A student appeared in an examination consisting of 8 true - false . The student guesses the answers with equal probability . The smallest value of n, so that the probability of guessing at least 'n' correct answer is less than `(1)/(2)` is

Options

A. 5
B. 6
C. 3
D. 4

Answer

To solve the problem, we need to determine the smallest value of \( n \) such that the probability of guessing at least \( n \) correct answers out of 8 true-false questions is less than \( \frac{1}{2} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
The student guesses the answers with equal probability, meaning the probability of getting each answer correct is \( \frac{1}{2} \). We need to find the smallest \( n \) such that the probability of getting at least \( n \) correct answers is less than \( \frac{1}{2} \).

2. **Setting Up the Probability**:
The total number of questions is 8. The number of ways to choose \( r \) correct answers out of 8 is given by the binomial coefficient \( \binom{8}{r} \). The probability of getting exactly \( r \) correct answers when guessing is:
\[ P(X = r) = \binom{8}{r} \left(\frac{1}{2}\right)^8 \]
Therefore, the probability of getting at least \( n \) correct answers is:
\[ P(X \geq n) = \sum_{r=n}^{8} P(X = r) = \sum_{r=n}^{8} \binom{8}{r} \left(\frac{1}{2}\right)^8 \]

3. **Finding the Cumulative Probability**:
We can express this as:
\[ P(X \geq n) = \frac{1}{2^8} \sum_{r=n}^{8} \binom{8}{r} \]
We need to find the smallest \( n \) such that:
\[ \frac{1}{2^8} \sum_{r=n}^{8} \binom{8}{r} < \frac{1}{2} \]
This simplifies to:
\[ \sum_{r=n}^{8} \binom{8}{r} < 2^7 = 128 \]

4. **Calculating the Binomial Coefficients**:
We need to calculate the cumulative sum of the binomial coefficients from \( n \) to 8 and find the smallest \( n \) for which this sum is less than 128.

- For \( n = 0 \):
\[ \sum_{r=0}^{8} \binom{8}{r} = 256 \]
- For \( n = 1 \):
\[ \sum_{r=1}^{8} \binom{8}{r} = 256 - 1 = 255 \]
- For \( n = 2 \):
\[ \sum_{r=2}^{8} \binom{8}{r} = 256 - 8 - 1 = 247 \]
- For \( n = 3 \):
\[ \sum_{r=3}^{8} \binom{8}{r} = 256 - 28 - 8 - 1 = 219 \]
- For \( n = 4 \):
\[ \sum_{r=4}^{8} \binom{8}{r} = 256 - 70 - 28 - 8 - 1 = 149 \]
- For \( n = 5 \):
\[ \sum_{r=5}^{8} \binom{8}{r} = 256 - 126 - 70 - 28 - 8 - 1 = 70 \]

5. **Conclusion**:
The smallest \( n \) such that the sum is less than 128 is \( n = 5 \).

Thus, the answer is:
\[ \boxed{5} \]