An examination paper has 150 multiple choice questions of one mark each, with each question having four choices. Each incorrect answer fetches -0.25 marks. Suppose 1000 students choose all their answers randomly with uniform probability.The sum total of the expected marks obtained by all these students isa)0b)2550c)7525d)9375Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Probability of choosing the correct option = 1/4
Probability of choosing a wrong option = 1/3
So, expected mark for a question for a student

Expected mark for a student for 150 questions

So, sum total of the expected marks obtained by all 1000 students

Answer

Answer

Total Marks Calculation
To find the expected marks obtained by the students, we need to calculate the expected score for one student first and then multiply it by the total number of students.
Expected Score for One Student
- Each question has 4 choices, so the probability of choosing the correct answer is $ \frac{1}{4} $.
- The probability of choosing an incorrect answer is $ \frac{3}{4} $.
Marks Breakdown
- For a correct answer, the student earns +1 mark.
- For an incorrect answer, the student loses -0.25 marks.
Expected Value Calculation
- The expected score for one question can be calculated as follows:
$$
\text{Expected Score} = \left( \frac{1}{4} \times 1 \right) + \left( \frac{3}{4} \times (-0.25) \right)
$$
- Simplifying this:
$$
\text{Expected Score} = \frac{1}{4} - \frac{3}{16} = \frac{4}{16} - \frac{3}{16} = \frac{1}{16}
$$
Expected Score for 150 Questions
- For 150 questions, the expected score becomes:
$$
\text{Total Expected Score for One Student} = 150 \times \frac{1}{16} = \frac{150}{16} = 9.375
$$
Total Expected Score for 1000 Students
- Finally, for 1000 students:
$$
\text{Total Expected Marks} = 1000 \times 9.375 = 9375
$$
Thus, the correct answer is option 'D', which is 9375.

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