Suppose that Julie and Kevin are both very skilled basketball players and decide to compete in some basketball shooting games. a) (26 pts.) Julie and Kevin start with a free throw contest, in which each player stands behind the freethrow line and attempts to shoot a basketball through the hoop. The contest consists of 20 attempts for each player, with the winner being the player who successfully made more shots than the other player; the contest ends in a tie if the players score the same amount of points. Suppose that Julie and Kevin are equally matched in shooting ability, such that they both make 70% of their attempts. i. Compute the probability that Kevin makes at least 15 shots out of 20 attempts. State any assumptions required to make this calculation. ii. Simulate 10,000 replicates of the described contest to estimate the probability of Julie winning, the probability of Kevin winning, and the probability of the contest ending in a tie. Briey describe the logic of the simulation and clearly comment your code. iii. Let I be the number of shots Julie makes, K the number of shots Kevin makes, and D the difference in number of shots made (where D : I K). Does D follow a binomial distribution? Explain your reasoning. Julie and Kevin decide to play a different game that involves taking shots from anywhere on the basketball court. This includes the free throw shots with 70% chance of success, as well as easier shots with higher success rate and harder shots with lower success rate. Remember that Julie and Kevin are equally matched and have the same success probability for any type of shot. The game consists of 20 rounds. In the rst 1 0 rounds, one player is the \"leader\" and has the chance to score points; the other player becomes the \"leader" in the subsequent 10 rounds. The leader chooses a type of shot to take and makes an attempt. The other player must choose the same type of shot to make; i.e., if Julie is the leader and chooses to attempt a shot with 70% success rate, Kevin must also attempt a shot with 70% success rate. If the leader successfully makes the shot and the other player fails to make the shot, the leader scores a point. If the leader fails to make the shot, the leader does not score a point. The other player may still attempt to make the same shot (for fun), but regardless of whether they fail or succeed, the leader does not score a point. Only the leader can score points. In other words, if Julie is the leader during the rst 10 rounds, Julie has the chance to score points; Kevin can prevent Julie from scoring points but cannot himself score points. The winner is the player with the most points at the end of the 20 rounds. Use the information about this game to answer the following questions. b) (14 pts.) Suppose that Julie and Kevin decide to always attempt free throw shots. Write a simulation with 10,000 replicates to estimate the probability of Julie winning, the prob ability of Kevin winning, and the probability of the contest ending in a tie. Write a brief paragraph outlining the logic of the simulation and clearly comment your code. c) (14 pts.) Suppose that Iulie will continue always attempting free throw shots, but Kevin is considering two possible different strategies: either to always attempt shots with 90% success rate or to always attempt shots with 40% success rate. Remember that a player can only decide what type of shots are being taken for the rounds in which they are the leader. Using a simulation approach with 10,000 replicates, determine whether always attempting easier shots (90% success rate) or always attempting harder shots (40% success rate) is the better choice for Kevin to maximize his chances of winning the contest. Write a brief paragraph outlining the logic of the simulation(s) and clearly comment your code. (1) (6 pts.) A shot can be attempted from (virtually) anywhere on the basketball court; you may assume that various types of shots exist with success rate between 0% and 100%. What type of shot (i.e., what shots with a certain success rate) should Kevin always attempt in order to maximize his chances of winning the contest? Explain your reasoning. Limit your answer to at most ve sentences