Q7 Jane, a minecraft speedrunner, is planning her schedule for the next few months. Here, we consider a simplified version of the game, where the speedrunner's strategy depends solely on how fast she can convert gold ingots into ender pearls by bartering with piglins. In the simplified game, each gold ingot, when bartered with a piglin, has a chance of 4% of being converted into an ender pearl. To end the game, she needs 14 ender pearls. 7.1 Let X denote the random number of ender pearls Jane has after bartering n gold ingots. Name the distribution of Xn, and identify its parameters. 7.2 Calculate the expectation and standard deviation of the number of ender pearls Jane will have after bartering 200 gold ingots. State clearly any formulas that you use from the notes. 7.3 How many gold ingots does she need to ensure that she has a chance of at least 1% of getting at least 14 ender pearls? You may use the normal approximation. State clearly any formulas that you use from the notes. 7.4 Let k* denote the answer to the previous question. Jane decides always to collect exactly k* gold ingots on her runs. Find the expectation of the number of these runs she needs until she can finish the game. You may find the following formula useful (you do not need to prove it): ;(1 - 2) = j=1 1-9 q which holds for any number q such that 0 < q < 1. 7.5 Joe, a competing speedrunner, has beaten Jane's world record and published a speedrun where he collected 150 gold ingots, which he converted into 14 ender pearls. Do you think Joe was cheating? Provide a clear probabilistic argument behind your answer.