5. In the popular app HQ Trivia, participants answer 12 multiple choice trivia questions with 3 possible answers for each question. At the end of the game, everyone who got all 12 questions right...

5. In the popular app HQ Trivia, participants answer 12 multiple choice trivia questions with 3 possible answers for each question. At the end of the game, everyone who got all 12 questions right splits the pot of money available. (a) If all choices are equally likely and I guess on every question, what's the probability that I win the game? (b) If you've ever played HQ Trivia, you probably noticed that the first three questions are quite obvious. If we assume I know the first three questions, and guess on all of the other questions, how much more likely am I to win than in part (a)? (Calculate the "new" probability of winning divided by the "old" probability of winning) (c) I'm actually not bad at HQ trivia, and I usually make it through the first six questions with no problem. If I assume that I get the first six questions right, and can eliminate one possible answer for the next four questions (so I only guess between two options), but still have to guess between all three options on the last two questions, what's the probability that I win? (d) This probability of winning is still too small for me. I want to guarantee a win, so I'm going to collude with some of my closest friends. If we assume we get the first three questions right as in (b), how many friends do I need to guarantee that at least one of us wins? (e) If there are 1,000,000 people playing for a S5, 000 prize, and they are all guessing as in part (a), how much should the winners expect to receive?