An exam consists of 20 multiple-choice questions with 5 possible answer choices per question. You haven't studied, so you decide to make random guesses for each answer choice. Assume each question's answer is independent from the others.
a. What is the probability that a person who randomly guesses on each question gets exactly 5 questions correct given that they got at least 1 question correct?
b. What is the expected number of correct answers on the exam? What is the standard deviation?
c. What is the probability I will have to grade more than 12 exams to find one with exactly 5 correct answers, assuming all of my students were randomly guessing on all the questions?
d. What is the expected number of correct answers if a person takes 6 exams? What is the standard deviation in the number of correct answers if a person takes 6 exams?
e. If you have to pay $2 to take the exam, but your parents pay you 50 cents for every correct answer, what is your expected net profit? What is the] standard deviation in your net profit?