Suppose an exam consists of multiple-choice questions, with each question having

Question

Suppose an exam consists of multiple-choice questions, with each question having 5 options. Exactly one of the five options is the correct answer. Suppose 20 points are awarded for a correctly answered question. a) How many points should be deducted for an incorrectly answered question, so that for a student guessing randomly, the expected score on a question is zero? b) If a student is able to correctly eliminate one option, as a possible correct answer but is still guessing randomly, what happens to his/her expected score for that question. Use the answe from a) as the number of points being deducted for an incorrect answer.

Explanation / Answer

a)

p(guessing correctly)=.2

Let s = score assigned to incorrect question.

E(score) = 0.2*5 + 0.8s = 1 + 0.8s

Now need to find s, so that E(score) = 0
1 + 0.8s = 0
0.8s = -1
s = -1/0.8
s = -1.25

So you should deduct 1.25 points for each incorrect answer.

b)

Student is now guessing out of 4 answers.
p(guessing correctly for this question) = 0.25

E(score) = p * score awarded for correct answer + (1-p) * score deducted for incorrect answer
E(score) = 0.25 * 5 + 0.75 * (-1.25)
E(score) = 1.25 - 0.9375
E(score) = 0.3125

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