1. Random Grades (5 points)
Every week, 20,000 students flip a 10,000-sided fair dice, numbered 1 to 10,000, to see if they can get their GPA changed to a 4.0. If they roll a 1, they win (they get their GPA changed). You may assume each student's roll is independent. Let X be the number of students who win.
(a) [1 Point] For any given week, give the appropriate probability distribution (including parameter(s)), and find the expected number of students who win.
(b) [2 Points] For any given week, find the exact probability that at least 2 students win. Give your answer to 5 decimal places.
(c) [2 Points] For any given week, estimate the probability that at least 2 students win, using the Poisson approximation. Give your answer to 5 decimal places.