Assume your class consists of you and another 39 students, all of you attended the final examination in an advanced mathematics Course. The exam was 40 multiple choice questions and any student needs to complete at least %50 of the questions right in order to pass the exam. Each question has the same weight in grading, no partial grading if didn’t answered a question right, and full credit for each question if a student chooses the right answer. Each one of you and all other students either knew the right answer for certain and will choose, or was unable to have a single hint or clue of how to solve the question and will choose a random answer. Assume that each student has a zero probability to place a right or a wrong answer by error (Meaning you just, either know it and get credit for it, or do not know it and randomly pick a choice. No unfilled questions).

Now assume that you and another 19 students (half of the class) knew for sure how to answer 10 of the questions right, the other half of the class knew less questions to answer right, half of them knew how to answer 6 questions right and the rest knew how to answer 4 questions right.

Answer all of these questions:

I. What is the non-weighted average probability that all of the students will pass the exam?

II. What is the weighted average probability that all of the students will pass the exam?

III. What is the non-weighted average probability that none of the students will pass the exam?

IV. What is the weighted average probability that none of the students will pass the exam?