Αn exam paper has 150 multiple choice questions of 1 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. Find the sum total of the expected marks obtained by all the students.

Αn exam paper has 150 multiple choice questions of 1 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. Find the sum total of the expected marks obtained by all the students. 

(A) 0 

(B) 2550 

(C) 7525 

(D) 9375 

(GATE 2004) 

Answer: (D) 9375 

Explanation: 

Let $X$ be a random variable that denotes the marks obtained for each question. 

Here, the probability distribution of $X$ is as follows:  

\[P(X = x) = \left\{ \begin{array}{ll} 0.25 & \mbox{if } x = 1 \\ 0.75 & \mbox{if } x = -0.25 \end{array} \right.\]

Now, the expected marks for one question

$= E(X) = 1 \times 0.25 + (-0.25) \times 0.75$ 

$= 0.0625$ 

Therefore, the expected marks for 150 questions  

$= 150 \times E(X)$ 

$= 150 \times 0.0625$ 

$= 9.375$ 

Therefore, the sum total of the expected marks obtained by all the students 

$= 1000 \times 9.375$ 

$= 9375$