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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 is

1. $0$
2. $2550$
3. $7525$
4. $9375$

edited | 2.2k views

Probability of choosing the correct option = $\frac{1}{4}$
Probability of choosing a wrong option = $\frac{3}{4}$

So, expected mark for a question for a student = $\frac{1}{4} \times 1 + \frac{3}{4}\times (-0.25) = 0.0625$

Expected mark for a student for $150$ questions = $0.0625 \times 150 = 9.375$

So, sum total of the expected marks obtained by all $1000$ students = $9.375 \times 1000 = 9375$.

by Veteran (424k points)
edited
+1
'Students choose all their answers randomly with uniform probability' Does this mean that it has unform distribution?
0
We want to calculate

$\bigcup_{i=1}^{1000}E[S_i]=\sum_{i=1}^{1000}E[S_i]$

where $S_i$ denotes expected marks of a Student i.

Now to calculate the expected marks of a student, we need to take Union over expectations of marks from each question

$E[S_i]=\bigcup_{j=1}^{150}E[Q_i]=\sum _{j=1}^{150}E[Q_i]$

where $E[Q_i]$ indicates expected marks per question.

Now with probabilty $\frac{1}{4}$ we can correctly mark a question and fetch +1 mark or with probability $\frac{3}{4}$ we can incorrectly mark a question and loose -0.25 marks.

So Expected marks per question=$\left ( \frac{1}{4}*1 \right )+\left ( \frac{3}{4} *-0.25\right )=0.0625=E[Q_i]$

So, $E[S_i]=150*0.0625=9.375$

And sum total of expected marks of all students=1000*9.375=9375.

It is a Binomial experiment as we are doing the same bernouli experiment repeatedly.

Which is the Bernouli experiment here ? Visiting a question and checking whether it is a correct answer or not. Why it is a Bernouli experiment ? Because a question can have either correct answer(success) or wrong answer(failure).

Total number of questions we are visiting,n = 1000*150 = 150000.

Let X be a random variable which indicates number of correct answer among the total of 150000 questions.

E(X) (i.e) Expectation of X (i.e) Expected number of correct answers = n * p  (since it is a binomial experiment)

where n  is no: of Bernouli experiments and p is probability of success (i.e) probability of correct answer.

Here n = 150000 , p = 0.25

So,

Expected number of correct answers = 150000 * .25 = 37500.

Expected number of wrong answers = 150000 - 37500 = 112500.

Expected Total marks = Expected number of correct answers * number of marks per correct answer + Expected number of wrong answers * number of marks per wrong answer.

= [37500 * 1] + [112500 * (-.25)]

= 37500 - 28125

= 9375

by Loyal (7.9k points)