https://r2.erweima.ai/mathbot/image/1726880836968uuu48hgt.jpg

To solve the problem described in the image, let's break it down step by step.

### Problem Restatement:
In a multiple-choice math test, each question contains four statements, and each statement is either true or false. A student receives points depending on how many correct statements they identify:
- Correctly identifying 1 statement earns 0.1 points.
- Correctly identifying 2 statements earns 0.25 points.
- Correctly identifying 3 statements earns 0.5 points.
- Correctly identifying all 4 statements earns a full 1 point.

The student is randomly guessing their answers for two questions. The task is to calculate the probability that the student scores exactly 1 point (which means they must answer all statements correctly on both questions).

### Solution:

1. **Step 1: Probability for 1 question to be completely correct**
   Since the student is randomly guessing, each statement has a \( \frac{1}{2} \) probability of being correct (true or false).
   
   - The probability of guessing all 4 statements correctly on one question is \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \).

2. **Step 2: Probability for two questions**
   The student must answer all statements on both questions correctly to achieve a total score of 1 point.
   
   - The probability of getting both questions completely correct (since the events are independent) is \( \left( \frac{1}{16} \right) \times \left( \frac{1}{16} \right) = \frac{1}{256} \).

### Final Answer:
The probability that the student will get exactly 1 point by correctly answering both questions is \( \frac{1}{256} \).

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Do you have any questions, or would you like a deeper explanation on any of the steps?

### 5 Related Questions:
1. What would the probability be if the student had to answer only 1 question correctly?
2. How does the probability change if there are more than two questions?
3. What is the expected score if the student guesses randomly for multiple questions?
4. Can the probability be improved if the student eliminates one incorrect option for each statement?
5. How does this scoring system compare to other common multiple-choice grading systems?

### Tip:
Understanding how probabilities compound when dealing with independent events is crucial in solving problems like these.

Trong một đề thi trắc nghiệm môn Toán theo cấu trúc mới nhất của Bộ Giáo Dục và Đào Tạo có dạng câu hỏi trả lời đúng sai. Một câu hỏi có 4 ý hỏi, mỗi ý hỏi học sinh chỉ cần trả lời là đúng hoặc sai. Nếu 1 ý trả lời đúng đáp án thì được 0,1 điểm, đúng đáp án 2 ý được 0,25 điểm, đúng đáp án 3 ý được 0,5 điểm và đúng đáp án cả 4 ý thì được trọn vẹn 1 điểm. Giả sử rằng một thí sinh làm bài bằng cách chọn phương án ngẫu nhiên để trả lời cho 2 câu hỏi dạng đúng sai này. Tính xác suất để học sinh đó được 1 điểm ở phần trả lời 2 câu hỏi dạng đúng sai này?

This problem calculates the probability that a student scores exactly 1 point by correctly answering two true/false questions with random guesses. Each question has four sub-statements, and the student's total score depends on correctly answering all sub-statements. This example covers key concepts in probability theory and independent events.

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