A high school history teacher gives a 50-question multiple-choice examination in which each question has four choices. The scoring includes a penalty for guessing. Each correct answer is worth 1 point, and each wrong answer costs 12 point. For example, if a student answers 35 questions correctly, 8 questions incorrectly, and does not answer 7 questions, the total score for this student will be 35  (1/2)(8) = 31.
a. What is the expected score of a student who answers 38 questions correctly and guesses on the other 12 questions? Assume that the student randomly chooses one of the four answers for each of the 12 guessed questions.
b. Does a student increase his expected score by guessing on a question if he has no idea what the correct answer is? Explain.
c. Does a student increase her expected score by guessing on a question for which she can eliminate one of the wrong answers? Explain.