A group of N people are invited to an algorithms conference. Guests are given an N-sided die and a watch as memorabilia. During a lunch gathering of the entire group, people keep accidentally talking over each other and rapidly grow frustrated. They realize they need a system such that, at least a good portion of the time, only one person will speak at a time. Refusing to simply assign a moderator that decides who gets to speak, they devise the following system: At each timeslot (lets say, 10 seconds), anyone who wishes to speak rolls their N-sided dice. If they roll a 1, they proceed to speak. Otherwise, they do nothing. Assume N to be arbitrarily large. (a) Assuming all N people try to speak in a given timeslot, what is the probability that no one speaks, assuming everyone follows the dice-roll convention? (b) Assuming all N people try to speak in a given timeslot, what is the probability that exactly one person speaks (with no one speaking over them)? (c) What if only half of our N attendees attempted to speak at the same time? What would be the approximate probability exactly one person speaks? (Hint: for this problem, you will need to use the limit definition of e x , which our first deathbed formula is based on).