Question

An examination paper has $150$ mutliple-choice questions of one mark each, with each question having four choices which are equally likely. Each incorrect answer fetches $-0.25$ mark. Suppose $1000$ students choose all their answers randomly with uniform probability. The sum total of the expected marks by all these students is

Answer 1

Let the marks obtained per question be a random variable $X$
For $x=1,p\left(x\right)=\frac{1}{4}$
For $x=-0.25,p\left(x\right)=\frac{3}{4}$

Expected marks per question,
$E\left(X\right)=\sum xp\left(x\right)=\frac{1}{4}-\frac{3}{16}=\frac{1}{16}$

Expected marks for $150$ questions
$=\frac{1}{16}\times 150=\frac{75}{8}$

Total Expected marks of $1000$ students
$=\frac{75}{8}\times 1000=9375$