The royal court of Prince Lemon had dukes, earls, and barons. At the beginning of the Prince's reign, there were 2020 members of the court and each day, one of them killed another one in a duel. Dukes only killed earls, earls only killed barons, and barons only killed dukes. No person won a duel twice. In the end, the only member of the court left alive was Baron Orange. What was the title of the first court member who was killed?

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?