In an examination, there are 3 multiple-choice questions and each question has 4 choices. If a student randomly select an answer for all the three question, what is the probability that the students will not answer all the three questions correctly?

Options

A. `1//64`
B. `63//64`
C. `1//12`
D. `11//12`

Answer

To solve the problem, we need to determine the probability that a student will not answer all three multiple-choice questions correctly when selecting answers randomly.

### Step-by-Step Solution:

1. **Identify the Total Number of Questions and Choices**:
- There are 3 multiple-choice questions.
- Each question has 4 choices.

2. **Calculate the Total Possible Outcomes**:
- For each question, there are 4 possible answers.
- Therefore, the total number of ways to answer all 3 questions is calculated as:
\[ \text{Total Outcomes} = 4 \times 4 \times 4 = 4^3 = 64 \]

3. **Determine the Favorable Outcomes for Not Answering All Correctly**:
- If a student answers all questions correctly, there is only 1 way to do this (choosing the correct answer for each question).
- Thus, the number of favorable outcomes for not answering all questions correctly is:
\[ \text{Favorable Outcomes} = \text{Total Outcomes} - \text{Correct Outcomes} = 64 - 1 = 63 \]

4. **Calculate the Probability of Not Answering All Questions Correctly**:
- The probability \( P \) of not answering all questions correctly is given by the formula:
\[ P(\text{not all correct}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{63}{64} \]

5. **Conclusion**:
- The probability that the student will not answer all three questions correctly is:
\[ P(\text{not all correct}) = \frac{63}{64} \]

### Final Answer:
The probability that the student will not answer all three questions correctly is \( \frac{63}{64} \).