3. Suppose that a student bubbled in answers to an exam incorrectly by g in the

Question

3. Suppose that a student bubbled in answers to an exam incorrectly by g in the next largest answer choice for each item. (For example, if he thought the answer was 2, then he bubbled in 3.) Suppose that there were 20 items on the exam, each having four answer choices, and each worth one point if correct and none if incorrect. The correct answers are ordered randomly throughout the exam. Given that the class average was 75%, what is the expected value of such a student's score? Some things to note: The bubble sheet has ten options for each item, so there is no "wrapping around" for an answer that would normally be 4. You may assume that this student is "average", in the sense that the student's expected score is 75%. You may assume that when the student is wrong, the chosen answer is randomly chosen among the three incorrect choices.

Explanation / Answer

The probability that the student knows an answer correctly is 0.75. Now anytime he knows the correct answer he marks it wrong and scores 0.

The probability that the student does not know the correct answer is 1-0.75 = 0.25. Now he chooses one of the three other options randomly. So the probability that he chooses the right answer from the remaining three is 0.33(i.e 1/3).

Thus the total probability of choosing a right answer is 0.25*0.33

Thus the expected value of his score = 20*0.25*0.33 = 1.667

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