Question

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) For any given week, give the appropriate probability distribution (including parameter(s)), and find the expected number of students who win.

(b) For any given week, find the exact probability that at least 2 students win. Give your answer to 5 decimal places.

(c) For any given week, estimate the probability that at least 2 students win, using the Poisson approximation. Give your answer to 5 decimal places.

Answer 1

The probability of winning is the probability of rolling a 1 and is equal to

a) Let X be the number of students who win out of 20000 students who try. X has a Binomial distribution with parameters n=20000 as the number of trials and p=0.0001 as the success probability (probability that a randomly selected student wins).

The probability of X=x is given by

The expected value of X is

The expected number of students who win each week is 2

b) The  probability that at least 2 students win is

ans: The  probability that at least 2 students win is 0.59401

c) Let Y be the Poisson approximation for the number of students who win each week. Y has a Poisson distribution with parameter

The probability of Y=y is

The probability that at least 2 students win is

The probability that at least 2 students win, using the Poisson approximation is 0.59399