GATE CS 2004 Q74 — Probability: Random Variables and Distributions

We can solve this problem step-by-step by calculating the expected marks for a single question, then for a single student, and finally for all $1000$ students.

Step 1: Expected marks of a student on a single question

Each multiple-choice question has $4$ choices, out of which only $1$ is correct. Since a student guesses randomly with uniform probability:

• Probability of a correct answer, $P(\text{Correct}) = \frac{1}{4} = 0.25$
• Probability of an incorrect answer, $P(\text{Incorrect}) = \frac{3}{4} = 0.75$

Let $X$ be the marks obtained on a single question. The expected value $E[X]$ is:

Step 2: Expected marks of a single student for the entire paper

The examination paper has $150$ questions. By the linearity of expectation, the expected total marks $E[S]$ for a single student is:

Step 3: Sum total of expected marks for all $1000$ students

Since there are $1000$ students, the sum total of their expected marks is:

Therefore, the correct option is D.