Consider an exam which has n multiple choice questions. Each question has k possible answers, among which only one answer is correct. (a) Consider a student who chooses at random the answers for all questions of the exam. Let X be the number of correct answers of this student. What is distribution of X? What is the expected value of X?

b) To eliminate the effect of guessing, the instructor decides to mark the exam according to the following rule: for each correct answer he gives 1 mark, but for each incorrect answer he subtracts x marks with 0 x 1. Find the value of x such that the average grade of a student who chooses at random the answers for all questions is 0.

Hint: Denote by Y the grade of a student who chooses at random all answers, under the new rule. Express E(Y ) as a function of x. Solve for x in the equation E(Y ) = 0.

c) Find the value x for a multiple choice exam as above with n = 25 and k = 5