While your advisors are running around with big nets trying to catch drones for your data, you decide to relax, by going incognito among your people and seeing how they're doing. You find that they are playing the following game: a player will show you two coins, one of which is fair (each outcome with equal probability), the other of which is loaded', giving Hyoid with probability 0.75, and Thyroid with probability 0.25. The player will select one of these coins at random and flip a number of times, at which point you have to guess, based on the results, which coin it is. You decide on the following strategy: out of N flips, if over half of them are H, you'll say it's the loaded coin, otherwise you'll say it's the fair coin. 1) Following this strategy, what's the probability you win in a game of 10 flips? 20 flips? (5 points) 2) If you want to maximize your probability of winning, what threshold should you set for your strategy in a game of 10 flips? 20 flips? Be thorough as to your methods, and precise. (5 points) 3) If the loaded coin had a probability p > 1/2 of giving H and 1 p of giving T, where should you set your threshold for your strategy, to maximize your probability of winning in a game of N flips? (13 points) 4) Taking p = 0.75 as before, if you could specify in advance how many flips you wanted before guessing - how many flips should you ask for if you wanted as few flips as possible, but still at least a 90% chance of winning? (Assume you use an optimal threshold.) (7 points) While your advisors are running around with big nets trying to catch drones for your data, you decide to relax, by going incognito among your people and seeing how they're doing. You find that they are playing the following game: a player will show you two coins, one of which is fair (each outcome with equal probability), the other of which is loaded', giving Hyoid with probability 0.75, and Thyroid with probability 0.25. The player will select one of these coins at random and flip a number of times, at which point you have to guess, based on the results, which coin it is. You decide on the following strategy: out of N flips, if over half of them are H, you'll say it's the loaded coin, otherwise you'll say it's the fair coin. 1) Following this strategy, what's the probability you win in a game of 10 flips? 20 flips? (5 points) 2) If you want to maximize your probability of winning, what threshold should you set for your strategy in a game of 10 flips? 20 flips? Be thorough as to your methods, and precise. (5 points) 3) If the loaded coin had a probability p > 1/2 of giving H and 1 p of giving T, where should you set your threshold for your strategy, to maximize your probability of winning in a game of N flips? (13 points) 4) Taking p = 0.75 as before, if you could specify in advance how many flips you wanted before guessing - how many flips should you ask for if you wanted as few flips as possible, but still at least a 90% chance of winning? (Assume you use an optimal threshold.) (7 points)
