Calls arrive at a switchboard according to a Poisson process with rate A42 per minute. The switchboard jams if more than 4 calls arrive in a 3 second interval. (a) Find the probability that the switchboard jams. (b) Prove that the time the first call arrives is exponentially distributed with mean 1/42, and find the probability the first call arrives within 4 seconds. (c) Find the probability that the 10th call arrives after 15 seconds and before 20 seconds have elapsed. (d) Find the probability the time between the 5th and the 10th call is greater than 10 seconds. (e) If 20% of calls are long-distance and 80% are local find the mean time between two long-distance calls, and the probability that the time between two consecutive long-distance calls is less than 20 seconds.