Question

A psychology quiz consists of ten multiple choice questions (each question is of worth 1 mark) with 5 options per question. A student knows the correct answer to 5 of the questions, can rule out 1 option in 3 questions, can rule out 2 options in 1 question, can not rule out any options in 1 question. What are the expected marks that the student would get?

Answer 1

Answer:

Let, Xi is the random variable which denotes the mark the student gets in ith question, i = 1(1)10

Since, each question is of worth 1 mark and there is no negative marking, then we have,

Xi = 1 when the student answers the question correctly

= 0 when the student answers the question wrong

we know, the student knows the correct answer to 5 of the questions, can rule out 1 option in 3 questions, can rule out 2 options in 1 question, can not rule out any options in 1 question.

Without loss of generality, let us assume he knows the correct answer to first 5 questions, can rule out 1 option in 6th, 7th and 8th questions, can rule out 2 options in 9th question, can not rule out any options in 10th question.

If P (Xi = 1) denotes the probability that the student will give correct answer to the ith question then we have,

P (X1 = 1) = P (X2 = 1) = P (X3 = 1) = P (X4 = 1) = P (X5 = 1) = 1 [since he knows the correct answer to first 5 questions]

For questions 6, 7 and 8, he can rule out 1 option. Since, each question has 5 options, to give a correct answer he has to choose one option out of the remaining 4 options. So, the probability of giving correct answer to question 6, 7 and 8 is:

P (X6 = 1) = P (X7 = 1) = P (X8 = 1) = 1/4

For question 9, he can rule out 2 options. In order to give a correct answer, he has to choose one option out of the remaining 3 options. So, the probability of giving correct answer to question 9 is:

P (X9 = 1) = 1/3

For question 10, he can not rule out any options. So, in order to give a correct answer, he has to choose one option out of the 5 options. So, the probability of giving correct answer to question 10 is:

P (X10 = 1) = 1/5

So the expected marks that the student would get is:

= 1 X 1 + 1 X 1 + 1 X 1 + 1 X 1 + 1 X 1 + 1 X 1/4 + 1 X 1/4 + 1 X 1/4 + 1 X 1/3 + 1 X 1/5

= 5 + 3/4 + 1/3 + 1/5

= (300 + 45 + 20 + 12)/60

= 377/60 = 6.2833 (approx)

Answer: The expected marks that the student would get is 6.2833 (approx).