Simplifying the multiplication step-by-step:
\[ = \frac{1 \times 2 \times 3 \times 7}{2 \times 3 \times 4 \times 8} \]
- Cancel \( 2 \) in the numerator and denominator:
\[ = \frac{1 \times 1 \times 3 \times 7}{1 \times 3 \times 4 \times 8} \]
- Cancel \( 3 \) in the numerator and denominator:
\[ = \frac{1 \times 1 \times 7}{1 \times 4 \times 8} = \frac{7}{32} \]

5. **Calculate the Probability that At Least One Hits the Target:**
- The probability that at least one of them hits the target is given by:
\[ P(\text{at least one hits}) = 1 - P(\text{none hit}) \]
\[ = 1 - \frac{7}{32} = \frac{32 - 7}{32} = \frac{25}{32} \]

### Final Answer:
The probability that at least one of the four persons hits the target is \( \frac{25}{32} \).