The probability that a man can hit a target is `3//4`. He tries 5 times. The probability that he will hit the target at least three times is

Options

A. 291/364
B. 371/464
C. 471/502
D. 459/512

Answer

To solve the problem, we need to find the probability that a man hits a target at least three times out of five attempts, given that the probability of hitting the target in a single attempt is \( \frac{3}{4} \).

### Step-by-Step Solution:

1. **Identify the parameters**:
- Probability of success (hitting the target), \( p = \frac{3}{4} \)
- Probability of failure (not hitting the target), \( q = 1 - p = 1 - \frac{3}{4} = \frac{1}{4} \)
- Number of trials, \( n = 5 \)

2. **Define the event**:
- We want to find the probability of hitting the target at least 3 times. This can be expressed as:
\[ P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5) \]

3. **Use the binomial probability formula**:
The binomial probability formula is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
where \( \binom{n}{r} \) is the binomial coefficient.

4. **Calculate \( P(X = 3) \)**:
\[ P(X = 3) = \binom{5}{3} \left( \frac{3}{4} \right)^3 \left( \frac{1}{4} \right)^{5-3} \]
\[ = \binom{5}{3} \left( \frac{3}{4} \right)^3 \left( \frac{1}{4} \right)^2 \]
\[ = 10 \cdot \left( \frac{27}{64} \right) \cdot \left( \frac{1}{16} \right) = 10 \cdot \frac{27}{1024} = \frac{270}{1024} \]

5. **Calculate \( P(X = 4) \)**:
\[ P(X = 4) = \binom{5}{4} \left( \frac{3}{4} \right)^4 \left( \frac{1}{4} \right)^{5-4} \]
\[ = 5 \cdot \left( \frac{81}{256} \right) \cdot \left( \frac{1}{4} \right) = 5 \cdot \frac{81}{1024} = \frac{405}{1024} \]

6. **Calculate \( P(X = 5) \)**:
\[ P(X = 5) = \binom{5}{5} \left( \frac{3}{4} \right)^5 \left( \frac{1}{4} \right)^{5-5} \]
\[ = 1 \cdot \left( \frac{243}{1024} \right) \cdot 1 = \frac{243}{1024} \]

7. **Sum the probabilities**:
\[ P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5) \]
\[ = \frac{270}{1024} + \frac{405}{1024} + \frac{243}{1024} \]
\[ = \frac{270 + 405 + 243}{1024} = \frac{918}{1024} \]

8. **Simplify the fraction**:
\[ = \frac{459}{512} \]

### Final Answer:
The probability that he will hit the target at least three times is \( \frac{459}{512} \).