A student is given a quiz with 10 true or false question, and he answers by sheer guessing. If X is the number of question answered correctly. If the student passes the quiz by getting 7 or maximum correct answer, then the probability that the student passes the quiz is

Options

A. `11/32`
B. `11/16`
C. `11/64`
D. `11/128`

Answer

To solve the problem, we need to find the probability that a student passes a quiz by answering at least 7 out of 10 true or false questions correctly, given that he answers by sheer guessing.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The student answers 10 questions, each with two possible answers (True or False). The probability of answering a question correctly (P) is 1/2, and the probability of answering incorrectly (Q) is also 1/2.

2. **Defining the Random Variable**:
Let \( X \) be the number of questions answered correctly. \( X \) follows a binomial distribution with parameters \( n = 10 \) (the number of questions) and \( p = 1/2 \) (the probability of a correct answer).

3. **Finding the Required Probability**:
We need to find the probability that the student passes the quiz, which is defined as answering 7 or more questions correctly. Mathematically, this is expressed as:
\[ P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \]

4. **Using the Binomial Probability Formula**:
The probability mass function for a binomial distribution is given by:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
For our case, \( n = 10 \) and \( p = 1/2 \). Therefore, the formula simplifies to:
\[ P(X = k) = \binom{10}{k} \left(\frac{1}{2}\right)^{10} \]

5. **Calculating Each Probability**:
- For \( k = 7 \):
\[ P(X = 7) = \binom{10}{7} \left(\frac{1}{2}\right)^{10} = \frac{120}{1024} \]
- For \( k = 8 \):
\[ P(X = 8) = \binom{10}{8} \left(\frac{1}{2}\right)^{10} = \frac{45}{1024} \]
- For \( k = 9 \):
\[ P(X = 9) = \binom{10}{9} \left(\frac{1}{2}\right)^{10} = \frac{10}{1024} \]
- For \( k = 10 \):
\[ P(X = 10) = \binom{10}{10} \left(\frac{1}{2}\right)^{10} = \frac{1}{1024} \]

6. **Summing the Probabilities**:
Now we add these probabilities together:
\[ P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \]
\[ P(X \geq 7) = \frac{120}{1024} + \frac{45}{1024} + \frac{10}{1024} + \frac{1}{1024} = \frac{176}{1024} \]

7. **Simplifying the Result**:
Simplifying \( \frac{176}{1024} \):
\[ \frac{176 \div 16}{1024 \div 16} = \frac{11}{64} \]

### Final Answer:
The probability that the student passes the quiz is \( \frac{11}{64} \).