Question

A student is taking a standardized test consisting of several​ multiple-choice questions. One point is awarded for each correct answer. Questions left blank neither receive nor lose points. If there are six options for each question and the student is penalized 1/3 point for each wrong​ answer, how many options must the student be able to rule out before the expected value of guessing is zero?

Answer 1

We are given a student will get 1 point for each correct answer and will get -1/3 point on each wrong answer.

Also we know that each question has 6 options.

Case-1:

Therefore, if the student guesses, then he has to guess an option out of 6 other options

Thus the probability of correct answer when students rule out 0 is:   

Also, the probability of wrong answer when students rule out 0 is:

Therefore, the expected value is given by:

This gives us

The expectation is negative, so one should not guess if only one option can be eliminated.

Case-2:

Thus the probability of correct answer when students rule out 1 is:   

Also, the probability of wrong answer when students rule out 0 is:

Therefore, the expected value is given by:

The expectation is negative, so one should not guess if only one
option can be eliminated.

Case-3:

Thus the probability of correct answer when students rule out 2 is:   

Also, the probability of wrong answer when students rule out 0 is:

Therefore, the expected value is given by:

Case-4:

Thus the probability of correct answer when students rule out 3 is:   

Also, the probability of wrong answer when students rule out 0 is:

Therefore, the expected value is given by:

The number of options the student must be able to rule out before the expected value of guessing becomes zero is 3.