Probability Take-Home Exam Questions

5. Coffee Contest (20pts)A group of n people meet at lunch for a cup of coffee. They play a game tosee who gets to pay for all the coffees. Each person flips a coin. If all thecoins come up the same except for one person, then that one person gets topay for all the coffee. If the coins do not result in this way, then everyoneflips again until there is exactly one person different. Obviously, the gamedoesn’t work for less than three people. Also, as n grows large it may takemany flips to decide the loser of the game.(a) For n = 3 people, how many flips will it take on average to find awinner?(b) For n = 10 people, how many flips will it take on average to find awinner?(c) Find a general formula that computes the average number of flips as afunction of n. Create a graph for n = 3...20 showing the number ofrequired flips.6. Chutes and Ladders (25 pts)Milton Bradley manufactures a children’s board game called ”Chutes andLadders”. They want to figure out if the game is too difficult for youngchildren; in particular, will the game last too long for the average child’sattention span? Studies indicate that young children of the age appropriatefor this game will be willing to play for approximately 15 minutes beforebecoming bored. To be safe, Milton Bradley wants to ensure that the averagegame of Chutes and Ladders lasts less than 12 minutes. They havealso determined that the average turn in the game takes about 7 seconds. Ashead mathematician at Milton Bradley, your job is to determine if the averagegame can be completed in less than 12 minutes.7. Birthdays in the Crowd (25 pts)Consider a room with n people. Assume there are 365 days in the year(ignore leap years).2