The probability that a man can hit a target is `3//4`. He makes 5 trials. The probability that he will hit the target every time he hits is

Options

A. `(243)/(1024)`
B. `(81)/(1024)`
C. `(243)/(256)`
D. `(241)/(256)`

Answer

To solve the problem, we need to find the probability that a man hits a target every time he attempts to hit it in 5 trials, given that the probability of hitting the target in a single trial is \( \frac{3}{4} \).

### Step-by-Step Solution:

1. **Identify the Probability of Hitting the Target**:
The probability of hitting the target in a single trial is given as \( P(H) = \frac{3}{4} \).

2. **Determine the Number of Trials**:
The man makes 5 trials. So, the number of trials \( n = 5 \).

3. **Calculate the Probability of Hitting the Target Every Time**:
To find the probability that he hits the target every time in all 5 trials, we need to raise the probability of hitting the target in a single trial to the power of the number of trials. This is calculated as follows:
\[ P(\text{hitting every time}) = P(H)^n = \left(\frac{3}{4}\right)^5 \]

4. **Calculate \( \left(\frac{3}{4}\right)^5 \)**:
\[ \left(\frac{3}{4}\right)^5 = \frac{3^5}{4^5} = \frac{243}{1024} \]

5. **Final Result**:
Therefore, the probability that he will hit the target every time in 5 trials is:
\[ P(\text{hitting every time}) = \frac{243}{1024} \]

### Answer:
The probability that he will hit the target every time he attempts is \( \frac{243}{1024} \).

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