Question

An 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 answer randomly with uniform probability. The sum total of the expected marks obtained by all the students is

• A. 0
• B. 2550
• C. 7525
• D. 9375

Answer 1

The correct option is D 9375

Let x = {marks obtained by single student in single question}

= {1, -1/4}

So, $p_1=p(x=1\text{marks})=1/4$

$p_2=p(x=-\frac{1}{4}\text{marks})=\frac{3}{4}$

So probability distribution is
$\begin{array}{ccc}x:& 1& -\frac{1}{4}\\ p\left(x\right):& \frac{1}{4}& \frac{3}{4}\end{array}$

So expected marks obtained by single student in single question

$E\left(x\right)=\sum p_i{X}_i=p_1{x}_1+p_2{x}_2$
$=1.\left(0.25\right)+(-0.25)\left(0.75\right)=0.0625$

Expected marks for 150 questions = 150.(0.0625)

Total expected marks for 1000 students
= 1000(150) (0.0625)
= 9375 marks