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Problem 1 (30 Points) . Consider three football teams . B and C. In a regular game, A beats B with probability p and beats C with probability q. C is a better team than B. so p > q. To win the tournament, A needs to win two games in a row out of three games against B and C, and it can't play two games in a row against the same team. A can either select to play these three games as B - C - B or C - B -C. Show that A maximizes the probability of winning the tournament by playing against the better team C, twice. (10 points) . Three friends, A. B and C are in a paintball shootout. On each round, until one player remains standing, the current shooter can choose one of the other players as a target and is allowed one shot. At the start of the game, who goes first, second and third are chosen randomly. A player who gets hit is eliminated. A is a perfect shot. he eliminates his targets with 100% probability. B has 80% accuracy. and C has 50% accuracy. We also assume that players are not required to aim at an opponent, they can simply shoot in the air if they desire. Show that C is the one that is most likely to survive this shootout. (20 points) Problem 2 (20 Points) Suppose you sample repeatedly from the continuous uniform distribution on [0, 1]. until you get a number that is less than your previous sample. What is the expected number of samples you obtain? Problem 3 (25 Points) Let X and Y be independent random variables uniformly distributed on [0, 1). Find the CDF and PDF of Z = [X3 - Y | (15 points). Also, find the correlation coefficient between Z and X (10 points). Problem 4 (25 Points) Assume that you are trying to assess if a coin is unbiased or not. You toss the coin 100 times, expecting that around 50 of these tosses will be heads if the coin is fair. However, you observe 60 heads in total, and then you claim that coin is biased. What is the probability that your conclusion is false? Bonus Computational Problem (25 Points) Verify the answers to Problem I by writing a script that simulates these random scenarios, and average multiple runs to estimate the desired probabilities