The GW Inn has a total of 300 identical rooms. Like many major airlines and hotels, it has adopted an overbooking policy in an effort to maximize the usage of its available lodging capacity. Assume that each potential customer holding a room reservation, independently of other customers, cancels the reservation or simply does not show up at the hotel on a given day with probability 0.15. (You can use R in your computations but be sure that you show your work)

(a) Find the largest number of room reservations that the hotel can book and still be at least 95% sure that everyone who shows up at the hotel will have a room on a given night.

(b) Given that the hotel books a total of 320 reservations, find the probability that at most 90% of the available rooms will be occupied on a given night.